Biased Random Walks in Biology - University of Essexprivateecodling/ecodlingthesis.pdf · Biased...

Biased Random Walks in Biology

Edward Alexander Codling

Submitted in accordance with the requirements for the degree of

Doctor of Philosophy

The University of Leeds,

Department of Applied Mathematics.

August 2003

The candidate confirms that the work submitted is his own and that appropriate credit

has been given where reference has been made to the work of others. This copy has been

supplied on the understanding that it is copyright material and that no quotation from

the thesis may be published without proper acknowledgment.

Acknowledgements

I would like to thank my supervisor Prof. Nick Hill for his calm guidance and unfailing

help and support throughout four years of research. He has always made time to help me

and has supported all my activities, whether part of my research or not. His enthusiasm

for the subject has kept me inspired throughout and I feel privileged to have been able to

share in his expert knowledge through the many discussions we have had.

While at Leeds, I have had many helpful and rewarding discussions with everyone involved

with the Biomaths group, in particular Prof. Brian Sleeman and Mike Plank. I have

enjoyed working with Dr. Jon Pitchford, who has been very encouraging and eager to help

with the fish larvae project, and always keen to discuss any other aspect of the rest of my

research. Dr. Steve Simpson has been extremely helpful in passing on his detailed and

expert knowledge of fish larvae behaviour.

I am grateful for the generous assistance of Chris Needham in helping me to program in C

and C++ — without such help I would not have been able to set up and run such detailed

numerical simulations.

Finally, I must give special thanks to the Carswell family: Neil, Eleanor, Annie and

Thomas, for supporting and accommodating me during the writing up period, and also

the rest of my family and friends (especially Becky) for all the help, support, and biscuits.

Financial support for this research has been provided by the E.P.S.R.C.

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Abstract

Random walks are used to describe the trajectories of many motile animals and micro-

organisms. They are a useful tool for both qualitative and quantitative descriptions of

the behaviour of such creatures. Simple diffusive random walk models, or position jump

processes, are unrealistic as they allow for effectively infinite propagation and do not take

into account correlation between steps. Othmer et al. (1988) use a generalised trans-

port equation to model biased and correlated velocity jump processes where the speed of

movement is finite. In two dimensions an equation for the underlying spatial distribution

of the velocity jump process model cannot be found, so Othmer et al. use a method of

calculating moments to derive and solve differential equations for the statistics of inter-

est. We extend the velocity jump process and method of moments used by Othmer et

al. to include reorientation models where the mean turning angle is dependent on the

previous direction of movement, as observed by Hill & Hader (1997) in experiments on

algae. Closure assumptions are made in order to derive and solve a system of differential

equations for the higher order moments and statistics of the underlying spatial distri-

bution. Numerical simulations are then used to compare the asymptotic solutions with

simulated data, and the fit is good for biologically realistic parameter values. Numerical

simulations of velocity jump processes are also used to investigate the method used by

Hill & Hader to calculate the reorientation parameters from the angular statistics of a

random walk, and also to investigate the effect of spatially dependent parameters or a

changing preferred direction on the spatial statistics. We look at the ratio between the

root of the mean squared displacement and the mean dispersal distance in both unbiased

and biased random walks and demonstrate how this can give us more information about

the spatial distribution. We give an application of the velocity jump process model to the

movement and recruitment of reef fish larvae. The variability in the movement is found

to be important if there is a low survival probability, while in simple reef environments

the survival probability appears to be highly sensitive to the reorientation parameters and

corresponding swimming behaviour.

ii

Contents

1 Introduction and background 1

1.1 General background to random walks . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 The isotropic random walk and the diffusion equation . . . . . . . . 2

1.1.3 Random walks to a barrier — a simple example . . . . . . . . . . . . 8

1.1.4 The telegraph equation . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Circular statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.1 The mean direction . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.2 The mean resultant length and the circular variance . . . . . . . . . 14

1.2.3 Probability distributions on the circle . . . . . . . . . . . . . . . . . 15

1.3 Modelling biological motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.1 The movement of animals and micro-organisms as a random walk . 18

1.3.2 Biased movement and taxis . . . . . . . . . . . . . . . . . . . . . . . 19

1.3.3 Other applications of the random walk in biology . . . . . . . . . . . 20

1.4 Properties of correlated random walks . . . . . . . . . . . . . . . . . . . . . 20

1.4.1 Mean squared displacement . . . . . . . . . . . . . . . . . . . . . . . 21

1.4.2 Sinuosity and mean dispersal distance . . . . . . . . . . . . . . . . . 23

1.5 The circular random walk and reorientation models arising from experi-

ments on algae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5.1 Deriving the Fokker–Planck equation for a circular random walk . . 26

1.5.2 Reorientation models and solutions to the Fokker–Planck equation . 27

1.5.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.6 Overview of subsequent chapters . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Simple two-dimensional random walk models 31

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Two-dimensional uncorrelated random walks . . . . . . . . . . . . . . . . . 32

2.3 Lattice model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.1 Turning probabilities independent of position . . . . . . . . . . . . . 32

2.3.2 Turning probabilities dependent on position . . . . . . . . . . . . . . 34

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iv

2.4 Multi-directional discrete direction model and continuous direction model . 35

2.4.1 Multi-directional discrete direction model . . . . . . . . . . . . . . . 35

2.4.2 Continuous direction model . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 Solution of the Fokker–Planck diffusion equation . . . . . . . . . . . . . . . 39

2.5.1 Solution for isotropic movement . . . . . . . . . . . . . . . . . . . . . 39

2.5.2 Solution for biased movement . . . . . . . . . . . . . . . . . . . . . . 40

2.6 The telegraph equation in higher dimensions . . . . . . . . . . . . . . . . . 44

2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Spatial statistics of two-dimensional velocity jump processes 48

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Generalized equation for velocity jump processes . . . . . . . . . . . . . . . 49

3.2.1 Generalized model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2.2 Velocity jump processes in one dimension — the telegraph equation 50

3.3 Velocity jump processes in two dimensions — random walks in external fields 50

3.3.1 Defining statistics of interest . . . . . . . . . . . . . . . . . . . . . . 51

3.3.2 Deriving equations for spatial statistics . . . . . . . . . . . . . . . . 53

3.3.3 Solving equations for spatial statistics . . . . . . . . . . . . . . . . . 58

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Velocity jump processes using sinusoidal reorientation 63

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 Reorientation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.1 Hill & Hader’s general reorientation model . . . . . . . . . . . . . . 64

4.2.2 The reorientation kernel T (θ, θ′) . . . . . . . . . . . . . . . . . . . . 65

4.2.3 Sinusoidal reorientation model . . . . . . . . . . . . . . . . . . . . . 66

4.2.4 The biological relevance of the turning angle distribution parameters 67

4.3 Defining statistics of interest . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4 Results and assumptions to be used in analysis . . . . . . . . . . . . . . . . 68

4.4.1 Integrals of the von Mises distribution . . . . . . . . . . . . . . . . . 68

4.4.2 Asymptotic expansions of the trigonometric functions . . . . . . . . 69

4.4.3 Previous results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4.4 Other assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.5 Differential equations for the spatial statistics and higher order moments . . 70


4.5.2 Deriving equations for the higher order moments . . . . . . . . . . . 72

4.6 Closing and solving the system of equations for H(t), V(t), Fn(t) and Yn(t) 78

4.6.1 Approximating the higher order moments . . . . . . . . . . . . . . . 78

4.6.2 The general solution to a linear system of differential equations . . . 79

4.6.3 Solving the final system of equations for H(t), V(t), Fn(t) and Yn(t) 80

v

4.7 Closing and solving the system of equations for D2(t), Gn(t) and Zn(t) . . . 83

4.7.1 Approximating the higher order moments . . . . . . . . . . . . . . . 84

4.7.2 Solving the final system of equations for D2(t), Gn(t) and Zn(t) . . 85

4.7.3 Equations for the spread about the mean position . . . . . . . . . . 88

4.8 Solution plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.8.1 Comment on solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.9 Working with the equations for the statistics of interest . . . . . . . . . . . 93

4.9.1 Limitations of the model and solutions . . . . . . . . . . . . . . . . . 93

4.9.2 Rescaling the equations . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.9.3 Limits on the parameters . . . . . . . . . . . . . . . . . . . . . . . . 95

4.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5 Velocity jump processes using linear reorientation 99

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2 Results and assumptions to be used in analysis . . . . . . . . . . . . . . . . 99

5.2.1 Reorientation model . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2.2 Defining higher order moments . . . . . . . . . . . . . . . . . . . . . 100

5.2.3 Integrals of the von Mises distribution . . . . . . . . . . . . . . . . . 102

5.2.4 Asymptotic expansions of the trigonometric functions . . . . . . . . 102

5.2.5 Previous results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.3 Differential equations for the spatial statistics and higher order moments . . 103


5.3.2 Deriving equations for the higher order moments . . . . . . . . . . . 104

5.3.3 System of equations for non-spatial moments . . . . . . . . . . . . . 126

5.3.4 System of equations for spatial moments . . . . . . . . . . . . . . . . 127

5.4 Solving the systems of equations . . . . . . . . . . . . . . . . . . . . . . . . 129

5.4.1 Solving for the non-spatial higher order moments . . . . . . . . . . . 129

5.4.2 Solving for V(t) and H(t) . . . . . . . . . . . . . . . . . . . . . . . . 130

5.4.3 Solving for the spatial higher order moments . . . . . . . . . . . . . 131

5.4.4 Solving for D2(t) and σ2(t) . . . . . . . . . . . . . . . . . . . . . . . 132

5.5 Final system of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.5.1 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.5.2 Solution plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.5.3 Comment on solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.5.4 Limitations of the model and solutions . . . . . . . . . . . . . . . . . 138

5.6 Comparing solutions of the sinusoidal and linear models . . . . . . . . . . . 139

5.6.1 Comparing solutions for H(t) . . . . . . . . . . . . . . . . . . . . . . 139

5.6.2 Comparing solutions for σ2(t) . . . . . . . . . . . . . . . . . . . . . . 140

5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

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6 Spatial statistics of simulated random walks 142

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.2 Computer simulations of random walks . . . . . . . . . . . . . . . . . . . . . 143

6.2.1 Simulation of an individual random walk . . . . . . . . . . . . . . . 143

6.2.2 Collecting average statistics for a set of random walks . . . . . . . . 147

6.3 Simulations to validate theoretical results . . . . . . . . . . . . . . . . . . . 150

6.3.1 Mean position — H(t) . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.3.2 Average velocity — V(t) . . . . . . . . . . . . . . . . . . . . . . . . 159

6.3.3 Measure of spread about the origin — D2(t) . . . . . . . . . . . . . 162

6.3.4 Measure of spread about the mean position — σ2(t) . . . . . . . . . 172

6.4 The effect of the reorientation parameters on fixed time solutions . . . . . . 185

6.4.1 Fixed time spatial distribution . . . . . . . . . . . . . . . . . . . . . 186

6.4.2 The effect of changing the reorientation parameters κ and dτ . . . . 186

6.5 Simulations with parameters from experimental data . . . . . . . . . . . . . 195

6.5.1 Data set C1 (Sinusoidal model) . . . . . . . . . . . . . . . . . . . . . 195

6.5.2 Data set C3 (Linear model) . . . . . . . . . . . . . . . . . . . . . . . 199

6.5.3 Data set C4 (Linear model) . . . . . . . . . . . . . . . . . . . . . . . 201

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

7 Angular statistics and the effect of sampling length 208

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

7.2 The long-time absolute angular distribution . . . . . . . . . . . . . . . . . . 208

7.2.1 Validating the approximation for M0(t) . . . . . . . . . . . . . . . . 210

7.2.2 Comparing theoretical distributions to simulation results . . . . . . 210

7.2.3 Moments of the long-time absolute angular distribution . . . . . . . 215

7.3 The effect of sampling length on the angular statistics of a velocity jump

process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

7.3.1 Examples of changing the sampling length . . . . . . . . . . . . . . . 220

7.3.2 Angular statistics of a velocity jump process with sinusoidal reori-

entation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

7.3.3 Angular statistics of a velocity jump process with linear reorientation226

7.4 Limitations of using the angular statistics to estimate the reorientation

parameters of a velocity jump process . . . . . . . . . . . . . . . . . . . . . 229

7.4.1 The effect of sampling length on the angular statistics of a velocity

jump process with a fixed time between turns . . . . . . . . . . . . . 230

7.4.2 Estimating the reorientation parameters for large and small values

of τ and τs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

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8 Further modelling with computer simulations 239

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

8.2 Simulations with parameter values outside the limits of the theoretical models239

8.2.1 The effect of the parameter κ . . . . . . . . . . . . . . . . . . . . . . 240

8.2.2 The effect of the parameter dτ . . . . . . . . . . . . . . . . . . . . . 242

8.2.3 Theoretical optimal value of dτ . . . . . . . . . . . . . . . . . . . . . 246

8.2.4 Biological relevance of larger reorientation parameter values . . . . . 251

8.3 Simulations with non-constant parameters . . . . . . . . . . . . . . . . . . . 251

8.3.1 Spatial dependence of κ . . . . . . . . . . . . . . . . . . . . . . . . . 251

8.3.2 Spatial dependence of dτ . . . . . . . . . . . . . . . . . . . . . . . . . 256

8.3.3 Biological relevance of spatially dependent reorientation parameters 262

8.4 Simulations with a changing preferred direction . . . . . . . . . . . . . . . . 264

8.4.1 Reorientation models for a changing preferred direction . . . . . . . 264

8.4.2 Examples of individual random walks . . . . . . . . . . . . . . . . . 265

8.4.3 Average position — Hy(t) . . . . . . . . . . . . . . . . . . . . . . . . 266

8.4.4 Spread about the mean position — σ2(t) . . . . . . . . . . . . . . . 267

8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

9 Mean dispersal distance of correlated random walks 273

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

9.2 The mean squared displacement . . . . . . . . . . . . . . . . . . . . . . . . . 274

9.2.1 Comparing the mean squared displacement for unbiased discrete

random walks and velocity jump processes . . . . . . . . . . . . . . . 274

9.2.2 Mean squared displacement for variable and fixed step lengths . . . 275

9.3 The mean dispersal distance of unbiased random walks . . . . . . . . . . . . 275

9.3.1 Calculating the mean dispersal distance from the mean squared dis-

placement in a discrete random walk . . . . . . . . . . . . . . . . . . 276

9.3.2 A better model for MDD(n) . . . . . . . . . . . . . . . . . . . . . . 277

9.3.3 The mean dispersal distance of an unbiased velocity jump process

with a variable time step . . . . . . . . . . . . . . . . . . . . . . . . 279

9.3.4 The mean dispersal distance in each direction for an unbiased ve-

locity jump process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

9.4 The mean dispersal distance of biased random walks . . . . . . . . . . . . . 284

9.4.1 The limiting value of the correction factor . . . . . . . . . . . . . . . 286

9.4.2 Simulated behaviour of the limiting value of the correction factor . . 286

9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

10 Random walks to a barrier and the recruitment of fish larvae 290

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

10.2 Background to fish larval movement and recruitment . . . . . . . . . . . . . 291

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10.2.1 Recruitment of fish larvae in the open sea . . . . . . . . . . . . . . . 291

10.2.2 Recruitment of reef fish larvae . . . . . . . . . . . . . . . . . . . . . 291

10.2.3 Theoretical models of fish larvae returning to a reef . . . . . . . . . 292

10.2.4 Experimental data for fish larvae returning to a reef . . . . . . . . . 292

10.3 The effect of variability on fish larvae recruitment . . . . . . . . . . . . . . . 294

10.3.1 Model 1: simple reef environment . . . . . . . . . . . . . . . . . . . . 294

10.3.2 Deterministic model for population dynamics . . . . . . . . . . . . . 295

10.3.3 Stochastic model for population dynamics . . . . . . . . . . . . . . . 296

10.3.4 Survival probabilities for the simple reef model . . . . . . . . . . . . 297

10.4 Optimal swimming behaviour for fish larvae attempting to recruit to a reef 300

10.4.1 Model 2: simple circular reef model . . . . . . . . . . . . . . . . . . 300

10.4.2 Model 3: simple current model . . . . . . . . . . . . . . . . . . . . . 303

10.4.3 Further models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

10.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

11 Concluding remarks 310

11.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

11.2 Possible future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

Bibliography 314

List of Figures

1.1 Plots showing P (x, t) for D = 1, 5 and 10, and t = 1 and 10. (The scales

on each plot are different). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Plots showing P (x, y, t) for D = 1 and 5, and t = 1 and 10. (The scales on

each plot are different). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Plots of P (x, t), the solution of the drift diffusion equation, for various u . . 7

1.4 Plots of p(x, t), the solution of the telegraph equation for various parameter

values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Example of a data set on a circle with R ≈ 0 but with a non-uniform spread

of points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6 Examples of the von Mises distribution for various values of κ, and µ = 0. . 16

1.7 Plot of κ against σ2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.8 Plot comparing µ0(θ) for sinusoidal (—) and linear reorientation (- -), for

−π ≤ θ < π and B−1 = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.1 Example of a two-dimensional lattice random walk. . . . . . . . . . . . . . . 32

2.2 Example of a multi-directional random walk. . . . . . . . . . . . . . . . . . 36

2.3 Plots showing f(x, y, t) for various parameter values at t = 10. . . . . . . . 43

3.1 Sketch of the probability distributions for h(δ) and k(θ) as used by Othmer

et al. (1988). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Sketch showing the difference between D2 and σ2 (H is the average position). 52

3.3 Plots of Vx1(t) and Hx1(t) for various values of CI . . . . . . . . . . . . . . . 60

3.4 Plots of D2(t) and σ2(t) for various values of CI . . . . . . . . . . . . . . . . 61

4.1 Plot of V(t) for dτ = 0.1 and dτ = 0.3 and various values of κ. (The scale

of each plot is different) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.2 Plot of H(t) for dτ = 0.1 and dτ = 0.3 and various values of κ. (The scale


4.3 Plot of D2(t) for dτ = 0.1 and dτ = 0.3 and various values of κ. (The scale


4.4 Plot of σ2(t) for dτ = 0.1 and dτ = 0.3 and various values of κ. (The scale


ix

x

4.5 Plot of D2x1(t) and D2

x2(t) for dτ = 0.3 and various values of κ. (The scale


4.6 Plot of σ2x1(t) and σ2

x2(t) for dτ = 0.3 and various values of κ. (The scale of

each plot is different) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.7 Plot of ζ1 against κ for dτ = 0, 0.1, 0.2, 0.3. . . . . . . . . . . . . . . . . . . . 96

4.8 Plot of ζ2 against κ for dτ = 0.1, 0.2, 0.3. . . . . . . . . . . . . . . . . . . . . 97

5.1 Plots comparing k1(µ, κ) to the exact integral for various values of κ. (The

scale of each plot is different). . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2 Plots comparing l1(µ, κ) to the exact integral for various values of κ. (The


5.3 Plots comparing m1(µ, κ) to the exact integral for various values of κ. (The


5.4 Plots comparing n1(µ, κ) to the exact integral for various values of κ. (The


5.5 Plots comparing k2(µ, κ) to the exact integral for various values of κ. (The


5.6 Plots comparing l2(µ, κ) to the exact integral for various values of κ. (The


5.7 Plots comparing m2(µ, κ) to the exact integral for various values of κ. (The


5.8 Plots comparing n2(µ, κ) to the exact integral for various values of κ. (The


5.9 Plot of V(t) for dτ = 0.1 and dτ = 0.3 and various values of κ. . . . . . . . 135

5.10 Plot of H(t) for dτ = 0.1 and dτ = 0.3 and various values of κ. . . . . . . . 135

5.11 Plot of D2(t) for dτ = 0.1 and dτ = 0.3 and various values of κ. (The scale


5.12 Plot of σ2(t) for dτ = 0.1 and dτ = 0.3 and various values of κ. (The scale


5.13 Plot of D2x1(t) and D2

x2(t) for dτ = 0.3 and various values of κ. (The scale


5.14 Plot of σ2x1(t) and σ2


each plot is different) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.1 Simple algorithm for an individual random walk. . . . . . . . . . . . . . . . 143

6.2 i) Random walk with κ = 0.1, dτ = 0. The random walk is close to being

completely random (Brownian) motion. . . . . . . . . . . . . . . . . . . . . 145

6.3 ii) Random walk with κ = 2, dτ = 0. The random walk appears more

correlated but there is no overall preferred direction. . . . . . . . . . . . . . 146

xi

6.4 iii) Random walk with κ = 0.5, dτ = 0.2. The random walk is less correlated

but there is a definite preferred direction (y-direction). . . . . . . . . . . . . 146

6.5 iv) Random walk with κ = 4, dτ = 0.3. The random walk is highly corre-

lated and the preferred direction is clear. . . . . . . . . . . . . . . . . . . . . 147

6.6 Algorithm used to calculate average statistics for a set of random walks. . . 148

6.7 Plots showing theoretical Hy(t) (—), and 95% confidence interval from

simulated (· · ·), against time for sinusoidal reorientation with dτ = 0.1.

(The scale used for each plot is different.) . . . . . . . . . . . . . . . . . . . 152








simulated (· · ·), against time for linear reorientation with dτ = 0.1. (The

scale used for each plot is different.) . . . . . . . . . . . . . . . . . . . . . . 156







6.13 Plots showing theoretical (—), and simulated (· · ·), absolute velocity (Hy(t)/t)

in the y-direction against time for sinusoidal reorientation with dτ = 0.1.


6.14 Plots showing theoretical (—), and simulated (· · ·) absolute velocity (Hy(t)/t)

in the y-direction against time for sinusoidal reorientation with dτ = 0.3.



in the y-direction against time for linear reorientation with dτ = 0.1. (The



in the y-direction against time for linear reorientation with dτ = 0.3. (The


6.17 Plots showing theoretical (—), and simulated (· · ·), D2(t) against time for

sinusoidal reorientation with dτ = 0.1. (The scale used for each plot is

different.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

xii



different.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166


linear reorientation with dτ = 0.1. (The scale used for each plot is different.)168



6.21 Plots showingD2x(t) andD2

y(t) against time for sinusoidal reorientation with

various values of the parameters. (The scale used for each plot is different.) 170

6.22 Plots showing D2x(t) and D2

y(t) against time for linear reorientation with

various values of the parameters. (The scale used for each plot is different.) 171

6.23 Plots showing theoretical (—), and simulated (· · ·), σ2(t) against time for

sinusoidal reorientation with dτ = 0. (The scale used for each plot is different.)173



different.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174



different.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175



different.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176







6.30 Plots showing σ2x(t) against time for sinusoidal reorientation with various

values of the parameters. (The scale used for each plot is different.) . . . . 181

6.31 Plots showing σ2y(t) against time for sinusoidal reorientation with various

values of the parameters. (The scale used for each plot is different.) . . . . 182

6.32 Plots showing σ2x(t) against time for linear reorientation with various values

of the parameters. (The scale used for each plot is different.) . . . . . . . . 183

6.33 Plots showing σ2y(t) against time for linear reorientation with various values

of the parameters. (The scale used for each plot is different.) . . . . . . . . 184

6.34 Example plots of the population position and spread at t = 100. . . . . . . 187

6.35 Plots showing Hy(100) against κ for sinusoidal and linear reorientation with

dτ = 0 (—), dτ = 0.1 (· · ·), dτ = 0.2 (−−) and dτ = 0.3 (· − ·). . . . . . . . 188

xiii

6.36 Plots showing D2(100) against κ for sinusoidal and linear reorientation with

dτ = 0 (—), dτ = 0.1 (· · ·), dτ = 0.2 (−−) and dτ = 0.3 (· − ·). . . . . . . . 189

6.37 Plots showing D2x(100) against κ for sinusoidal and linear reorientation with

dτ = 0 (—), dτ = 0.1 (· · ·), dτ = 0.2 (−−) and dτ = 0.3 (· − ·). . . . . . . . 190

6.38 Plots showing D2y(100) against κ for sinusoidal and linear reorientation with

dτ = 0 (—), dτ = 0.1 (· · ·), dτ = 0.2 (−−) and dτ = 0.3 (· − ·). . . . . . . . 192

6.39 Plots showing σ2(100) against κ for sinusoidal and linear reorientation with

dτ = 0 (—), dτ = 0.1 (· · ·), dτ = 0.2 (−−) and dτ = 0.3 (· − ·). . . . . . . . 193

6.40 Plots showing σ2y(100) against κ for sinusoidal and linear reorientation with

dτ = 0 (—), dτ = 0.1 (· · ·), dτ = 0.2 (−−) and dτ = 0.3 (· − ·). . . . . . . . 194

6.41 Plots showing (a) final position and spread (t = 100), (b) Hy(t), (c) σ2x(t)

and (d) σ2y(t) for reorientation parameters from data set C1:a. . . . . . . . . 197


and (d) σ2y(t) for reorientation parameters from data set C1:b. . . . . . . . . 198









7.1 Plots of M0(t) against t. Legend: (- -) simulation κ = 1, (· · ·) simulation

κ = 2, (−·−) simulation κ = 4, (+) approximation κ = 1, (*) approximation

κ = 2, (♦) approximation κ = 4. . . . . . . . . . . . . . . . . . . . . . . . . 210

7.2 Plots showing theoretical and simulated long-time p.d.f., f(θ), with param-

eter values taken from Hill and Hader’s experiments with data set C1. . . . 211





7.5 Plots showing theoretical and simulated long-time p.d.f., f(θ), for data set

C1 with τ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

7.6 Plots showing theoretical and simulated long-time p.d.f., f(θ), for data set

C4 with τ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

xiv

7.7 Plots showing the first angular moment a1 against k0 for the sinusoidal re-

orientation model, with (a) B−1 = 0.1, (b) B−1 = 0.5. Legend: theoretical

results (—), simulation results with τ = 0.1 s (- -), simulation results with

τ = 1 s (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

7.8 Plots showing the first angular moment a1 against k0 for the linear reori-

entation model, with (a) B−1 = 0.1, (b) B−1 = 0.5. Legend: theoretical

results (—), simulation results with τ = 0.1 s (- -), simulation results with

τ = 1 s (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

7.9 Plots showing the third angular moment a3 against k0 with B−1 = 0.5, for

(a) sinusoidal reorientation model (b) linear reorientation model. Legend:

theoretical results (—), simulation results with τ = 0.1 s (- -), simulation

results with τ = 1 s (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

7.10 Plots showing the fourth angular moment a4 against k0 with B−1 = 0.5, for

(a) sinusoidal reorientation model (b) linear reorientation model. Legend:

theoretical results (—), simulation results with τ = 0.1 s (- -), simulation

results with τ = 1 s (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

7.11 Plots showing the effect of changing the sampling length τs of an individual

random walk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

7.12 Plots showing how µδ(θ) changes with θ for the sinusoidal model with vari-

ous sampling lengths τs. Simulation results for angular bins of π9 rads (—),

and functions fitted by inspection to the data (- -). . . . . . . . . . . . . . . 222

7.13 Plots showing how σ2δ (θ) changes with θ for the sinusoidal model with vari-

ous sampling lengths τs. Simulation results for angular bins of π9 rads (—),

and the mean from the data averaging over all θ (- -). . . . . . . . . . . . . 223

7.14 Plots showing (a) the amplitude of the mean turning angle dτs , (b) vari-

ance of the turning angle σ2δ , against rescaled sampling length τs/τ for the

sinusoidal model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

7.15 Plots showing how µδ(θ) changes with θ for the linear model with various

sampling lengths τs. Simulation results for angular bins of π9 rads (—), and

functions fitted by inspection to the data (- -). . . . . . . . . . . . . . . . . 227

7.16 Plots showing how σ2δ (θ) changes with θ for the linear model with various

sampling lengths τs. Simulation results for angular bins of π9 rads (—), and

the mean from the data averaging over all θ (- -). . . . . . . . . . . . . . . . 228

7.17 Plots showing (a) the amplitude of the mean turning angle dτs , (b) variance

of the turning angle σ2δ , against rescaled sampling length τs/τ for the linear

model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

7.18 Plots showing how µδ(θ) and σ2δ change with θ for the sinusoidal model

with fixed time between turns. . . . . . . . . . . . . . . . . . . . . . . . . . 231

xv

7.19 Plots showing how µδ(θ) and σ2δ change with θ for the linear model with

fixed time between turns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

7.20 Plots showing (a) the amplitude of the mean turning angle dτs , (b) vari-

ance of the turning angle σ2δ , against rescaled sampling length τs/τ for the

sinusoidal model with fixed time between turns. . . . . . . . . . . . . . . . . 233

7.21 Plots showing (a) the amplitude of the mean turning angle dτs , (b) variance

of the turning angle σ2δ , against rescaled sampling length τs/τ for the linear

model with fixed time between turns. . . . . . . . . . . . . . . . . . . . . . . 233

7.22 Log-plot of − log10(τ) against ρ. . . . . . . . . . . . . . . . . . . . . . . . . 236

8.1 Plots showing Hy(100) against κ for sinusoidal and linear reorientation for

dτ = 0.1 (—), and dτ = 0.3 (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . 240

8.2 Plots showing σ2x(100) against κ for sinusoidal and linear reorientation for

dτ = 0.1 (—), and dτ = 0.3 (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . 241

8.3 Plots showing σ2y(100) against κ for sinusoidal and linear reorientation for

dτ = 0.1 (—), and dτ = 0.3 (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . 242

8.4 Plots showing distribution at t = 100 for sinusoidal and linear reorientation

for dτ = 0.1 and κ = 0.1, κ = 10 and κ = 50. . . . . . . . . . . . . . . . . . 243

8.5 Plots showing Hy(100) against dτ for sinusoidal and linear reorientation for

κ = 1 (—), and κ = 4 (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

8.6 Plots showing σ2x(100) against dτ for sinusoidal and linear reorientation for

κ = 1 (—), and κ = 4 (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

8.7 Plots showing σ2y(100) against dτ for sinusoidal and linear reorientation for

κ = 1 (—), and κ = 4 (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . . . . 245


for κ = 4 and dτ = 0.1, dτ = 1 and dτ = 2. . . . . . . . . . . . . . . . . . . . 247

8.9 Plot of dopt against κ for sinusoidal reorientation. . . . . . . . . . . . . . . . 249

8.10 Plot of κ(y) against y with κI = 1, for p = 0.01 (—), p = 0.05 (· · ·), and

p = 0.1 (- -). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

8.11 Plots showing individual random walks for sinusoidal reorientation with

κ(y) for various parameter values. (The scale of each plot is different) . . . 253

8.12 Plots showing Hy(100) against p for sinusoidal and linear reorientation for

dτ = 0.1 (—), and dτ = 0.3 (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . 254

8.13 Plots showing σ2x(100) against p for sinusoidal and linear reorientation for

dτ = 0.1 (—), and dτ = 0.3 (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . 255

8.14 Plots showing σ2y(100) against p for sinusoidal and linear reorientation for

dτ = 0.1 (—), and dτ = 0.3 (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . 255


for dτ = 0.1 and p = 0.05 and p = 0.5. . . . . . . . . . . . . . . . . . . . . . 257

xvi

8.16 Plot of dτ (y) against y with dint = 0.1 and dopt = 1, for q = 0.01 (—),

q = 0.05 (· · ·), and q = 0.1 (- -). . . . . . . . . . . . . . . . . . . . . . . . . 258

8.17 Plots showing individual random walks for sinusoidal reorientation with

dτ (y) for various parameter values. (The scale of each plot is different) . . . 259

8.18 Plots showing Hy(100) against q for sinusoidal and linear reorientation for

κ = 1 (—), and κ = 4 (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

8.19 Plots showing σ2x(100) against q for sinusoidal and linear reorientation for

κ = 1 (—), and κ = 4 (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

8.20 Plots showing σ2y(100) against q for sinusoidal and linear reorientation for

κ = 1 (—), and κ = 4 (· · ·). . . . . . . . . . . . . . . . . . . . . . . . . . . . 261


for κ = 4 and q = 0.01 and q = 0.1. . . . . . . . . . . . . . . . . . . . . . . . 263

8.22 Plots showing individual random walks for sinusoidal and linear reorienta-

tion where the preferred direction is to a point. . . . . . . . . . . . . . . . . 265

8.23 Plots showing the average position in the y-direction, Hy(t), against t for

sinusoidal and linear reorientation. . . . . . . . . . . . . . . . . . . . . . . . 266

8.24 Plots showing the spread in the x-direction, σ2x(t), against t for sinusoidal

and linear reorientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

8.25 Plots showing the spread in the y-direction, σ2y(t), against t for sinusoidal

and linear reorientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269


where the preferred direction is to a point. . . . . . . . . . . . . . . . . . . . 270

9.1 Plots of the spread of a population of 500 walkers after t = 100, moving

as an unbiased and correlated velocity jump process with (a) κ = 1, (b)

κ = 50. The dotted circle shows the maximum possible displacement at

t = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

9.2 Plots comparing expected values of Z(c, t) (—) to simulated results (+) for

(a) κ = 1, (b) κ = 4, (c) κ = 10, (d) κ = 20. . . . . . . . . . . . . . . . . . . 281

9.3 Plots of MDD(t) v t for the velocity jump process model (—), Kareiva &

Shigesada’s model (· · ·), Bovet & Benhamou’s model (- -), and simulation

results (+). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

9.4 Plots of MDD(t) v t for velocity jump process model with Z(c, t) (—),

Z = 0.89 (- -), and simulation results (+). . . . . . . . . . . . . . . . . . . . 283

9.5 Simulated plots of the spread of a population of 500 walkers after t = 100,

moving as a biased and correlated velocity jump process with dτ = 0.1 and

(a) sinusoidal reorientation, κ = 1, (b) sinusoidal reorientation, κ = 50, (c)

linear reorientation, κ = 1, (d) linear reorientation, κ = 50. . . . . . . . . . 285

xvii

9.6 Plots of values of Z(κ, dτ , t), Zx(κ, dτ , t), and Zy(κ, dτ , t) as a function of κ

at t = 1000 from numerical simulations of sinusoidal and linear reorientation

with dτ = 0.1 (- -), dτ = 0.5 (· · ·), and dτ = 1 (- · -). The solid lines (—)

correspond to Z = 0.798 or Z = 0.89 respectively, the expected values if

the distribution is Normal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

10.1 Simple ‘infinite’ reef model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

10.2 Plots showing (a) survival probability PR(VF , γ) against death rate for (a)

0.0001 ≤ µ ≤ 0.0002, and (b) 0.0002 ≤ µ ≤ 0.0004. Legend: deterministic

model (—), stochastic model (- -), simulation model (+). . . . . . . . . . . 298

10.3 Plots of relative survival probability RSP against PR(VF , 0) from theoreti-

cal (—) and simulation (+) results. . . . . . . . . . . . . . . . . . . . . . . . 299

10.4 Simple circular reef model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

10.5 Plots showing survival probability PR(dτ , κ) for sinusoidal reorientation and

Model 2 against (a) κ, for dτ = 0.1 (—), dτ = 0.3 (- -), dτ = 0.5 (· · ·), and

dτ = 1.0 (- · -); (b) dτ , for κ = 0.4 (—), κ = 1.0 (- -), κ = 2.0 (· · ·), and

κ = 4.0 (- · -). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

10.6 Plots showing survival probability PR(dτ , κ) for linear reorientation and


dτ = 1.0 (- · -); (b) dτ , for κ = 0.4 (—), κ = 1.0 (- -), κ = 2.0 (· · ·), and

κ = 4.0 (- · -). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

10.7 Circular reef with a constant current. . . . . . . . . . . . . . . . . . . . . . . 304

10.8 Plots showing survival probability PR(dτ , κ) for sinusoidal reorientation and


dτ = 1.5 (- · -); (b) dτ , for κ = 1.8 (—), κ = 2.0 (- -), κ = 3.0 (· · ·), and

κ = 5.0 (- · -). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

10.9 Plots showing survival probability PR(dτ , κ) for linear reorientation and


dτ = 1.0 (- · -); (b) dτ , for κ = 1.4 (—), κ = 2.0 (- -), κ = 3.0 (· · ·), and

κ = 5.0 (- · -). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

10.10Plots showing survival probability PR(U, κ) v U with dτ = 0.8 for (a) si-

nusoidal reorientation and (b) linear reorientation. Legend: κ = 1.0 (—),

κ = 2.0 (- -), κ = 3.0 (· · ·), and κ = 5.0 (- · -). . . . . . . . . . . . . . . . . . 306

List of Tables

1.1 Swimming speed and reorientation parameters estimated by Hill & Hader

for the data sets C1, C3 and C4. . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1 Long-time numerical solutions for V(t) with linear reorientation . . . . . . . 133

5.2 Long-time numerical solutions for D2x1(t) with linear reorientation . . . . . 134

5.3 Long-time numerical solutions for D2x2(t) with linear reorientation . . . . . 134

5.4 Long-time numerical solutions for σ2x1(t) with linear reorientation . . . . . . 134

5.5 Comparing long-time numerical solutions for H(t) . . . . . . . . . . . . . . 139

5.6 Comparing long-time numerical solutions for σ2(t) . . . . . . . . . . . . . . 140

7.1 Estimated value for the amplitude of the mean turning angle µδ(θ), and

calculated mean value of σ2δ , for the sinusoidal model with rescaled sampling

length τs/τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224


calculated mean value of σ2δ , for the linear model with rescaled sampling

length τs/τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

7.3 Estimated value for the amplitude of the mean turning angle µδ(θ), and cal-

culated mean value of σ2δ , for the sinusoidal model with fixed time between

turns and with rescaled sampling length τs/τ . . . . . . . . . . . . . . . . . . 231


calculated mean value of σ2δ , for the linear model with fixed time between

turns and with rescaled sampling length τs/τ . . . . . . . . . . . . . . . . . . 232

7.5 Values of ρ, the ratio between the expected and observed values of B−1 with

the corresponding average time step between turns in the original random

walk, τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

xviii

Chapter 1

Introduction and background

1.1 General background to random walks

1.1.1 History

1.1.1.1 Brownian motion

The endless irregular motion of individual pollen particles in liquid was famously studied

by the English botanist Brown (1828), and such random movement has been subsequently

known as Brownian motion. At the turn of the century many eminent physicists such

as Einstein (1905, 1906) and Smoluchowski (1916) were drawn to the subject. During

the course of research on Brownian motion, not only random walk theory (Uhlenbeck

& Ornstein, 1930), but also such important fields as random processes, random noise,

spectral analysis, and stochastic equations were developed.

1.1.1.2 The random walk

Classical works on probability have been in existence for centuries so it is somewhat

surprising that the first ‘random walk’ problem only appeared in the literature in 1905

when the journal Nature (Vol. 72, p.294) published ‘The problem of the random walker’

by Karl Pearson. The question posed was this:

‘A man starts from a point 0 and walks l yards in a straight line: he then turns

through any angle whatever and walks another l yards in a second straight line. He

repeats this process n times. I require the probability that after these n stretches

he is at a distance between r and r + δr from his starting point 0. The problem is

one of considerable interest, but I have only succeeded in obtaining an integrated

solution for two stretches. I think, however, that a solution ought to be found, if

only in the form of a series in powers of 1/n, where n is large.’

Lord Rayleigh responded (Nature, Vol. 72, p.318, 1905):

1

CHAPTER 1: Introduction and background 2

‘The problem, proposed by Prof. Karl Pearson in the current number of Nature, is

the same as that of the composition of n isoperiodic vibrations of unit amplitude

and of phases distributed at random, considered in Philosophical Magazine, Vol. 10,

p.73, 1880; Vol. 47, p.246, 1889 (Scientific Papers, I, p.491; IV, p.370). If n be very

great, the probability sought is

2n−1e−r2/nr dr.

Probably methods similar to those employed in the papers referred to would avail for

the development of an approximate expression applicable when n is only moderately

great.’

In fact, Rayleigh had been studying similar problems to the random walk but under

different names.

The first simple models of movement using random walks are uncorrelated, meaning that

each step taken is completely independent of previous steps taken and as the direction

moved at each step is completely random the motion is Brownian. Such models can be

shown to produce the standard diffusion equation (sometimes called the heat equation).

Bias can be introduced by making the probability of moving in a certain direction greater

and one can derive the drift-diffusion equation. These models have been classed as ‘position

jump processes’ (Othmer et al., 1988), and in general are only valid for large time scales

as their solutions allow for effectively infinite propagation speeds. They can be thought of

as an asymptotic approximation to the true equations governing movement that include

correlation effects.

1.1.2 The isotropic random walk and the diffusion equation

The simple isotropic random walk model is the basis of most of the theory of diffusive

processes. The derivation of the probability distribution is a standard procedure (see for

example Chandreskar (1943), Lin & Segel (1974), Okubo (1980), Murray (1993) etc.). The

main points are presented here.

1.1.2.1 Deriving an equation for the probability density

For the simple isotropic one-dimensional random walk it is straightforward to derive an

equation for the probability density function by considering the limit as the number of

steps gets very large. Consider a one dimensional uniform lattice, and suppose we have a

walker moving along the lattice. The walker moves a short distance δ either left or right

in a short time τ . The motion is assumed to be completely random (isotropic) so that the

probability of moving left or right is 12 . After one time interval, τ , the walker can either

be a distance of δ to the left of the origin with probability 12 , or a distance of δ to the


right of the origin with probability 12 . After the next time interval, the walker will either

be a distance of 2δ to the left of the origin with probability 14 , or a distance of 2δ to the

right of the origin with probability 14 , or will have returned to the origin with probability

12 (but the walker cannot still be a distance δ from the origin — the walker can only be

an even distance from the origin). Continuing in this way, the probability that a walker

will be at a distance of mδ to the right of the origin after N time steps (where m and N

are even), is given by

p(m,N) = (1

2)N

N !

[(N +m)/2]![(N −m)/2]!= (

1

2)N

NN−m

2

. (1.1)

This is the binomial distribution, which for large N converges to the Gaussian (or Normal)

distribution, see for example Clarke & Cooke (1992). Thus,

limN→∞

p(m,N) =

(

2

πN

) 1

2

e−m2/2N . (1.2)

Let x = mδ, and t = τN , and since m is even we set

P (x, t) dx ≡ p

(

x

δ,t

τ

)

dx

2δ. (1.3)

Then the probability of being between x and x+ dx is given by

P (x, t) dx =1

√

2πδ2t/τe−x

2τ/2δ2t dx, (1.4)

and if we take limits such that τ, δ → 0, while δ2/τ = constant ≡ 2D, then

P (x, t) =1√

4πDte−x

2/4Dt. (1.5)

For x ∈ R and t ∈ R+, equation (1.5) is the fundamental solution to the diffusion equation

∂P

∂t= D

∂2P

∂2x, (1.6)

where P (x, 0) = δ(x) (where δ is the Dirac delta function). If we multiply equation (1.6)

by N , the number of individual walkers in a population, then we get a special case (where

D is constant) of Fick’s equation (1.7) for the concentration (C), or number density of the

population (see Okubo (1980))

∂C

∂t=

∂

∂x

(

D∂C

∂x

)

. (1.7)

Solution plots for (1.5) are shown in Figure 1.1.

Useful statistics of this process are the mean position, < x >, and the mean squared

displacement, < x2 >, defined as

< x >=

∫

∞

−∞

xP (x, t) dx, (1.8)


D=1D=5D=10

0

0.05

0.1

0.15

0.2

0.25

P(x,t)

–40 –20 20 40x

(a) P (x, 1)

D=1D=5D=10

0

0.02

0.04

0.06

0.08

P(x,t)

–40 –20 20 40x

(b) P (x, 10)

Figure 1.1: Plots showing P (x, t) for D = 1, 5 and 10, and t = 1 and 10. (The scales on

each plot are different).

and

< x2 >=

∫

∞

−∞

x2P (x, t) dx. (1.9)

For the one-dimensional diffusion solution, < x >= 0 (as we have no bias or preferred

direction), and < x2 >= 2Dt. It is a standard result for a diffusion process that the mean

squared displacement increases in proportion to time, < x2 >∼ t.

1.1.2.2 Solutions in higher dimensions

A similar derivation can be completed in higher dimensions. In s dimensions, the diffusion

equation is given by (Montroll & Shlesinger, 1984)

∂P

∂t= D

(

∂2

∂x21

+ ...+∂2

∂x2s

)

P. (1.10)

If we assume an initial delta function distribution P (x1, ..., xs, 0) = δ(x1)...δ(xs), then

(1.10) has solution

P (x, t) =1

(4πDt)s/2e−r

2/4Dt, (1.11)

where r2 = x21 + ...+ x2

s.

From (1.10) the two-dimensional diffusion equation is

∂P

∂t= D

(

∂2P

∂x2+∂2P

∂y2

)

, (1.12)


where P (x, y, 0) = δ(x)δ(y). From (1.11), the solution is

P (x, y, t) =1

4πDte−(x2+y2)/4Dt. (1.13)

Plots of example solutions for (1.13) are shown in Figure 1.2 for different diffusion coeffi-

cients, D.

–20

–10

0

10

20

x

–20

–10

0

10

20

y

0

0.02

0.04

0.06

0.08

P(x,y,t)

(a) P (x, y, 1) for D = 1

–20

–10

0

10

20

x

–20

–10

0

10

20

y

0

0.002

0.004

0.006

0.008

P(x,y,t)

(b) P (x, y, 10) for D = 1

–20

–10

0

10

20

x

–20

–10

0

10

20

y

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

P(x,y,t)

(c) P (x, y, 1) for D = 5

–20

–10

0

10

20

x

–20

–10

0

10

20

y

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

P(x,y,t)

(d) P (x, y, 10) for D = 5

Figure 1.2: Plots showing P (x, y, t) for D = 1 and 5, and t = 1 and 10. (The scales on

each plot are different).

For the two-dimensional diffusion solution, the mean position is < (x, y) >= (0, 0) , and

the mean squared displacement is < r2 >=< (x2 + y2) >= 4Dt.

Looking at the solutions in (1.5) and (1.13), one can see that P (x, t) > 0 for any t > 0

and any x, y ∈ R. The diffusion process predicts a non-zero probability for arbitrarily

large displacements at arbitrarily small times, and in this sense the underlying speed

of propagation is infinite. Because of this, and because in (1.2) and (1.5) we assumed

that N → ∞ and δ → 0, the solution of the diffusion equation can be considered as an


asymptotic approximation, valid for large time, of equations that more accurately describe

the correlations in movement that are likely to be present at shorter time scales.

1.1.2.3 Using a difference equation to derive the diffusion equation

In the previous section, we showed how to derive an equation for the solution to the

diffusion equation P (x, t) directly. Working the other way, it is possible to set up a

difference equation and then derive the diffusion equation by completing a Taylor series

expansion and taking appropriate limits, see for example Lin & Segel (1974), Okubo

(1980) etc. With this method it is easy to introduce different probabilities for left and

right movement, and derive a form of the diffusion equation that includes drift.

Consider a walker moving along a one-dimensional lattice, where at each time step τ it

either moves a distance δ to the left with probability l, or a distance δ to the right with

probability r, or stays in the same position with probability 1−r− l (the isotropic random

walk has r = l = 1/2). We define the probability that an individual walker is at a position

x at time t by P (x, t). One time step earlier, at time t− τ , the walker must have been at

position x − δ and then moved to the right, or at position x + δ and then moved to the

left, or at position x and then not moved at all. Thus

P (x, t) = P (x, t− τ)(1 − l − r) + P (x− δ, t− τ)r + P (x+ δ, t− τ)l. (1.14)

We now assume that τ and δ are small when compared to t and x respectively, so that

(1.14) can be expanded as a Taylor series in x and t. Writing P for P (x, t), we have

P =

(

P − τ∂P

∂t

)

(1 − l − r) +

(

(P − τ∂P

∂t− δ

∂P

∂x+δ2

2

∂2P

∂2x

)

r

+

(

(P − τ∂P

∂t+ δ

∂P

∂x+δ2

2

∂2P

∂2x

)

l +O(δ3) +O(τ2). (1.15)

Rearranging this gives

∂P

∂t= −δǫ

τ

∂P

∂x+kδ2

2τ

∂2P

∂x2+O(δ3) +O(τ2), (1.16)

where ǫ = r − l and k = l + r. We now let δ, τ, ǫ → 0 in such a way that the following

limits are finite:

u = limδ,ǫ,τ→0

δǫ

τ, (1.17)

D = k limδ,ǫ,τ→0

δ2

2τ. (1.18)

Taking these limits, the O(δ3) and O(τ2) terms in (1.16) go to zero yielding

∂P

∂t= −u∂P

∂x+D

∂2P

∂x2. (1.19)


This is a form of the diffusion equation that includes drift. Note that if we set r = l = 1/2

as in the isotropic random walk, then u = 0, giving (1.6).

The fact that we have introduced waiting into the random walk by allowing a walker to

stay in the same position (with probability 1 − r − l), does not change the final diffusion

equation solution. However, from (1.18) one can see that k = l + r < 1 in this case, and

hence the value of the diffusion constant D will be smaller.

For x ∈ R and t ∈ R+, Montroll & Shlesinger (1984) give the solution that satisfies (1.19)

with initial condition P (x, 0) = δ(x) as

P (x, t) =1√

4πDte−(x−ut)2/4Dt. (1.20)

Solution plots for (1.20) are shown in Figure 1.3.

t=1t=10t=50

0.05

0.1

0.15

0.2

0.25

P(x,t)

–20 20 40 60 80 100 120x

(a) P (x, t) with D = 1, u = 1

t=1t=10t=50

0.05

0.1

0.15

0.2

0.25

P(x,t)

–20 20 40 60 80 100 120x

(b) P (x, t) with D = 1, u = 2

Figure 1.3: Plots of P (x, t), the solution of the drift diffusion equation, for various u

Similar derivations can be completed for two-dimensional random walks, and results are

presented in Chapter 2.

It is possible to calculate the statistics < x > and < x2 > directly from the drift-diffusion

equation without having to solve to find P (x, t). This is a technique that we will exploit

in later chapters. To find < x >, multiply (1.19) by x and integrate over R to give

∫

∞

−∞

x∂P

∂tdx = −u

∫

∞

−∞

x∂P

∂xdx+D

∫

∞

−∞

x∂2P

∂t2dx. (1.21)

Using integration by parts and making the assumption that P (x, t) and its first two x

derivatives tend to zero as |x| → ∞ gives

d < x >

dt= u, (1.22)


which with the initial condition < x > (0) = 0, has solution

< x > (t) = ut. (1.23)

In a similar manner we can derive a differential equation for < x2 >,

d < x2 >

dt= 2u2t+ 2D, (1.24)

which with the initial condition < x2 > (0) = 0, has solution

< x2 > (t) = u2t2 + 2Dt. (1.25)

The same solutions can be obtained by multiplying (1.20) by x or x2 and then integrating

over R. The statistic

σ2(t) =

∫

∞

−∞

[x− < x >]2P (x, t) dx, (1.26)

that measures the dispersal about the mean position, is more appropriate for a diffusion

drift process than < x2 >, which measures the dispersal about the origin. For (1.19),

σ2(t) = 2Dt, the same value as < x2 > for a diffusion process without drift.

The limiting process in (1.18) is such that terms of the form δ2/τ tend to be finite as

δ, τ → 0, which means that δ/τ → ∞ as δ, τ → 0, implying an infinite propagation

speed, see Okubo (1980). Othmer et al. (1988) classified this way of modelling movement

using an uncorrelated random walk as a ‘position jump process’.

1.1.3 Random walks to a barrier — a simple example

Using the simple random walk models described above it is possible to investigate the

effect of placing a barrier into the random walk. To model movement in a confined domain

(for example fish swimming in a tank), then one can impose a ‘repelling’ or ‘reflecting’

barrier — a walker reaching the barrier will turn around and move away in the opposite

direction. To model movement where walkers leave the system upon reaching a given

point (for example larval fish recruiting to a reef — see Chapter 10), then one can impose

an ‘absorbing’ barrier — a walker reaching the barrier will be absorbed and is no longer

part of the system. The following simple example of an absorbing barrier is adapted from

an example in Grimmett & Stirzaker (2001).

Suppose we have a random walk process that satisfies the diffusion equation with drift

(1.19), i.e. we have∂g

∂t= −u∂g

∂x+σ2

2

∂2g

∂x2x > 0, (1.27)

where g is the probability density, u is the drift in the x direction, and σ2 is the variance

about the x position. Suppose the walker starts at position x = d > 0, and we have an

‘absorbing barrier’ at x = 0 — if a walker reaches the point x = 0 it is ‘absorbed’ and


removed from the system. This gives the boundary conditions

g(t, 0) = 0 t ≥ 0, (1.28)

g(0, x) = δ(x− d) x ≥ 0. (1.29)

From (1.20), the solution to (1.27) with boundary condition (1.29) is

g(t, x) =1

σ√

2πtexp

(

−(x− d− ut)2

2σ2t

)

. (1.30)

From Grimmett & Stirzaker (2001), the solution to (1.27) that takes into account both

boundary conditions (1.28) and (1.29) is

g(t, x) =1

σ√

2πt

[

exp

(

−(x− d− ut)2

2σ2t

)

− exp

(

−(x+ d− ut)2

2σ2t− 2ud

σ2

)]

. (1.31)

It is a simple step to derive the density function of the time ta until the absorption of the

walker. At time t, either the walker has been absorbed, or its position has density function

given by (1.31), and hence

P (ta ≤ t) = 1 −∫

∞

0g(t, x)dx. (1.32)

Differentiation with respect to t of the cumulative distribution in (1.32) gives the proba-

bility density function fa(t) of the absorbing time ta,

fa(t) =d

σ√

2πt3exp

(

−(d+ ut)2

2σ2t

)

. (1.33)

The probability of absorption taking place (ta <∞) is given by

P (ta <∞) =

1 if u ≤ 0

e−2ud if u > 0.(1.34)

1.1.4 The telegraph equation

In the previous section we derived diffusion equations as the governing equations behind

the spatial distribution of a simple random walk. The random walks did not include

correlation — the direction of motion chosen at a certain step was independent of the

previous direction of movement. As a consequence, solutions of the diffusion equation can

have infinite propagation speed. The solutions are satisfactory for large time-scales but

for smaller time-scales we must look for a better model that includes correlation effects.

The one-dimensional telegraph equation was first derived by Goldstein (1951), (see also

Kac (1974), Okubo (1980), Othmer et. al (1988), etc), and we present the derivation here.


1.1.4.1 The unbiased one-dimensional telegraph equation

We restrict the population to moving left or right along an infinite line at a constant speed

v. We split the population into left-moving individuals β and right-moving individuals α,

where the total population is given by p = α + β. At each time step τ the individuals

either move a distance δ in the direction they were previously moving (with a probability

given by q = 1 − λτ) or they change direction and then move a distance δ in this new

direction (with the turning probability given by r = λτ).

If we take a forward time step then the number density of individuals at position x moving

right and left respectively is given by

α(x, t+ τ) = qα(x− δ, t) + rβ(x− δ, t),

β(x, t+ τ) = rα(x+ δ, t) + qβ(x+ δ, t).

We can expand these equations as Taylor series to give

α+ τ∂α

∂t+O(τ2) = q(α− δ

∂α

∂x+O(δ2)) + r(β − δ

∂β

∂x+O(δ2)),

β + τ∂β

∂t+O(τ2) = q(β + δ

∂β

∂x+O(δ2)) + r(α+ δ

∂α

∂x+O(δ2)).

Substituting for q and r gives

α+ τ∂α

∂t+O(τ2) = α− δ

∂α

∂x− λτα+ λτδ

∂α

∂x+ λτβ − λτδ

∂β

∂x+O(δ2),

β + τ∂β

∂t+O(τ2) = β + δ

∂β

∂x− λτβ − λτδ

∂β

∂x+ λτα+ λτδ

∂α

∂x+O(δ2).

Now divide through by τ and take the limit such that δ/τ → v as δ → 0 and τ → 0 (where

v is the constant speed), giving

∂α

∂t= −v∂α

∂x+ λ(β − α), (1.35)

∂β

∂t= v

∂β

∂x− λ(β − α). (1.36)

Adding (1.35) and (1.36) gives

∂(α + β)

∂t= v

∂(β − α)

∂x, (1.37)

which can be differentiated to give

∂2(α+ β)

∂t2= v

∂2(β − α)

∂x∂t. (1.38)

Subtracting (1.36) from (1.35) gives

∂(β − α)

∂t= v

∂(α + β)

∂x− 2λ(β − α), (1.39)


∂2(β − α)

∂x∂t= v

∂2(α+ β)

∂x2− 2λ

∂(β − α)

∂x. (1.40)


Substituting (1.40) into (1.38), and using (1.37) and the fact that α+ β = p, gives

∂2p

∂t2+ 2λ

∂p

∂t= v2 ∂

2p

∂x2. (1.41)

This is the telegraph equation. The equation can be solved if given initial conditions

specified by the initial distribution p(x, 0). We have a fixed speed v and so (unlike the

diffusion equation) we cannot have an arbitrarily large propagation speed.

Multiplying (1.41) by x, and integrating over R gives the mean < x >, which for the

unbiased telegraph equation is zero. Multiplying (1.41) by x2, and integrating over R

gives a differential equation for the mean squared displacement < x2 > ,

d2 < x2 >

dt2+ 2λ

d < x2 >

dt= 2s2. (1.42)

Assuming that p(x, 0) = δ(x) and ∂p∂t (x, 0) = 0, then the appropriate initial conditions for

(1.42) are < x2(0) >= ddt < x2(0) >= 0, and the solution is

< x2(t) >=v2

λ

(

t− 1

2λ(1 − e−2λt)

)

. (1.43)

For small t, < x2(t) >∼ v2t2, which is characteristic of a wave propagation process, and

for large t, < x2(t) >∼ v2t/λ, which is characteristic of a diffusion process with diffusion

coefficient D = v2/2λ.

When deriving the uncorrelated random walk we showed that D = δ2/2τ , and these two

results can be related. In a Poisson process of intensity λ the mean time between events

is 1/λ, see Grimmett & Stirzaker (2001). Thus the average distance travelled between

reversals is δ = v/λ, and therefore

D =v2

2λ=δ2λ

2. (1.44)

Since τ = 1/λ, the ‘diffusion limit’ of the telegraph process consists of letting λ → ∞(equivalent to τ → 0 for uncorrelated process), and v → ∞ (equivalent to δ/τ → ∞for uncorrelated process), while maintaining v2/λ constant (equivalent to δ2/τ constant

in uncorrelated process). Thus we can argue that when λ → ∞ both the uncorrelated

random walk and the telegraph process tend to the same limit. This is equivalent to the

large time limit of both processes being the same — correlation effects become less evident

as t→ ∞.

Morse & Feshbach (1953) give a solution to the one-dimensional telegraph equation (1.41),

subject to the initial conditions p(x, 0) = δ(x), ∂p∂t (x, 0) = 0 as

p(x, t) =

e−λt

2

δ(x− vt) + δ(x+ vt) + λv

[

I0(Z) + λtZ I1(Z)

]

for |x| < vt,

0 for |x| ≥ vt,(1.45)

where Z = λ√

t2 − x2/v2 and I0 and I1 are the modified Bessel functions of the first kind.

Plots of (1.45) are shown in Figure 1.4.


v=1, l=1v=1, l=5v=5, l=1

Legend

0

0.05

0.1

0.15

0.2

0.25

p(x,t)

–20 –10 10 20x

Figure 1.4: Plots of p(x, t), the solution of the telegraph equation for various parameter

values

From Abramowitz and Stegun (1965), the Bessel functions have the following asymptotic

expansions

I0(Z) ∼ eZ√2πZ

+O(1/Z),

I1(Z) ∼ eZ√2πZ

+O(1/Z),

for Z → ∞. The solution of (1.45) for x → ∞, t → ∞ and for |x| ≪ vt (so terms of the

form α = x2/v2t2 ∼ 0), can be shown to be

p(x, t) ∼ 1√4πDt

e−x2/4Dt + e−λtO(α2). (1.46)

Thus far from the boundaries |x| = vt, the solution of (1.45) reduces as t → ∞ to the

solution of the diffusion equation, as expected.

1.1.4.2 The biased one-dimensional telegraph equation

The derivation of the biased one-dimensional telegraph equation is similar to the derivation

for the unbiased telegraph equation, except that we now have different turning probabilities

depending on which way the individual is moving. This introduces a bias to the direction

of movement. We split the population into right-moving individuals α and left-moving


individuals β where the total population is given by p = α+ β. At each time step τ each

individual either moves a distance δ in the direction they were previously moving (with

a probability given by q1 = 1 − λ1τ if they are right-moving or q2 = 1 − λ2τ if they are

left-moving) or they change direction and then move a distance δ in this new direction

(with the turning probability given by r1 = λ1τ if they are right-moving or r2 = λ2τ if

they are left-moving).

Using a difference equation in a similar way to the unbiased case we get the following

equation for the

population density

∂2p

∂t2+ (λ1 + λ2)

∂p

∂t+ v(λ2 − λ1)

∂p

∂x= v2 ∂

2p

∂x2. (1.47)

This is the biased telegraph equation. It is of similar form as the unbiased telegraph

equation (1.41) but has an additional ‘drift’ term due to the bias. If λ1 > λ2 then an

individual is more likely to turn if right-moving and hence there will be a drift to the left.

If λ2 > λ1 then the opposite is true and we get drift to the right. One can see that for the

biased case with equal turning probability (λ1 = λ2) we get the original unbiased equation

again.

As with the unbiased telegraph equation, it is possible to calculate the moments < x >

and < x2 > directly from (1.47) but we do not do so here.

1.2 Circular statistics

In general, the simple random walk models discussed previously were restricted to ‘lattices’

so that there are only a finite number of choices of direction at each move. A more realistic

model will allow for a continuous choice of direction. In two dimensions this means we

allow a walker to move in any direction θ on the unit circle, where −π < θ ≤ π, and where

−π and π correspond to the same direction. Such models generate data and statistics on

the direction of movement. Linear statistical measures cannot be used because of the fact

that the angles on a unit circle have modulus 2π (e.g. π = 3π = 5π etc), and the fact that

−π and π correspond to the same direction.

Fisher (1993), and Fisher, Lewis & Embleton (1987) provide a general introduction and

methodology for dealing with statistics of circular data and spherical data respectively,

while Batschelet (1981) uses circular statistics to model particular problems that occur in

biology, and Mardia & Jupp (1999) provide a large amount of theoretical background and

models for use with directional data. The main relevant results that we will require are

presented here.


1.2.1 The mean direction

Suppose that we are given unit vectors x1, . . . ,xn with corresponding angles θi, i =

1, . . . , n. The mean direction θ of θ1, . . . , θn is the direction of the resultant x1 + . . .+ xn

of x1, . . . ,xn. It is also the direction of the centre of mass x of x1, . . . ,xn. Since the

Cartesian coordinates of xj are (cos θj, sin θj) for j = 1, . . . , n, the Cartesian coordinates

of the centre of mass are (C, S), where

C =1

n

n∑

j=1

cos θj and S =1

n

n∑

j=1

sin θj. (1.48)

Therefore θ is the solution of the equations

C = R cos θ, S = R sin θ (1.49)

(provided that R > 0), where the mean resultant length R is given by

R =√

C2 + S2. (1.50)

Note that θ is not defined when R = 0. When R > 0, θ is given explicitly by

θ =

tan−1(S/C), if C ≥ 0,

tan−1(S/C) + π, if C < 0,(1.51)

where the inverse tangent function takes values in the range [−π/2, π/2].In the context of circular statistics θ does not mean (θ1 + . . . + θn)/n, which is not well

defined. The mean direction of θ1 −α, . . . , θn− α is θ−α, i.e. the sample mean direction

is equivariant under rotation.

1.2.2 The mean resultant length and the circular variance

The mean resultant length R was introduced above as the length of the centre of mass

vector x, and is given by

R =√

C2 + S2.

Since x1, . . . ,xn are unit vectors, we have

0 ≤ R ≤ 1.

If the directions θ1, . . . , θn are tightly clustered then R will be almost 1. If θ1, . . . , θn are

widely dispersed then R will be almost 0. Thus R is a measure of concentration of a data

set. Note that any data set of the form θ1, . . . , θn, θ1 + π, . . . , θn + π has R = 0. It follows

that R ≈ 0 does not imply that the directions are spread almost evenly round the circle,

see Figure 1.5.

The resultant length R is the length of the vector x1 + . . .+ xn. Thus

R = nR.


Figure 1.5: Example of a data set on a circle with R ≈ 0 but with a non-uniform spread

of points.

By analogy with the standard deviation in linear statistics, we can define the angular

deviation s to be

s =√

2(1 − R), (1.52)

and then s2 is the circular variance. An alternative definition is

s0 =√

−2ln(R), (1.53)

which arises from the analogy between a circular distribution called the wrapped normal

distribution and the normal distribution of linear statistics. Both s and s0 tend to the

same limit, 0, as R tends to one, but s0 becomes infinite as R tends to zero.

1.2.3 Probability distributions on the circle

Probability distribution functions f(θ) on a circle satisfy

f(θ) ≥ 0 ∀ − π ≤ θ < π, (1.54)

and the normalization condition that the total probability is one, i.e.

∫ π

−πf(θ) dθ = 1. (1.55)

Corresponding to the moments of a linear probability distribution function (p.d.f.), the

angular (or trigonometric) moments are defined by

an =

∫ π

−πcos(nθ)f(θ) dθ, bn =

∫ π

−πsin(nθ)f(θ) dθ, (1.56)

where n = 1, 2, 3, . . .. The polar forms of these moments are written as ρn and φn where

ρneiφn = an + ibn. (1.57)

ρ1 and φ1 are simply interpreted as the mean length and mean angle of the distribution.

The following three distributions are particularly useful.


1.2.3.1 The uniform distribution

When the distribution is uniform, points are distributed with equal probability around

the unit circle and

f(θ) = U(θ) ≡ 1

2π. (1.58)

For the uniform distribution, ρ1 = 0, φ is undefined and s =√

2.

1.2.3.2 The von Mises distribution

The von Mises distribution, M(θ;µ, κ), has probability density function

f(θ) = M(θ;µ, κ) =1

2πI0(κ)eκ cos(θ−µ), (1.59)

where I0 denotes the modified Bessel function of the first kind and order 0, which is defined

by

I0(κ) =1

2π

∫ π

−πeκ cos θdθ. (1.60)

The parameter µ is the mean angle and the parameter κ is known as the concentration

parameter, where κ ≥ 0. The distribution is unimodal and is symmetrical about θ = µ.

The mode is at θ = µ and the antimode is at θ = µ+ π.

When κ = 0 the von Mises distribution equals the uniform distribution, and as κ → ∞the distribution becomes sharply peaked about the mean angle µ, see Figure 1.6.

k=0.5k=1k=2k=4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

–3 –2 –1 1 2 3

θ

Figure 1.6: Examples of the von Mises distribution for various values of κ, and µ = 0.


For the von Mises distribution, φ1 = µ and ρn = An(κ) where

An(κ) =In(κ)

I0(κ), (1.61)

which is readily calculated numerically and tabulated (Batschelet 1981; Mardia & Jupp

1999).

1.2.3.3 The wrapped normal distribution

The wrapped normal distribution is the linear normal distribution wrapped around the

unit circle. It too is unimodal and defined by

f(θ) = W (θ;µ, σ) ≡ 1

σ√

2π

k=∞∑

k=−∞

exp

[

−(θ − µ+ 2πk)2

2σ2

]

, (1.62)

where µ and σ > 0 are parameters. For the wrapped normal distribution, ρ1 = e−σ2

2 and

φ1 = µ. For this distribution,

σ =√

−2lnρ1, (1.63)

which motivates the alternative definition of the angular deviation in (1.53).

An important point to note is that when σ is related to κ by

A1(κ) = e−σ2

2 , (1.64)

then the von Mises and the wrapped normal distributions only differ by a few percent so

that in applications it is convenient to treat their properties as being the same.

This relation between κ and σ2 is plotted in Figure 1.7.

0

2

4

6

8

10

12

σ2

1 2 3 4 5

κ

Figure 1.7: Plot of κ against σ2.

The function A1(κ) and its inverse A−1(κ) are readily computed numerically using com-

puter software (e.g. Maple 8), or can be found by looking at tables of the inverse Bessel


functions (Batschelet 1981; Mardia & Jupp 1999). As κ→ 0, σ2 → ∞ asymptotically and

as κ→ ∞, σ2 → 0 asymptotically.

1.3 Modelling biological motion

The random walk models described in Sections 1.1.2 and 1.1.4 have been applied to a wide

range of different problems in the biological sciences at many different spatial scales. Many

of the results can be generalised and are applicable to more than one situation. Skellam

(1951) was one of the first in the literature to use random walks to specifically model the

dispersal of animal populations. Skellam (1973) and Levin (1986) both discuss the validity

and limitations of the simple isotropic random walk model, in particular the problem of

infinite propagation at small time scales arising from the lack of correlation, and the

fact that this model cannot account for interaction of individuals or habitat variability.

Okubo (1980) discusses a wide range of diffusion problems in many different biological

settings, and provides a detailed discussion of the derivation and limitations of the various

random walk models. Murray (1993) gives a brief overview of some biological diffusion

problems that can be modelled using the simple random walk models discussed previously.

A useful glossary of terms that are associated with directed movement and random walks

by Tranquillo & Alt is published in Alt & Hoffmann (1990), which also contains many

other models of biological motion and includes discussions on experimental limitations of

observing motion at different scales.

1.3.1 The movement of animals and micro-organisms as a random walk

Patlak (1953) developed an extended version of the random walk model, accounting for

correlation between successive steps, nonhomogeneity in the environment, and external

forces, as well as allowing for a variable speed and time interval between steps. Unfortu-

nately, the model is too general and is only practical when several simplifying assumptions

are made about the motion, see Okubo (1980). Siniff & Jessen (1969) were one of the first

in the literature to use a simulation model of a correlated and unbiased random walk that

included the von Mises distribution as the probability distribution for the turning angle at

each step, see Section 1.2.3.2. The task of judging the statistical accuracy of the results of

simulations when compared to data of telemetrically observed fox movement is difficult as

there is no criterion of goodness of fit — the problem is one of pattern recognition. Lovely

& Dahlquist (1975) used a random walk model to describe the motion of the bacteria

Escherichia coli when there is no preferred direction and showed how to derive a diffusion

coefficient for the flux of bacteria. They also discussed some bulk statistical measures

when there is a preferred direction, but did not link this to a random walk model for the

motion of individual cells.

Correlated random walks have been used by Hall (1977) when studying the movement of


Dictyostelium discoideum amoeba, Dunn (1983) to model the motion of chick heart fibrob-

lasts, Kareiva & Shigesada (1983) to model cabbage butterflies ovipositing or searching

for nectar sites, and many others. Bovet & Benhamou (1988) set up a simulation model

of a correlated random walk and suggest sinuosity as a spatial measure of the trajectory

that is independent of the sampling length imposed. More recently, Byers (2000) has used

simulations of correlated random walks to study the dispersal of bark beetles in forests,

and Byers (2001) uses the same simulation model to find a correction factor between the

root of the mean squared displacement and the mean dispersal distance. Details of the

main results of all these correlated random walk models are given in Section 1.4.

1.3.2 Biased movement and taxis

Berg & Brown (1972) and Berg (1983) studied the movement of bacteria, in particular

E. coli. The motion of the bacteria can be described as a ‘run and tumble’ or ‘run and

twiddles’ — long straight moves are separated by periods of very short random turns. An

uncorrelated random walk was used to model the movement, but bias is introduced by

making the probability of a long straight run more likely if the direction is preferable.

Berg also introduced the idea of rotational diffusivity to help describe the periods of rapid

random turning (tumbles).

Keller & Segel (1971a) also modelled the movement of E. coli as an uncorrelated random

walk. The rate of turning of the bacteria depends on the concentrations of chemical

attractants — this is known as klinokinesis (see Tranquillo & Alt in Alt & Hoffmann,

1990). With this model, the chemotaxis of bacteria is determined by diffusion, where

the diffusivity depends on the concentration of chemical substance, and advection, which

depends on the gradient of the concentration of the chemical substance. Using this model

of chemotaxis, Keller & Segel (1971b) were able to explain the phenomenon of wavelike

propagation of bands of certain species of bacteria under the influence of a chemical

substrate. A similar chemotaxis problem was also studied by Alt (1980). Othmer et al.

(1988) looked at a wide number of applications of the random walk in different biological

systems, both with and without bias. They classified two types of process — position jump

processes, which are essentially the simple uncorrelated random walk models that allow

for infinite propagation as described in Section 1.1.2; and velocity jump processes, where

the random walk is governed by a master transport equation and infinite propagation is

avoided. The simplest example of a one-dimensional velocity jump process is the telegraph

equation, as described in Section 1.1.4. We will discuss the velocity jump process model

of Othmer et al. in more detail in Chapter 3.

More recent models of chemotaxis have included interactions between the walkers and the

chemical substrate — the simplest example being when the initial walkers leave a trail

for other walkers to follow. This type of movement is modelled as a reinforced random

walk. A reinforced random walk is a position jump process where the weight of the


transition probabilities (the probability of moving from one point on the lattice to any of

the neighbouring lattice points, see for example Okubo (1980)) change when the position

has been occupied by a cell — the walk is ‘reinforced’ as cells cross a point. This type

of random walk also has the advantage that it allows easy transition between the discrete

model and its continuum limit. When modelling the movement and aggregation of bacteria

such as E. Coli, or slime mould amoeba such as D. discoideum, Othmer & Stevens (1997)

derived the continuum limit for several choices of the transition probabilities called the

local model, the barrier model, the nearest neighbour model and the gradient model.

Similar reinforced random walk models have been used by Levine & Sleeman (1997) and

by Plank & Sleeman (2003) when modelling angiogenesis leading to the development of

tumours. More recent work by Othmer & Hillen (2002) has looked at using velocity

jump processes in place of the standard position jump process to derive the chemotaxis

equations.

Hill & Hader (1997) studied the motion of algae such as C. nivalis and modelled their

movement as a continuous correlated and biased random walk, with two methods of re-

orientation depending on whether the motion was governed by gyrotaxis or phototaxis.

Section 1.5 explores their results in more detail.

1.3.3 Other applications of the random walk in biology

The random walk models discussed do not have to be restricted to animals or cells moving

in space. Branching models using correlated random walks have been used to model the

growth of the root structure of the Sitka spruce by Henderson et al. (1984), as well as

the growth of polymer chains (Tchen, 1952; Flory, 1969). Pitchford et al. (2003) use a

random walk in the weight of a fish larva with an absorbing barrier at a critical survival

size (recruitment size), to show how the variability in the environment and foraging rate

is important to the survival probability. Hillen (1996) used a correlated random walk in a

reaction-diffusion system and compared results to the standard reaction diffusion system

with diffusion through a standard isotropic random walk (Turing, 1952).

1.4 Properties of correlated random walks

We have shown in Section 1.1.2 that it is straightforward to derive expressions for the

probability density function, p(x, t) of an uncorrelated random walk in one or more di-

mensions, that has purely random (Brownian) motion. If we know the distribution function

p(x, t) it is simple to calculate the various moments of interest or to investigate various

modifications of the model (for example introducing an absorbing or reflecting barrier).

Unfortunately, with a correlated random walk it is not always as simple to calculate the

probability density function, p(x, t). In one dimension we have the telegraph equation

(Section 1.1.4), but it is not possible to complete a similar derivation in two dimensions,


as we shall demonstrate in Chapter 2. However, it is possible to derive equations for

certain statistics of the random walk directly, namely the mean squared displacement and

subsequently the mean dispersal distance.

1.4.1 Mean squared displacement

The theory of branching processes is highly applicable to the growth of polymer chains and

other large molecules, and such growth has been modelled by a correlated random walk

by for example, Tchen (1952) and Flory (1969). Both Tchen and Flory derived equations

for the mean squared displacement, while Tchen showed that the spatial distribution of

Xn and Yn is Normal if n is large (where Xn and Yn are the position in the X and Y

directions after n steps respectively). A similar equation to those derived by Tchen and

Flory for the mean squared displacement in a one-dimensional random walk is shown in

the next section.

1.4.1.1 Simple one-dimensional model

We demonstrated earlier that it is possible to derive equations for the moments (< x > and

< x2 >) of a one-dimensional random walk using either the probability density function

p(x, t), or the underlying diffusion or telegraph equation for the p.d.f. It is also possible

to calculate some statistics of simple random walks directly.

Okubo (1980) gives the following one-dimensional example of a correlated random walk to

demonstrate a result given by Skellam (1973), that the square of the distance of dispersal

(mean squared displacement) is asymptotically proportional to time.

Let xj, (j = 1, 2, ..., n), be the displacement of an organism after a succession of time

intervals τ . We assume that the length of each step is constant, |xj | = λ. The square of

the distance from the origin after nτ time has elapsed is given by

R2n = (x1 + x2 + ...+ xn)

2. (1.65)

Taking the average of this quantity gives,

R2n = (x1 + x2 + ...+ xn)2 = x2

1 + x22 + ...+ x2

n + 2(x1x2 + x2x3 + ...)

= nλ2 + 2(x1x2 + x2x3 + ...+ xrxs + ...). (1.66)

If we assume that there is only correlation between successive steps and define xjxj+1/λ2 =

γ, then xjxj+2/λ2 = γ2, xjxj+s/λ

2 = γs and so on, then (1.66) becomes

R2n = nλ2 + 2λ2[nγ + (n− 1)γ2 + (n− 2)γ3 + ...+ γn]

= λ2

(

n+2nγ

1 − γ− 2γ2(1 − γn

(1 − γ)2

)

, (1.67)

where we assume that γ < 1. Thus (1.67) reduces to

R2n = λ2 1 + γ

1 − γn− 2λ2γ2(1 − γn)

(1 − γ)2. (1.68)


As n becomes large, γn becomes small, and the second term on the right hand side of

(1.68) becomes negligible so that

R2n ∼ λ2 1 + γ

1 − γn. (1.69)

So we have the result that R2n is asymptotically proportional to the time in the limit as n

gets large. If γ = 0 and there is no correlation, then (1.68) reduces to R2n = λ2n, which is

comparable to the result < x2 >= 2Dt for an isotropic one-dimensional random walk.

1.4.1.2 Two-dimensional models

Nossal & Weiss (1974) modelled the movement of chemotactic cells and derived a similar

expression for the mean squared displacement of a random walk that included bias in the

step length but not the turning angle distribution.

Kareiva and Shigesada (1983) looked at data on butterfly flight and set up a correlated

random walk model to describe the movement. They complete a more complicated analysis

than the above example from Okubo (1980) as, similar to the model of Nossal & Weiss,

they do not insist that there is a fixed step length. The flight of a butterfly can easily be

broken down into a series of straight line steps with endpoints corresponding to landing

sites. Each step has a direction and turning angle, and a move length. They argue

that their method is equally applicable to continuous movement as this type of motion is

generally observed as a decomposition of straight line moves. In their model they designate

the mth move as (xm, ym), and θm as the turning angle at the end of the mth move. They

assume that the length of the mth move, lm =√

(x2m + y2

m), and the size of each turning

angle θm, are independent random variables where p(l) dl is the probability that the length

of each flight (step) has a value between l and l + dl, and g(θ) dθ is the probability that

and angle between two consecutive flights (steps) measured clockwise has a value between

θ and θ + dθ. Associated with these random variables are the expected values

E(l) =

∫

∞

0lp(l) dl, (1.70)

E(l2) =

∫

∞

0l2p(l) dl, (1.71)

c = E(cos θ) =

∫ π

−πcos θg(θ)dθ, (1.72)

s = E(sin θ) =

∫ π

−πsin θg(θ)dθ. (1.73)

Their random walk then consists of a series of random draws from p(l) and g(θ) for each

step, where each random draw is independent of the preceding draws (i.e. the process is

a first order Markov chain, see Grimmett & Stirzaker, 2001). The distribution g(θ) thus

provides a measure of the degree to which the direction of movement is correlated. If we

have a purely random walk (isotropic), g(θ) is uniform and c and s above are both zero.


Using this model they derive the following equation for the expected mean square dis-

placement after n steps

E(R2n) = nE(l2) + 2E(l)2

(c− c2 − s2)n− c

(1 − c)2 + s2+ 2E(l)2

2s2 + (c+ s2)(n+1)/2

[(1 − c)2 + s2]2γ, (1.74)

where

γ = [(1 − c)2 − s2] cos((n + 1)α) − 2s(1 − c) sin((n+ 1)α),

α = tan−1s/c.

This reduces to a much simpler form in particular cases. If g(θ) has a uniform density,

then (1.74) reduces to E(R2n) = nE(l2). A more realistic case is when organisms exhibit

equal probability of moving left or right and so g(θ) is symmetric about θ = 0. In this

case E(sin θ) ≡ 0, and (1.74) reduces to

E(R2n) = nE(l2) + 2E(l)2

c

1 − c

(

n− 1 − cn

1 − c

)

. (1.75)

This formula directly relates changes in g(θ) or p(l) with consequent changes in the mean

squared displacement.

1.4.2 Sinuosity and mean dispersal distance

Bovet and Benhamou (1988) suggested a purely spatial index of sinuosity, which expresses

the amount of turning associated with a given path length. They set up a simple random

walk model with steps of fixed length P between turns, and investigated the effect on the

sinuosity of rediscretizing the random walk with different step lengths. Using a similar

movement model to Siniff and Jessen (1969), they set the von Mises distribution with

mean angle 0 and parameter κ as the turning angle distribution. The parameter κ is

related to σ2δ , the variance of the turning angle, by (1.64).

By simulating many random walks, rediscretizing each of these paths with new step lengths

and measuring the new turning angle standard deviation σ∗R, they found the following

relation between the rediscretized step length R and σ∗R

σ∗R = 0.85σδ

√

R

P, (1.76)

which holds for R 6= P and assumes that R > 0.5P to avoid artefactual alignments of

points. Rearranging this equation they defined sinuosity as

S∗ =σδ√P

= 1.18σ∗R√R, (1.77)

where the first part of the equation is a theoretical expression for the sinuosity (as in

experiments the true value of P and σδ are usually not known).

An important point to note is that equation (1.77) was only found to hold when σ∗R < 1.2

rads. If R is too large then, as there is no bias in the original turning angle distribution,


the distribution of the changes in direction becomes uniform. In this case σ∗R becomes

large (> 1.2 rads) and no longer increases proportionally to√R, so the sinuosity cannot

be calculated. They make the point that it is not surprising that many animal paths

that may have not been recorded in enough detail (with too large distances between

successive locations) approximately fit the uniform random walk model. This provides no

information about the original path and for this reason they restricted σ∗R < 1.2 rads for

their simulations.

Bovet and Benhamou also related their measure of sinuosity to the expected diffusion.

The diffusion is characterized by the beeline distance D between the first and last point of

a walk, the mean squared displacement being D2. A more sinuous path will involve more

turns and thus the beeline distance will be smaller. Using the fact that the random walk

is correlated but not biased (so that there is no preferential orientation in space), they

calculated the expected value of D, given by

E(D) = P

√

0.79N(1 + r)

(1 − r), (1.78)

where P is the step length, N is the number of steps and r is a measure of the correlation

between the direction of successive steps and is given by r = exp(−σ2δ/2), where σ2

δ is the

variance of the turning angle. From results of simulations they concluded that the beeline

distance D is on average proportional to the square root of the path length and inversely

proportional to the sinuosity. Further details of their derivation of (1.78) are given in

Chapter 9.

A direct calculation of the beeline distance, D (or mean dispersal distance, MDD), was

also attempted by McCulloch & Cain (1989), who derived an approximate formula which

was complex to compute and only valid for a limited number of moves. Byers (2000) has

extended the work of Kareiva & Shigesada and Bovet & Benhamou to find an equation for

the mean dispersal distance (MDD), related to (1.75) and (1.78), where MDD = E(D)

from Bovet & Benhamou. The mean dispersal distance is a more realistic measure for

dispersal as it measures the absolute displacement not the displacement squared. There is

an error factor between the square root of the mean squared displacement and the mean

dispersal distance, so that√

R2n 6= MDD. Details of Byers’ work and our extensions to

the models and results are presented in Chapter 9.

1.5 The circular random walk and reorientation models aris-

ing from experiments on algae

Suspensions of algae such as Chlamydomonas nivalis and Peridinium gatunense are known

to form cooperative bioconvection patterns on length-scales of millimetres, much greater

than the size of individual cells. In order to understand the fluid dynamics of such macro-

scopic patterns, the suspension has been modelled mathematically as a continuum, see for


example Childress et al. (1975), Pedley et al. (1988), Hill et al. (1989) and Pedley &

Kessler (1990). An important feature of such models is the fact there exists a preferred

direction of swimming influenced by external factors such as gravity or light (Kessler et

al., 1992), and at the same time there is a degree of ‘randomness’ in the behaviour of

individuals and across the whole population.

Experiments tracking individual algae such as described in Hader & Lebert (1985), have

limitations due to the size of the algae and resolution of available camera equipment, and

because heat from the camera can cause convection currents in the fluid — results are

likely to be subject to experimental noise. More recent experiments with laser tracking

may prove a better alternative (Vladimirov et al., 2000).

From experimental observations, algae such as C. nivalis are subject to i) gyrotaxis which

orients the cells to move upwards on average and is thought to be due to a passive,

mechanical torque due to the cells being bottom heavy (Kessler, 1986); ii) phototaxis

which in contrast, is an active internal mechanism by which the cells move towards the

light (presumably to increase their rate of photosynthesis), or away from the light if it

is too bright, see for example Hill & Vincent (1993) and Vincent & Hill (1996). Kessler

was the first to discover the role of the gravitational torque in producing macroscopic

bioconvection patterns. He shows that the rate of reorientation of a spherical cell, when it

is not swimming vertically, is determined by the balance between the gravitational torque,

due to its offset centre of mass, and the viscous torque as it rotates in the fluid, giving

ω = dθ/dt = −B−1 sin θ, where B represents the typical reorientation time. This balance

between the viscous and gravitational torques is what is known as gyrotaxis.

In experimental observations, the motion of the cells is not deterministic and subject to

variations in the speed and direction of movement for several reasons: i) the population

of cells is not cloned so there are intrinsic differences in shape and behaviour between

individual cells; ii) the internal biochemistry of cells is such that they can seemingly

change their direction at random or occasionally not move at all; and iii) cells collide with

each other (at high concentrations) and with the walls of the container which introduces

further randomness into an individual cell trajectory (Kessler et al., 1992).

Hill and Hader (1997) analysed the results of experiments looking at the motion of swim-

ming micro-organisms, in particular C. nivalis which exhibits both gyrotaxis and photo-

taxis. By modelling the motion as a continuous limit of a biased and correlated random

walk on a circle, they derived a Fokker–Planck equation (see Chandreskar, 1943; Risken,

1989) for the probability distribution function of the orientation of the cells. They then

checked that the solutions of the Fokker–Planck equation fitted experimental data. Their

methods are described in detail in the following sections. More recently, Vladimirov et al.

(2000) used lasers to track individual algae and found the average velocity to be directed

upwards. However, they use a much larger timescale than Hill & Hader, and initial results

could not be used to calculate orientation statistics.


1.5.1 Deriving the Fokker–Planck equation for a circular random walk

We are interested in the p.d.f. for θ, the angle that a particular swimming organism is

moving in. Following Hill & Hader (1997), we set up a difference equation for the random

walk on a circle and then take appropriate limits to derive a Fokker–Planck equation for

the p.d.f. of θ. This is effectively a position jump process (Othmer et al., 1988) on the

unit circle.

Suppose that there is a small fixed time step τ between changes in direction and that at

time t the direction equals θ(t). At time t + τ , the direction either changes by a small

angle ±δ or is unchanged with probabilities given by

P (δ) = p(θ), (1.79)

P (−δ) = q(θ), (1.80)

P (0) = 1 − p(θ) − q(θ), (1.81)

where p(θ) and q(θ) are small, continuous functions of θ. Now noting the distinction

between the random variable Θ and its value θ, we define the p.d.f. f(θ, t) of Θ by

f(θ, t)δ = P [θ ≤ Θ(t) < θ + δ]. (1.82)

By considering the previous time step, we see that

f(θ, t) = f(θ − δ, t− τ)p(θ − δ) + f(θ, t− τ)[1 − p(θ) − q(θ)]

+f(θ + δ, t− τ)q(θ + δ). (1.83)

Expanding (1.83) as a Taylor series and rearranging gives

τ∂f

∂t= −δ(p − q)

∂f

∂θ− δ

(

∂p

∂θ− ∂q

∂θ

)

f

+δ2

2(p+ q)

∂2f

∂θ2+ δ2

(

∂p

∂θ+∂q

∂θ

)

∂f

∂θ

+δ2

2

(

∂2p

∂θ2+∂2q

∂θ2

)

f +O(τδ) +O(τ2) +O(δ3), (1.84)

where f = f(θ, t), p = p(θ) and q = q(θ).

The mean and variance of Θ(t+ τ)−Θ(t) = ∆(θ, τ), the random variable for the turning

angle are given by

E[∆(θ, τ)] = µδ(θ, τ) = δ(p − q), (1.85)

Var[∆(θ, τ)] = σ2δ (θ, τ) = δ2(p+ q) − (E[∆(θ, τ)])2. (1.86)

We now insist that the following relations hold asymptotically as τ → 0

µδ(θ, τ) = µ0(θ)τ (1.87)

σ2δ (θ, τ) = σ2

0(θ)τ , (1.88)


so that the mean and variance have a linear dependence on τ and both tend to zero as

τ → 0. We now have

µ0(θ) =δ

τ(p − q), (1.89)

σ20(θ) =

δ2

τ(p+ q) − µ2

0τ2. (1.90)

These conditions are satisfied if

p(θ) = [σ20(θ) + µ0(θ)δ]/2A, (1.91)

q(θ) = [σ20(θ) − µ0(θ)δ]/2A, (1.92)

and δ2/τ = A as τ → 0, where A is a positive constant. Taking this limit as τ → 0 in the

Taylor series expansion in (1.84) yields the Fokker–Planck equation for f(θ, t)

∂

∂tf(θ, t) = − ∂

∂θ[µ0(θ)f(θ, t)] +

1

2

∂2

∂θ2[σ2

0(θ)f(θ, t)]. (1.93)

In experiments Hill & Hader verified that the distribution of the turning angle, ∆, tended

to the correct limits, (1.87) and (1.88), as the time step is decreased.

The parameter µ0(θ) is referred to as the orientation (or drift) coefficient for the Fokker–

Planck equation, while

D =σ2

0

2(1.94)

is the effective rotational diffusivity, see Berg (1983).

The Fokker–Planck equation (1.93) is solved subject to the conditions that f(θ, t) and the

probability flux

j(θ, t) ≡ µ0(θ)f(θ, t)− 1

2

∂

∂θ[σ2

0(θf(θ, t)] (1.95)

are both periodic, and specifically that

f(−π, t) = f(π, t) and j(−π, t) = j(π, t) (1.96)

for all t > 0. In addition, for f(θ, t) to be a valid p.d.f. it must be non-negative

f(θ) ≥ 0 ∀ − π ≤ θ < π, (1.97)

and normalized∫ π

−πf(θ) dθ = 1, (1.98)

and for the time-dependent solution of (1.93), must satisfy a suitable initial condition.

All the solutions of (1.93) ultimately decay to a steady state independent of the initial

conditions, which is the state observed in experiments completed by Hill & Hader.

1.5.2 Reorientation models and solutions to the Fokker–Planck equation

To solve (1.93) Hill & Hader considered two possible models for the orientation coefficient

µ0(θ) and assumed that σ0(θ) is a constant. The first reorientation response model is

sinusoidal corresponding to the gravitational torque present in gyrotaxis, while the second

is linear corresponding to phototaxis, see Figure 1.8.


Figure 1.8: Plot comparing µ0(θ) for sinusoidal (—) and linear reorientation (- -), for

−π ≤ θ < π and B−1 = 0.1.

1.5.2.1 Sinusoidal reorientation

Suppose that the orientation coefficient takes the form

µ0(θ) = −B−1 sin(θ − θ0) (−π ≤ θ, θ0 < π), (1.99)

where θ0 is a constant corresponding to the preferred direction of movement (direction of

the bias), and B−1 is the amplitude of the orientation coefficient and is assumed to be

a positive constant. When σ0 is a constant, then the normalized solution of the steady

state Fokker–Planck equation (1.93) plus boundary conditions can be shown to be the von

Mises distribution (see Section 1.2.3.2).

f(θ) = M(θ; θ0, 2B−1/σ2

0) =1

2πI0(2B−1/σ20)

exp

(

2B−1

σ20

cos(θ − θ0)

)

, (1.100)

where I0(2B−1/σ2

0) is the modified Bessel function of the first kind and zero order. This

corresponds to the sinusoidal response observed by Kessler (1986) in gyrotactic algae.

1.5.2.2 Linear reorientation

A second model for the orientation coefficient is

µ0(θ) =

−B−1θ, −π < θ < π,

0, θ = ±π,(1.101)

where B−1 is the amplitude of the orientation coefficient and is a positive constant, and

θ0 = 0 is assumed to be the preferred direction. If σ20 is a positive constant, then the

normalized solution of the steady state Fokker–Planck equation can be shown to be

f(θ) = B(λ)e−λθ2

, λ = B−1/σ20 , (1.102)


where B(λ) is the normalization function defined by

B(λ) =

(∫ π

−πe−λθ

2

dθ

)

−1

=√λ(√

πerf(π√λ))

−1. (1.103)

1.5.3 Experimental results

Hill and Hader carried out a number of experiments to validate the conclusions from the

above analysis. In particular they tested the validity of the relations given in (1.87) and

(1.88), and the long time solutions given by (1.100) and (1.102). They also collected

data on the speed of swimming algae and found values for the mean speed v and standard

deviation of the speed vsd. The sampling time steps were not fixed due to the experimental

set-up, but the smallest average time step between turns τ that they could measure was

τ = 0.08 s. (1.104)

Although the experiments contained a lot of ‘noise’, they found that (1.87) fitted the data

well. Values of B−1, the amplitude of the orientation coefficient, were extrapolated back

from plotted graphs of the data. As there was some doubt over how to extrapolate the

data, different values for B−1 were suggested. One estimate was made over all the sampling

time steps observed, while another was made using only those sampling time steps that

were less than 0.4 s. It was found harder to verify (1.88) due to a lot of experimental

noise. Using the expected long-time angular distributions, (1.100) and (1.102), and the

estimates of the parameter B−1 they calculated a value of σ20 . Hence, for each estimate of

B−1 a different value for σ20 was calculated. Using this estimate the rotational diffusivity,

D = σ20/2, can also be calculated.

The parameter estimates from Hill & Hader’s experimental data for each data set are

given in Table 1.1.

Data Set v vsd B−1 σδ(τ)

C1 55 µms−1 31 µms−1 0.37 (0.80) s−1 1.3 (2.0)√τ rad

C3 60 µms−1 41 µms−1 0.44 (0.62) s−1 1.8 (2.1)√τ rad

C4 59 µms−1 47 µms−1 0.19 (0.61) s−1 0.9 (1.7)√τ rad

Table 1.1: Swimming speed and reorientation parameters estimated by Hill & Hader for

the data sets C1, C3 and C4.

Data set C1 corresponds to C. nivalis moving in a vertical plane and exhibiting gyrotaxis,

while C2 and C3 correspond to C. nivalis moving in a horizontal plane and exhibiting

phototaxis due to a light source of 80 klux and 200 klux repsectively. In each data set,

the figures inside the brackets show possible different values for the parameters if the

extrapolation from the data is completed for sampling time steps less than 0.4 s only. We

will henceforth refer to the results for the data sets using all sampling time steps as C1:a,


C3:a and C4:a; while the alternative results using only sampling time steps less than 0.4

s (the estimates given in brackets) will be referred to as C1:b, C3:b and C4:b.

1.6 Overview of subsequent chapters

In Chapter 2 we show how to derive the two-dimensional diffusion equation using a dif-

ference equation. We discuss the limitations of the diffusion model when attempting to

incorporate the sinusoidal and linear reorientation models of Hill & Hader (1997). We also

show how it is not possible to extend the telegraph process to two dimensions.

In Chapter 3 we explain the two-dimensional velocity jump process of Othmer et al. (1988),

and show how it is possible to derive and solve differential equations for the higher order

moments and statistics of interest, even though it is not possible to find an equation for

the probability density of the underlying spatial distribution.

In Chapters 4 and 5 we use the method of calculating moments from Othmer et al. to

derive asymptotic solutions for the higher order moments and statistics of interest for

velocity jump processes that include the reorientation models of Hill & Hader. Several

assumptions have to be made to close and solve the derived system of differential equations.

In Chapter 6 we introduce an algorithm to simulate the movement of a population of

walkers moving with the velocity jump models set up in Chapters 4 and 5. We compare

results of simulations to the expected results given by the asymptotic equations derived

in Chapters 4 and 5.

We revisit the method of analysing the angular statistics used by Hill & Hader to estimate

the reorientation parameters from experimental data in Chapter 7. Simulations are used

to test the validity of their method and to discuss the limitations when used on data from

a velocity jump process.

In Chapter 8 we extend the simulation algorithm introduced in Chapter 6 to investigate

velocity jump processes where the asymptotic solution equations from Chapters 4 and 5

are no longer valid — including simulations over a full range of reorientation parameter

values and non-homogenous environments.

We discuss in Chapter 9 the various correlated random walk models in the literature and

show that it is possible to calculate the mean dispersal distance from a correction of the

root of the mean squared displacement. We suggest a simpler model for the correction

factor in an unbiased but correlated random walk. Simulations are then used to look at the

correction factor in a biased random walk and the its relevance to the spatial distribution

of the population is discussed.

The simulations and results from previous chapters are used in Chapter 10 to investigate

a particular biological problem — the recruitment of reef fish larvae. We explain how

variability is important if there is a low survival probability and investigate the optimal

swimming behaviour in various reef environments.

Chapter 2

Simple two-dimensional random

walk models

2.1 Introduction

In the previous chapter (Section 1.1.2) we discussed two methods of deriving the diffusion

equation for an isotropic random walk in one dimension. One can derive the solution

directly as the limiting solution of a binomial process (1.5) or by using a difference equation

to derive the diffusion equation (1.19). Similar methods can be used to derive equations

and solutions for a random walk process in two dimensions. The case of an isotropic

random walk in two dimensions has been discussed previously (Section 1.1.2.2), but it is

also possible to use a difference equation derivation and introduce bias into the solution

by having non-equal probabilities of moving in different directions. In this chapter we

introduce several discrete (lattice) models, and then show that using the same methods

it is straightforward to develop this to a continuous model. All these methods can be

described as ‘position jump processes’, see Othmer et al. (1988).

The solutions to the two-dimensional diffusion process have the same major problem as

the one-dimensional model — the solution allows for a non-zero probability of being an

arbitrarily large distance away from the start point after an arbitrarily small time. In this

sense the underlying speed of propagation is infinite. The solution of the diffusion equation

can be considered as an asymptotic approximation, valid for large time, of equations that

more accurately describe the correlations in movement that are present at shorter time

scales.

In Section 1.1.4 we showed how by introducing correlation into the one-dimensional model

a different equation and solution can be derived. This ‘velocity jump process’ (Othmer et

al., 1988) results in the ‘telegraph equation’, which does not allow for infinite propagation

as the underlying speed is fixed. In this chapter we attempt a similar derivation in two

dimensions and show that this does not lead to a closed form equation for the population

31

CHAPTER 2: Simple two-dimensional random walk models 32

distribution for our simple random walk model.

2.2 Two-dimensional uncorrelated random walks

We are looking for an equation that describes how a population of individuals moves and

spreads out in two-dimensional space. From this we want to be able to calculate spatial

statistics for all time, which can then be used in continuum models. In all the following

models we assume that there is no interaction between individuals and that more than

one individual can occupy a given position — this is reasonable if we assume a low density

of individuals per unit area. We also assume that the population is moving around in a

static medium — there are no flow effects.

2.3 Lattice model

2.3.1 Turning probabilities independent of position

We restrict the population to moving on a two-dimensional infinite lattice (see Figure

2.1). At each time step τ an individual can move a distance δ either up, down, left or

right with probabilities independent of position, given by u, d, l and r respectively, where

u+ d+ l + r ≤ 1, or remain at the same position with probability 1 − u− l − d− r.

x

y

Figure 2.1: Example of a two-dimensional lattice random walk.

If we take a forward time step from time t− τ to time t, then the number density f(x, y, t)

of individuals at position (x, y) is given by

f(x, y, t) = f(x, y, t− τ)[1 − l − r − u− d]

+f(x− δ, y, t − τ)r + f(x+ δ, y, t − τ)l

+f(x, y − δ, t− τ)u+ f(x, y + δ, t− τ)d. (2.1)

We can expand each term as a Taylor series up to O(δ2) and O(τ), giving

f(x, y, t) = (f − τ∂f

∂t)[1 − l − r − u− d]


+(f − τ∂f

∂t− δ

∂f

∂x+

1

2δ2∂2f

∂x2)r

+(f − τ∂f

∂t+ δ

∂f

∂x+

1

2δ2∂2f

∂x2)l

+(f − τ∂f

∂t− δ

∂f

∂y+

1

2δ2∂2f

∂y2)u

+(f − τ∂f

∂t+ δ

∂f

∂y+

1

2δ2∂2f

∂y2)d+O(δ3) +O(τ2),

which when multiplied out gives

τ∂f

∂t= δ

(

(l − r)∂f

∂x+ (d− u)

∂f

∂y

)

+δ2

2

(

(r + l)∂2f

∂x2+ (u+ d)

∂2f

∂y2

)

+O(δ3) +O(τ2). (2.2)

We define the following parameters:

b1 = limδ,τ,ǫ1→0

ǫ1δ

τ, (2.3)


ǫ2δ

τ, (2.4)

a11 = limδ,τ→0

k1δ2

2τ, (2.5)

a22 = limδ,τ→0

k2δ2

2τ, (2.6)

where ǫ1 = r − l, ǫ2 = u− d, k1 = r+ l, and k2 = u+ d. Now take equation (2.2), divide

through by τ and take the limit as δ, τ, ǫ1, ǫ2 → 0 such that ǫ1δ/τ , ǫ2δ/τ , k1δ2/2τ , and

k1δ2/2τ all tend to a constant, giving

∂f

∂t= −b1

∂f

∂x− b2

∂f

∂y+ a11

∂2f

∂x2+ a22

∂2f

∂y2, (2.7)

which can be written as∂f

∂t= −∇.bf + ∇.(∇Df), (2.8)

where

b =

b1

b2

and D =

a11 0

0 a22

.

If b1 = b2 = 0 then this is just the two-dimensional diffusion equation (1.12) which has a

solution given in (1.13). When either of b1 or b2 are non-zero then the solution includes

drift. If we let a11 = a22 = D, so that diffusion is equal in the x and y directions, then

(2.8) has solution

f(x, y, t) =1

4πDte−((x−b1t)2+(y−b2t)2)/4Dt. (2.9)

The statistics of interest can be calculated from (2.9) or directly from (2.8), and assuming

a11 = a22 = D, are given by

< (x, y) > = (b1t, b2t), (2.10)

< r2 > = < x2 + y2 >= b21t2 + b22t

2 + 4Dt, (2.11)


and

σ2 =< r2 > − < (x, y) >2= 4Dt. (2.12)

These solutions are comparable to the one-dimensional solutions in (1.23) and (1.25).

2.3.2 Turning probabilities dependent on position

As before, we restrict the population to moving on a two-dimensional infinite lattice. At

each time step τ an individual can move a distance δ either up, down, left or right with

probabilities dependent on position, given by u(x, y), d(x, y), l(x, y) and r(x, y) respec-

tively, where u(x, y) + d(x, y) + l(x, y) + r(x, y) ≤ 1, or remain at the same position with

probability 1 − u(x, y) − l(x, y) − d(x, y) − r(x, y).

If we take a forward time step, then the number density of individuals at position (x, y)

is given by

f(x, y, t) = f(x, y, t− τ)[1 − l(x, y) − r(x, y) − u(x, y) − d(x, y)]

+f(x− δ, y, t − τ)r(x− δ, y) + f(x+ δ, y, t − τ)l(x+ δ, y)

+f(x, y − δ, t− τ)u(x, y − δ) + f(x, y + δ, t− τ)d(x, y + δ).

We can expand each term as a Taylor series up to O(δ2) and O(τ), giving

f(x, y, t) = (f − τ∂f

∂t)[1 − l − r − u− d]

+(f − τ∂f

∂t− δ

∂f

∂x+δ2

2

∂2f

∂x2)(r − δ

∂r

∂x+δ2

2

∂2r

∂x2)

+(f − τ∂f

∂t+ δ

∂f

∂x+δ2

2

∂2f

∂x2)(l + δ

∂l

∂x+δ2

2

∂2l

∂x2)

+(f − τ∂f

∂t− δ

∂f

∂y+δ2

2

∂2f

∂y2)(u− δ

∂u

∂y+δ2

2

∂2u

∂y2)

+(f − τ∂f

∂t+ δ

∂f

∂y+δ2

2

∂2f

∂y2)(d + δ

∂d

∂y+δ2

2

∂2d

∂y2)

+O(δ3) +O(τ2) +O(δτ),

which when multiplied out gives

τ∂f

∂t= δ(

∂l

∂x− ∂r

∂x)f + δ(

∂d

∂y− ∂u

∂y)f +

δ2

2(∂2r

∂x2+∂2l

∂x2)f

+δ2

2(∂2d

∂y2+∂2u

∂y2)f + δ(l − r)

∂f

∂x+ δ(d− u)

∂f

∂y

+δ2(∂l

∂x+∂r

∂x)∂f

∂x+ δ2(

∂d

∂y+∂u

∂y)∂f

∂y+δ2

2(l + r)

∂2f

∂x2

+δ2

2(d+ u)

∂2f

∂y2+O(δ3) +O(τ2) +O(δτ). (2.13)



ǫ1δ

τ, (2.14)



ǫ2δ

τ, (2.15)

a11 = limδ,τ→0

k1δ2

2τ, (2.16)

a22 = limδ,τ→0

k2δ2

2τ, (2.17)

where ǫ1 = r − l, ǫ2 = u− d, k1 = r + l and k2 = u+ d. Similarly, we have

∂b1∂x

= limδ,τ,ǫ1→0

∂ǫ1∂x

δ

τ,

∂2b1∂x2

= limδ,τ,ǫ1→0

∂2ǫ1∂x2

δ

τetc.

We now take equation (2.13), divide through by τ and take the limit as δ, τ, ǫ1, ǫ2 → 0

such that ǫ1δ/τ , ǫ2δ/τ , k1δ2/2τ , and k1δ

2/2τ all tend to a constant, giving

∂f

∂t= −∂(fb1)

∂x− ∂(fb2)

∂y+∂2(fa11)

∂x2+∂2(fa22)

∂y2, (2.18)


∂t= −∇.(bf) + ∇.(∇(Df)), (2.19)

where

b =

b1(x, y)

b2(x, y)

and D =

a11(x, y) 0

0 a22(x, y)

.

This is the same result as the previous case (2.8), except that b and D are now dependent

on position. This is a special case of the Fokker–Planck diffusion equation.

In principle equation (2.19) can be solved if the dependence of the parameters in D and

b on (x, y) are known but this can be extremely complex so we do not try and solve for

general solutions here.

2.4 Multi-directional discrete direction model and continu-

ous direction model

2.4.1 Multi-directional discrete direction model

Suppose we now allow the population to move around freely in any direction and not

restrict ourselves to a lattice. Suppose at each time step τ an individual can move a

distance δ in one of N possible directions given by θi, where θi ∈ θ1, ..., θN : 0 ≤ θi < 2π,with probability dependent on position given by pi(x, y), where

N∑

i=1

pi(x, y) ≤ 1.


θ

θ θ

i

j k

t0

t1 t2

θ = 0

Figure 2.2: Example of a multi-directional random walk.

If we take a forward time step then the number density, f(x, y), of individuals at position

(x, y) is given by

f(x, y, t) =N∑

i=1

(

f(x− δ sin θi, y − δ cos θi, t− τ)pi(x− δ sin θi, y − δ cos θi))

+f(x, y, t− τ)(

1 −N∑

i=1

pi(x, y))

. (2.20)

We can expand each term in the above as a Taylor series up to O(δ2) and O(τ), which

gives

f = (f − τ∂f

∂t)(

1 −N∑

i=1

pi)

+N∑

i=1

(

(f − τ∂f

∂t− δ sin θi

∂f

∂x− δ cos θi

∂f

∂y

+δ2 sin θi cos θi∂2f

∂x∂y+δ2 sin2 θi

2

∂2f

∂x2+δ2 cos2 θi

2

∂2f

∂y2)

(pi − δ sin θi∂pi∂x

− δ cos θi∂pi∂y

+ δ2 cos θi sin θi∂2pi∂x∂y

+δ2 sin 2θi

2

∂2pi∂x2

+δ2 cos2 θi

2

∂2pi∂y2

))

+O(δ3) +O(τ2) +O(δτ). (2.21)

We can expand this out to give

τ∂f

∂t= δ

(

−N∑

i=1

(sin θi∂pi∂x

) −N∑

i=1

(cos θi∂pi∂y

))

f

+δ2

2

(

N∑

i=1

(2 cos θi sin θi∂2pi∂x∂y

) +N∑

i=1

(sin2 θi∂2pi∂x2

) +N∑

i=1

(cos2 θi∂2pi∂y2

))

f

+ δ(

−N∑

i=1

(sin θipi))∂f

∂x+ δ2

(

N∑

i=1

(sin2 θi∂pi∂x

) +N∑

i=1

(sin θi cos θi∂pi∂y

))∂f

∂x

+ δ(

−N∑

i=1

(cos θipi))∂f

∂y+ δ2

(

N∑

i=1

(cos2 θi∂pi∂y

) +N∑

i=1

(sin θi cos θi∂pi∂x

))∂f

∂y

+ δ2(

N∑

i=1

(sin θi cos θipi)) ∂2f

∂x∂y+δ2

2

(

N∑

i=1

(sin2 θipi))∂2f

∂x2


+δ2

2

(

N∑

i=1

(cos2 θipi))∂2f

∂y2+O(δ3) +O(τ2) +O(δτ). (2.22)



(

ǫ1δ

τ

)

, (2.23)


(

ǫ2δ

τ

)

, (2.24)

a11 = limδ,τ→0

(

N∑

i=1

(pi sin2 θi)

δ2

2τ

)

, (2.25)

a22 = limδ,τ→0

(

N∑

i=1

(pi cos2 θi)

δ2

2τ

)

, (2.26)

a12 = limδ,τ→0

(

N∑

i=1

(pi sin θi cos θi)δ2

2τ

)

, (2.27)

where ǫ1 =∑Ni=1(pi sin θi), ǫ2 =

∑Ni=1(pi cos θi). In a similar way we have

∂b1∂x

= limδ,τ,ǫ1→0

(

∂ǫ1∂x

δ

τ

)

,

∂2b1∂x2

= limδ,τ,ǫ1→0

(

∂2ǫ1∂x2

δ

τ

)

etc.


such that ǫ1δ/τ , ǫ2δ/τ , and δ2/τ all tend to a constant.

After rearranging, we get

∂f

∂t= −∂(fb1)

∂x− ∂(fb2)

∂y+ 2

∂2(fa12)

∂x∂y+∂2(fa11)

∂x2+∂2(fa22)

∂y2(2.28)


∂t= −∇.(bf) + ∇.(∇(Df)) (2.29)

where

b =

b1

b2

and D =

a11 a12

a12 a22

.

This is the same equation as (2.19), except that the diffusion matrix D now has non-zero

off-diagonal terms. These off-diagonal terms arise from the covariance we introduced by

allowing movement in both the x and y directions during one step.

2.4.2 Continuous direction model

We can extend the above result to allow movement in any direction rather than just one

of N directions. Suppose at each time step τ an individual can either move a distance δ

in any direction θ, where −π ≤ θ < π, with probability dependent on position given by

p(θ;x, y), where∫ π

−πp(θ;x, y) dθ ≤ 1,


or remain in the same position with probability

1 −∫ π

−πp(θ;x, y) dθ.

If we take a forward time step then the number density, f(x, y), of individuals at position

(x, y) is given by

f(x, y, t) =

∫ π

−πf(x− δ sin θ, y − δ cos θ, t− τ)p(θ;x− δ sin θ, y − δ cos θ) dθ

+f(x, y, t− τ)(

1 −∫ π

−πp(θ;x, y)dθ

)

. (2.30)

We can expand each term in the above as a Taylor series up to O(δ2) and O(τ), which

gives

f =

∫ π

−π

(

(f − τ∂f

∂t− δ sin θ

∂f

∂x− δ cos θ

∂f

∂y+ δ2 sin θ cos θ

∂2f

∂x∂y

+δ2 sin2 θ

2

∂2f

∂x2+δ2 cos2 θ

2

∂2f

∂y2)(p − δ sin θ

∂p

∂x− δ cos θ

∂p

∂y

+δ2 cos θ sin θ∂2p

∂x∂y+δ2 sin 2θ

2

∂2p

∂x2+δ2 cos2 θ

2

∂2p

∂y2))

dθ

+(f − τ∂f

∂t)(

1 −∫ π

−πp dθ

)

+O(δ3) +O(τ2) +O(δτ). (2.31)

Multiplying out gives

τ∂f

∂t= δ

(

−∫ π

−π(sin θ

∂p

∂x)dθ −

∫ π

−π(cos θ

∂p

∂y)dθ)

f

+δ2

2

(

∫ π

−π(2 cos θ sin θ

∂2p

∂x∂y)dθ +

∫ π

−π(sin2 θ

∂2p

∂x2)dθ +

∫ π

−π(cos2 θ

∂2p

∂y2)dθ)

f

+δ(

−∫ π

−π(sin θp)dθ

)∂f

∂x+ δ2

(

∫ π

−π(sin2 θ

∂p

∂x)dθ +

∫ π

−π(sin θ cos θ

∂p

∂y)dθ)∂f

∂x

+δ(

−∫ π

−π(cos θp)dθ

)∂f

∂y+ δ2

(

∫ π

−π(cos2 θ

∂p

∂y)dθ +

∫ π

−π(sin θ cos θ

∂p

∂x)dθ)∂f

∂y

+δ2(

∫ π

−π(sin θ cos θp)dθ

) ∂2f

∂x∂y+δ2

2

(

∫ π

−π(sin2 θp)dθ

)∂2f

∂x2

+δ2

2

(

∫ π

−π(cos2 θp)dθ

)∂2f

∂y2+O(δ3) +O(τ2) +O(δτ). (2.32)



(

ǫ1δ

τ

)

, (2.33)


(

ǫ2δ

τ

)

, (2.34)

a11 = limδ,τ→0

(

∫ π

−π(p sin2 θ)dθ

δ2

2τ

)

, (2.35)

a22 = limδ,τ→0

(

∫ π

−π(p cos2 θ)dθ

δ2

2τ

)

, (2.36)

a12 = limδ,τ→0

(

∫ π

−π(p sin θ cos θ)dθ

δ2

2τ

)

, (2.37)


where ǫ1 =∫ π−π(p sin θ)dθ and ǫ2 =

∫ π−π(p cos θ)dθ. In a similar way we have

∂b1∂x

= limδ,τ,ǫ1→0

(

∂ǫ1∂x

δ

τ

)

,

∂2b1∂x2

= limδ,τ,ǫ1→0

(

∂2ǫ1∂x2

δ

τ

)

etc.


such that ǫ1δ/τ , ǫ2δ/τ , and δ2/τ all tend to a constant. As with the previous models, for

the limit to be valid we must assume that the ǫ1 and ǫ2 terms and their derivatives tend to

zero in the same limit. Thus we need∫ π−π(p sin θ)dθ ∼ O(δ) and

∫ π−π(p cos θ)dθ ∼ O(δ),

which is only true if the distribution p(θ;x, y) is close to uniform when considering the

probability of moving with a particular angle θ. Thus this model will only be valid if there

is a small amount of bias and the angular distribution is close to uniform.

After rearranging, we get

∂f

∂t= −∂(fb1)

∂x− ∂(fb2)

∂y+ 2

∂2(fa12)

∂x∂y+∂2(fa11)

∂x2+∂2(fa22)

∂y2, (2.38)


∂t= −∇.(bf) + ∇.(∇Df), (2.39)

where

b =

b1

b2

and D =

a11 a12

a12 a22

.

This is exactly the same equation that we derived for the multi-directional case (2.29). The

parameters given in (2.33)—(2.37) can be calculated if we know the form of the probability

distribution for the direction of motion, p(θ;x, y).

2.5 Solution of the Fokker–Planck diffusion equation

One can solve (2.39) after choosing an appropriate probability distribution for p(θ;x, y),

the probability of moving in a certain direction θ given the current position (x, y). Note

that p(θ;x, y) will thus be independent of the previous direction of movement — there is

no directional correlation as this is a position jump process.

2.5.1 Solution for isotropic movement

Suppose that p(θ;x, y) = 1/2π, so that there is no bias and the probability of move-

ment is independent of the (x, y) position. We would expect (2.39) to reduce to the

two-dimensional isotropic diffusion equation (1.12) with corresponding solution (1.13).

Recalling that θ = 0 in the y direction, then from (2.33)—(2.37) we have


(

ǫ1δ

τ

)

= 0, (2.40)



(

ǫ2δ

τ

)

= 0, (2.41)

a11 = limδ,τ→0

(

∫ π

−π

sin2 θ

2πdθδ2

2τ

)

=δ2

4τ, (2.42)

a22 = limδ,τ→0

(

∫ π

−π

cos2 θ

2πdθδ2

2τ

)

=δ2

4τ, (2.43)

a12 = limδ,τ→0

(

∫ π

−π

sin θ cos θ

2πdθδ2

2τ

)

= 0 (2.44)

which are the same parameters as in (2.8) with r = l = u = d = 1/4, i.e. the isotropic

two-dimensional random walk. The corresponding diffusion equation is given in (1.12),

with solution in (1.13).

2.5.2 Solution for biased movement

We have discussed previously how the solution of the diffusion equation is only valid as

a long-time approximation. If we assume that this is the case, we can use the steady

state solution independent of position (so p(θ;x, y) = p(θ)) that is given by Hill & Hader’s

solutions to the Fokker–Planck equation for a circular random walk (1.100) and (1.102).

Assuming the preferred direction is given by θ0 = 0 then for sinusoidal reorientation we

have

p(θ) =1

2πI0(γ)eγ cos θ, (2.45)

where γ = 2B−1/σ20 , and for linear reorientation we have

p(θ) = B(λ)e−λθ2

, (2.46)

where λ = B−1/σ20 .

We can now calculate the parameters given in (2.33)—(2.37).

2.5.2.1 Sinusoidal model

In the following parameter equations we use p(θ) = 12πI0(γ)

eγ cos θ, where γ = 2B−1/σ20 and

I0(γ) is the modified Bessel function of the first kind and zero order. We use the standard

integral of the von Mises distribution from Mardia & Jupp (1999)

1

2πI0(κ)

∫ π

−πsinnθeκ cos θdθ = 0, (2.47)

1

2πI0(κ)

∫ π

−πcosnθeκ cos θdθ =

In(κ)

I0(κ). (2.48)

From (2.33) we have


(

ǫ1δ

τ

)

= 0, (2.49)

since ǫ1 =∫ π−π(p sin θ)dθ = 0 from (2.47).


From (2.34) we have


(

ǫ2δ

τ

)

= limδ,τ,γ→0

(

I1(γ)

I0(γ)

δ

τ

)

, (2.50)

since ǫ2 =∫ π−π(p cos θ)dθ = I1(γ)/I0(γ), using (2.48).

From (2.35) we have

a11 = limδ,τ→0

(

∫ π

−π(p sin2 θ)dθ

δ2

2τ

)

(2.51)

= limδ,τ→0

(

(

1

2− I2(γ)

2I0(γ)

)

δ2

2τ

)

, (2.52)

since∫ π−π(p sin2 θ)dθ = 1/2(1 − I2(γ)/I0(γ)), using (2.48).

From (2.36) we have

a22 = limδ,τ→0

(

∫ π

−π(p cos2 θ)dθ

δ2

2τ

)

(2.53)

= limδ,τ→0

(

(

1

2+

I2(γ)

2I0(γ)

)

δ2

τ

)

, (2.54)

since∫ π−π(p cos2 θ)dθ = 1/2(1 + I2(γ)/I0(γ)), using (2.48).

From (2.37) we have

a12 = limδ,τ→0

(

∫ π


δ2

2τ

)

= 0, (2.55)

since∫ π−π(p sin θ cos θ)dθ = 0, using (2.47).

For this analysis to be valid we need ǫ2 ∼ O(δ), and thus I1(γ)/I0(γ) ∼ O(δ). This is

true only if γ is small, which in turn means that the reorientation parameter B−1 must

be small or the variance per unit time, σ20 , must be large. In either case this implies that

there cannot be a large amount of bias in the system and the angular distribution p(θ)

must be close to the uniform distribution.

2.5.2.2 Linear model

In the following parameter equations we use p(θ) = B(λ)e−λθ2

, where λ = d0/σ20 and B(λ)

is the normalization function defined in (1.103).

From (2.33) we have


(

ǫ1δ

τ

)

= 0, (2.56)

since ǫ1 =∫ π−π(p sin θ)dθ = 0 as p(θ) is symmetric about θ = 0.

From (2.34) we have


(

ǫ2δ

τ

)

, (2.57)


where ǫ2 = B(λ)∫ π−π cos θe−λθ

2

dθ.

From (2.35) we have

a11 = limδ,τ→0

(

∫ π

−π(p sin2 θ)dθ

δ2

2τ

)

(2.58)

= limδ,τ→0

(

(

1 −∫ π

−πcos 2θe−λθ

2

dθ

)

δ2

4τ

)

. (2.59)

From (2.36) we have

a22 = limδ,τ→0

(

∫ π

−π(p cos2 θ)dθ

δ2

2τ

)

(2.60)

= limδ,τ→0

(

(

1 +

∫ π

−πcos 2θe−λθ

2

dθ

)

δ2

4τ

)

. (2.61)

From (2.37) we have

a12 = limδ,τ→0

(

∫ π


δ2

2τ

)

= 0, (2.62)

since p(θ) is symmetric about θ = 0.

As in the sinusoidal model, this analysis will only be valid if ǫ2 ∼ O(δ), which consequently

means that p(θ) must be close to the uniform distribution.

2.5.2.3 Solution of the Fokker–Planck equation

For both the sinusoidal and linear models b1 = a12 = 0, and (2.39) simplifies to

∂f

∂t= −b2

∂f

∂y+ a11

∂2f

∂x2+ a22

∂2f

∂y2, (2.63)

where one can substitute for b2 from either (2.50) or (2.57) depending on which model is

being used, and similarly for the parameters a11 and a22.

By inspection, we see that the following is a solution of (2.63)

f(x, y, t) =1

4π√a11

√a22t

exp

(

− x2

4a11t− (y − b2t)

2

4a22t

)

, (2.64)

where the parameters b2, a11 and a22 are dependent on the reorientation model chosen.

Solution plots for (2.64) are shown in Figure 2.3 (note that the parameter values are chosen

to illustrate the behaviour of the solution and are not necessarily ‘realistic’).

The moments < (x, y) >, < r2 >=< x2 + y2 > and σ2 can be calculated directly from

the differential equation (2.63) or from the solution (2.64), assuming that f(x, y, t) and its

derivatives tend to zero as |x|, |y| → ∞. Using either method, we have

< (x, y)(t) >= (0, b2t), (2.65)

< r2(t) >=< x2(t) + y2(t) >= b22t2 + 2(a11 + a22)t, (2.66)


–100

–50

0

50

100

x

–50

0

50

100

150

y

0

0.0005

0.001

0.0015

0.002

0.0025

f(x,y,t)

(a) b2 = 1, a11 = 1, a22 = 10.

–100

–50

0

50

100

x

–50

0

50

100

150

y

0

0.0005

0.001

0.0015

0.002

0.0025

f(x,y,t)

(b) b2 = 10, a11 = 10, a22 = 1.

Figure 2.3: Plots showing f(x, y, t) for various parameter values at t = 10.

and

σ2(t) = 2(a11 + a22)t. (2.67)

In the simple random walk models presented previously, the walkers can have a bias in

their movement which is either fixed or dependent on the spatial position, but not de-

pendent on the previous direction of movement, i.e. there is no correlation. As suggested

in the introduction chapter, the diffusion equation and corresponding solutions can be

considered as the long-time limiting solution of the underlying process that includes cor-

relation effects. The main problem with all the diffusion-type equations we have derived

is that the solutions allow for effectively infinite propagation speeds. This comes about

because of the way we take certain limits, see for example Okubo (1980). We assume that

limδ,τ→0

δ2

τ= constant.

One can easily see that as a consequence of this we have

limδ,τ→0

δ

τ→ ∞.

Over large time scales the solution reaches a steady state and this effectively infinite

propagation can be ignored but over small timescales it is not valid. If we introduce

correlation into the model we can avoid this problem of infinite propagation and make the

model more biologically realistic.

Henderson et al. (1984) modelled the growth of the roots of the Sitka spruce as a two-

dimensional correlated but unbiased random walk on a square lattice. In their model the

probability of moving forward f, back b, left l and right r at each step was fixed (where

the directions are relative to the previous direction of movement). They derived equations


for the mean position (which was found to be zero) and the variance in position (spread

from the origin). The variance was found to be dependent on the differences between the

probabilities of moving forward and back, (f − b), and left or right, (r − l). The variance

was found to be minimum when (f − b) was large and negative, decreasing as (r − l)

increased, and a maximum when (f − b) was large and positive. As might be expected,

the larger spread in spatial position was found to be for the motion that was most like a

straight line (corresponding to f ≈ 1).

By taking appropriate limits such that δt = δx2/σ2 for some fixed constant σ, they showed

that the limiting equation for the motion was the two-dimensional diffusion equation

(1.12). Thus even though they included correlation in the model, the limits chosen to

close the system results in a diffusion solution with the inherent ‘infinite propagation’

problem.

By modelling the random walk as the limit of a telegraph process that includes correlation

and a fixed velocity we showed that, in one dimension we can avoid the problem of infinite

propagation (see Section 1.1.4). We now attempt a similar derivation in two dimensions.

2.6 The telegraph equation in higher dimensions

The solution to the two-dimensional diffusion process with or without drift has been shown

to be only valid as a long-time approximation to the true underlying solution that includes

correlation effects. One can introduce correlation by completing a similar derivation as in

Section 1.1.4 but working with an infinite two-dimensional lattice rather than a line.

We now have four possible directions of movement — right, left, up and down. To start

with, we assume that the probability of turning is independent of the direction of movement

(so we have no bias). As before we assume a constant speed v. We split the population

into up-moving individuals α1, right-moving individuals α2, down-moving individuals α3

and left moving individuals α4, where α1 +α2 +α3 +α4 = p, the total population. At each

time step τ an individual can either move a distance δ in the direction it was previously

moving (with probability f = 1− (λ1 +λ2 +λ3)τ); or turn 90 to the left (so right-moving

becomes up-moving etc) and move a distance δ in this new direction (with probability

l = λ1τ); or turn 90 to the right (so right-moving becomes down-moving etc) and move

a distance δ in this new direction (with probability r = λ2τ); or turn 180 around (so

right-moving becomes left-moving etc) and move a distance δ in this new direction (with

probability b = λ3τ).

If we take a forward time step then the number of individuals at position x moving up,

right, down and left respectively is given by

α1(x, y, t+ τ) = fα1(x, y − δ, t) + lα2(x, y − δ, t) + bα3(x, y − δ, t) + rα4(x, y − δ, t),

α2(x, y, t+ τ) = fα2(x− δ, y, t) + lα3(x− δ, y, t) + bα4(x− δ, y, t) + rα1(x− δ, y, t),


α3(x, y, t+ τ) = fα3(x, y + δ, t) + lα4(x, y + δ, t) + bα1(x, y + δ, t) + rα2(x, y + δ, t),

α4(x, y, t+ τ) = fα4(x+ δ, y, t) + lα1(x+ δ, y, t) + bα2(x+ δ, y, t) + rα3(x+ δ, y, t).

We can expand the above equations as Taylor series to give

α1 + τ∂α1

∂t+O(τ2) = f(α1 − δ

∂α1

∂y) + l(α2 − δ

∂α2

∂y) + b(α3 − δ

∂α3

∂y) + r(α4 − δ

∂α4

∂y)

+O(δ2),

α2 + τ∂α2

∂t+O(τ2) = f(α2 − δ

∂α2

∂x) + l(α3 − δ

∂α3

∂x) + b(α4 − δ

∂α4

∂x) + r(α1 − δ

∂α1

∂x)

+O(δ2),

α3 + τ∂α3

∂t+O(τ2) = f(α3 − δ

∂α3

∂y) + l(α4 − δ

∂α4

∂y) + b(α1 − δ

∂α1

∂y) + r(α2 − δ

∂α2

∂y)

+O(δ2),

α4 + τ∂α4

∂t+O(τ2) = f(α4 − δ

∂α4

∂x) + l(α1 − δ

∂α1

∂x) + b(α2 − δ

∂α2

∂x) + r(α3 − δ

∂α3

∂x)

+O(δ2).

We can now substitute for f, l, b and r. After dividing through by τ and taking the limit

as τ → 0 and δ → 0 such that δτ → v we get

∂α1

∂t= −v∂α1

∂y− (λ1 + λ2 + λ3)α1 + λ1α2 + λ3α3 + λ2α4, (2.68)

∂α2

∂t= −v∂α2

∂x− (λ1 + λ2 + λ3)α2 + λ1α3 + λ3α4 + λ2α1, (2.69)

∂α3

∂t= v

∂α3

∂y− (λ1 + λ2 + λ3)α3 + λ1α4 + λ3α1 + λ2α2, (2.70)

∂α4

∂t= v

∂α4

∂x− (λ1 + λ2 + λ3)α4 + λ1α1 + λ3α2 + λ2α3. (2.71)

Adding (2.68) + (2.69) + (2.70) + (2.71) gives

∂p

∂t= v

(∂(α4 − α2)

∂x+∂(α3 − α1)

∂y

)

, (2.72)


∂2p

∂t2= v

(∂2(α4 − α2)

∂x∂t+∂2(α3 − α1)

∂y∂t

)

. (2.73)

One can differentiate (2.69) and (2.71) with respect to x, which after rearranging gives

∂(α4 − α2)

∂x∂t= v

∂2(α2 + α4)

∂x2− (λ1 + λ2 + 2λ3)

(α4 − α2)

∂x+ (λ2 − λ1)

∂(α3 − α1)

∂x. (2.74)

Similarly, after differentiating (2.68) and (2.70) with respect to y and rearranging we get

∂(α3 − α1)

∂y∂t= v

∂2(α1 + α3)

∂y2− (λ1 + λ2 + 2λ3)

(α3 − α1)

∂y+ (λ2 − λ1)

∂(α2 − α4)

∂y. (2.75)

We can substitute (2.74) and (2.75) back into (2.73) and use (2.72) to give

∂2p

∂t2= v2

(∂2(α2 + α4)

∂x2+∂2(α1 + α3)

∂y2

)

− v(λ1 + λ2 + 2λ3)∂p

∂t

+(λ2 − λ1)(∂(α3 − α1)

∂x− ∂(α4 − α2)

∂y

)

. (2.76)


One can see that the two-dimensional equation we have derived has a very similar form to

our original one-dimensional telegraph equation (1.41), but the system cannot be written

in terms of the total population p. We could solve equations (2.68)—(2.71) simultaneously

to find solutions for α1-α4 but we cannot solve directly to find a solution for p.

It is possible to do a similar analysis for models with more than four directions. However,

as above, this results in a system of equations that cannot be closed to find an equation

for p, see also the discussions in Othmer & Hillen (2000; 2002).

The two-dimensional telegraph equation does exist and is straightforward to solve in princi-

ple, see Morse & Feshbach (1953). However it is not the limiting equation to the correlated

random walk process we have described above.

2.7 Conclusions

In this chapter we have shown how it is straightforward to extend the one-dimensional

uncorrelated random walk process (a position jump process) to a two-dimensional lattice

and derive the Fokker–Planck diffusion equation. It is possible to make the probabilities

of moving in a certain direction dependent on position, extend the allowed moves from a

lattice to a discrete number of movement directions with differing probabilities, or allow a

continuous range of directions of movement on the unit circle with probability distribution

p(θ), all of which result in similar Fokker–Planck equations, albeit with slightly different

parameter definitions. We can use the experimental results of Hill & Hader and substitute

p(θ) for their predicted long-time angular distribution and calculate the parameters in the

Fokker–Planck equation, but this is only valid if we have a small amount of bias in the sys-

tem and p(θ) is close to uniform. All the solutions of the Fokker–Planck equation allow for

effectively infinite propagation and thus can only be considered large time approximations

to the true solutions.

If we try and extend the one-dimensional telegraph process (a velocity jump process) to

two dimensions with a simple correlated random walk on a lattice we cannot derive a

closed equation for the probability distribution.

To find out information about the underlying spatial distribution we need to approach the

problem in a slightly different manner. In Chapters 3, 4 and 5 we discuss a method of

calculating the moments of the spatial distribution rather then trying to find an equation

that describes the spatial distribution directly.

The main results of this chapter are summarised below:

• It is straightforward to develop the one-dimensional uncorrelated random walk model

to a two-dimensional lattice and using difference equations, the two-dimensional

Fokker–Planck diffusion equation can be derived.


• Bias is introduced by setting the probability of moving in a particular direction to be

greater than the probability of moving in any other direction. The Fokker–Planck

equation will then include drift terms.

• The probabilities of moving in a certain direction can be made dependent on position,

which results in a similar Fokker–Planck equation that has parameters that are

spatially dependent.

• The model can easily be extended from a lattice to a continuous angle probability

distribution p(θ) which results in the same Fokker–Planck diffusion equation.

• The long-time angular distribution for sinusoidal or linear reorientation from Hill &

Hader can be substituted for p(θ) and the parameters in the Fokker–Planck equation

calculated. However, this is only valid if there is a small amount of bias and p(θ) is

close to uniform.

• All the solutions of the Fokker–Planck diffusion equation allow for effectively infinite

propagation and thus can only be considered as long-time approximations to the

true solutions.

• It is not possible to extend the one-dimensional correlated random walk telegraph

process to two dimensions to derive a closed equation for the population distribution.

Chapter 3

Spatial statistics of

two-dimensional velocity jump

processes

3.1 Introduction

We have seen in the previous chapter (Section 2.6) that it is not possible to derive the

telegraph equation in two dimensions for our simple random walk model. A different

method must be used to get information about the system. Although it does not seem

possible to find a simple equation that describes the spatial distribution for all time, it

is possible to calculate the moments of this distribution. Equations (1.22) and (1.24) are

simple to solve and were easy to derive from the differential equation that describes the

underlying spatial distribution.

When looking at models of dispersal in biological systems, Othmer et al. (1988) suggested

a generalized equation to model velocity jump processes and derived a ‘linear transport

equation’ for the underlying spatial and velocity distribution p(x,v, t). Although they

did not try to solve directly for p(x,v, t), they showed how to derive equations for the

moments of the distribution from the linear transport equation.

In this chapter we present the velocity jump process model of Othmer et al. (1988) and

explain all their working and methods (that were unpublished in their paper). We shall

then extend this velocity jump process model and method of taking moments in later

chapters.

48

CHAPTER 3: Spatial statistics of two-dimensional velocity jump processes 49

3.2 Generalized equation for velocity jump processes

3.2.1 Generalized model

Let p(x,v, t) be the density function for individuals in a 2n-dimensional phase space with

coordinates (x,v), where x ∈ Rn is the position of an individual, and v ∈ Rn is its

velocity. Then p(x,v, t)dxdv is the number density of individuals with position between

x and x + dx and velocity between v and v + dv, and

n(x, t) =

∫

p(x,v, t)dv (3.1)

is the number density of individuals at x, whatever their velocity. As p(x,v, t) is a density

function we have p(x,v, t) ≥ 0 ∀ x,v, t. We assume that p(x,v, t) is integrable, and that

p(x,v, t) → 0 as |x| → ∞. The evolution of p is governed by the partial differential

equation∂p

∂t+ ∇

x.vp+ ∇

v.Fp = R, (3.2)

where F denotes the external force acting on the individuals and R is the rate of change of

p due to reaction, random choice of velocity, etc. We assume that F ≡ 0 and that the only

process that contributes to the changes on the right hand side of (3.2) is a process that

generates random velocity changes. We assume that the random velocity changes are the

result of a Poisson process of intensity λ, where λ may depend upon other variables (see

for example Grimmett & Stirzaker (2001)). Thus λ−1 is a mean time between the random

choices of direction. The net rate at which individuals enter the phase-space volume at

(x,v) is given by(

∂p

∂t

)

sp= −λp+ λ

∫

T (v,v′)p(x,v′, t)dv′ (3.3)

where ‘sp’ denotes the change due to the stochastic process. The kernel T (v,v′) is the

probability of a change in velocity from v′ to v , given that a reorientation occurs, and

therefore T (v,v′) is non-negative and normalised so that

∫

T (v,v′)dv = 1. (3.4)

This normalization condition just means that no individuals are lost during the process

of changing velocity. We assume that T (v,v′) is independent of the time between jumps.

Applying the above assumptions means that (3.2) becomes

∂p

∂t+ ∇

x.vp = −λp+ λ

∫

T (v,v′)p(x,v′, t)dv′. (3.5)

Equation (3.5) is the linear transport equation first introduced to model the velocity jump

process by Othmer et al. (1988), and also discussed more recently by Othmer & Hillen

(2000; 2002).


We are interested in the first few velocity moments, including the number density n(x, t)

introduced in (3.1), and the average velocity u(x, t), which is defined by

n(x, t)u(x, t) =

∫

p(x,v, t)v dv. (3.6)

Integrating (3.5) over v gives∂n

∂t+ ∇

x.nu = 0. (3.7)

Similarly, multiplying (3.5) by v and integrating over v gives

∂(nu)

∂t+

∫

v∇.(vp) dv = λ

∫

T (v,v′)vp(x,v′, t) dv′ dv− λnu. (3.8)

3.2.2 Velocity jump processes in one dimension — the telegraph equa-

tion

In one space dimension we can define

T (v,v′) =

0 if v = v′

1 if v 6= v′

and thus demand that individuals change direction each time a choice is made.

We assume the speed is constant, |v| = s and nu = s(α − β), where α ≡ p(x, s, t) and

β ≡ p(x,−s, t) are defined as right-moving and left-moving individuals respectively.

Also we have∫

v∇.(vp) dv = s2∂n

∂x= s2

∂(α+ β)

∂x.

Using the above choice of T (v,v′) the integral term in (3.8) reduces to −λs(α− β), and

therefore (3.7) and (3.8) reduce to

∂(α + β)

∂t+ s

∂(α− β)

∂x= 0, (3.9)

s∂(α− β)

∂t+ s2

∂(α+ β)

∂x= −2λs(α− β). (3.10)

These are just the equations given in (1.37) and (1.39) written in a slightly different form

and one can derive the telegraph equation given in (1.41), as done previously.

3.3 Velocity jump processes in two dimensions — random

walks in external fields

Suppose we have a two-dimensional phase space with a taxis-inducing gradient directed

along the positive x1 axis of the plane, under the assumption that the gradient only

influences the turn angle distribution T . We have a population of individuals p moving

with constant speed s and turning with constant turning frequency λ. The appropriate

density function is now p(x, θ, t) where θ is the angle between the current direction of


motion and the positive x1-axis. Thus the direction of travel is ξ = (cos θ, sin θ) and the

direction of the gradient is ξ1 = (1, 0). The evolution equation (3.5) reduces to

∂p

∂t+ sξ.∇

xp = −λp+ λ

∫ π

−πT (θ, θ′) p(x, θ′, t) dθ′. (3.11)

When modelling the motion of leukocytes in a constant chemotactic gradient, Othmer et

al. assumed that the turning angle distribution T is the sum of a symmetric probability

distribution h(δ), where δ = θ − θ′, and a bias term k(θ) that results from the taxis-

inducing gradient. Since the gradient is directed along the x1-axis, the bias term takes its

maximum at θ = 0, and is symmetric about θ = 0, see Figure 3.1. Thus

T (θ, θ′) = h(θ − θ′) + k(θ), (3.12)

where T ≥ 0 for all (θ, θ′) and h and k are normalized as follows:

∫ π

−πh(δ) dδ = 1,

∫ π

−πk(θ) dθ = 0,

∫ π

−πT (θ, θ′) dθ = 1, (3.13)

and also

∫ π

−πh(δ) cos δ dδ = ψd is defined as the directional persistence, (3.14)

and

∫ π

−πk(θ) cos θ dθ = χ is defined as the taxis coefficient. (3.15)

π−π

h

−π π

k

θ

(θ)

δ

(δ)

Figure 3.1: Sketch of the probability distributions for h(δ) and k(θ) as used by Othmer et

al. (1988).

3.3.1 Defining statistics of interest

Othmer et al. used (3.12) to calculate the statistics of interest, which include the mean

location of cells H(t), their mean squared displacement D2(t), and their mean velocity


V(t). They then used these to calculate a further statistic, the mean squared deviation

σ2(t), which is a measure of the fluctuations of the individual’s path around the expected

path. The statistics of interest are defined as

H(t) =1

N0

∫

R2

∫ π

−πx p(x, θ, t) dθ dx, (3.16)

V(t) =s

N0

∫

R2

∫ π

−πξ p(x, θ, t) dθ dx, (3.17)

and D2(t) =1

N0

∫

R2

∫ π

−π‖x‖2 p(x, θ, t) dθ dx. (3.18)

We shall need the following auxiliary function

B(t) =1

N0

∫

R2

∫ π

−π(x.ξ) p(x, θ, t) dθ dx. (3.19)

We also define the mean squared deviation

σ2(t) =1

N0

∫

R2

∫ π

−π‖x − H(t)‖2p(x, θ, t) dθ dx. (3.20)

The mean squared deviation, σ2(t), is different to the mean squared displacement, D2(t).

The first measures the spread about the mean position of the population while the latter

measures the spread about the origin. For a random walk with bias, it is more useful to

look at the statistic σ2(t). Note that one always has σ2(t) < D2(t), except in the case of

zero bias when the two statistics are equal (because the mean position of the population

is at the origin in this case). See Figure 3.2.

O

H

D2

σ 2

Figure 3.2: Sketch showing the difference between D2 and σ2 (H is the average position).

Othmer et al. assumed that at t = 0, all the population start at the origin (0, 0), with

initial directions uniformly distributed around the unit circle. Hence all the statistics

defined above are equal to zero at t = 0.


3.3.2 Deriving equations for spatial statistics

3.3.2.1 Changing the order of integration

We use Fubini’s theorem (see for example Cox (1998)) so that the order of integration can

be changed from dθ dx to dx dθ and dθ′ dθ to dθ dθ′.

If a(θ) and b(x) are bounded over the integral range (−π to π) and R2 respectively, then

the following result holds

∫

R2

∫ π

−πa(θ)b(x) dθ dx =

∫ π

−π

∫

R2

a(θ)b(x) dx dθ, (3.21)

and if f(θ) and g(θ′) are bounded over the integral range (−π to π) then the following

holds∫ π

−π

∫ π

−πf(θ)g(θ′) dθ′ dθ =

∫ π

−π

∫ π

−πf(θ)g(θ′) dθ dθ′. (3.22)

3.3.2.2 The divergence theorem

We shall make use of the divergence theorem (see for example Cox (1998)). The divergence

theorem states that in three-dimensional space the integral of the gradient of a function

is the same as the integral around the surface of the normal to the function.

∫

V∇

x.u dV =

∫

∂Vu.n dS.

The divergence theorem can be shown to hold in two dimensions as well, and we can write

∫

R2

∇x.u dx =

∫

Su.n dS,

where S is now the boundary surrounding our domain R2. Since we have assumed an

infinite domain, S will be the boundary at |x| → ∞. Now, because we have a finite

speed of propagation and start at x = 0, we make the assumption that within finite time

p(x, θ, t) = 0 on S. Hence all functions of the form up(x, θ, t), where u is a bounded vector

function, are zero on S. Thus

∫

R3

∇x.(up) dx =

∫

S(up).n dS = 0. (3.23)

In the subsequent analysis we will need to use the following equations that are straight-

forward to derive using the properties of the gradient and divergence functions. If a is a

scalar function then

∫

R2

−a(ξ.∇xp) dx =

∫

R2

(−∇x.(ξap) + pξ.(∇

xa)) dx =

∫

R2

pξ.(∇xa) dx (3.24)

since∫

R2 −∇x.(ξap) dx = 0 from (3.23).

If v is a vector function then

∫

R2

−v(ξ.∇xp) dx =

∫

R2

(−ξ.∇x(vp) + p(ξ.∇

x)v) dx. (3.25)


Rewriting the second term in the right hand side of the above gives

∫

R2

−v(ξ.∇xp) dx =

∫

R2

(−∇x.(ξv1p)ξ1 −∇

x.(ξv2p)ξ2 + p(ξ.∇

x)v) dx, (3.26)

where ξ1 = (1, 0) and ξ2 = (0, 1). Using (3.23), the first two terms in the right hand side

of the above are zero and we get

∫

R2

−v(ξ.∇xp) dx =

∫

R2

(p(ξ.∇x)v) dx. (3.27)

3.3.2.3 Deriving equation for H(t)

Multiply the original equation (3.11) by x, integrate over θ and x, and divide by N0, to

give

1

N0

∫

R2

∫ π

−πx∂p

∂tdθ dx = − s

N0

∫

R2

∫ π

−πx (ξ.∇

xp) dθ dx

− λ

N0

∫

R2

∫ π

−πx p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−πx T (θ, θ′) p(θ′) dθ′ dθ dx.

In the final term, we can use the fact that the order of integration of dθ and dθ′ can be

changed to give the inner integral

∫ π

−πT (θ, θ′) dθ = 1.

Substituting for H(t) from (3.16) and using the divergence theorem (3.27) gives

dH

dt=

s

N0

∫

R2

∫ π

−π(pξ.∇

x)x dθ dx − λH +

λ

N0

∫

R2

∫ π

−πx p(θ′) dθ′ dx.

The last two terms now cancel, giving

dH

dt=

s

N0

∫

R2

∫ π

−πξp dθ dx.

Substituting for V(t) from (3.17) gives the final differential equation for H(t)

dH

dt= V. (3.28)

3.3.2.4 Deriving equation for V(t)

Multiply the original equation (3.11) by sξ, integrate over θ and x, and divide by N0, to

give

s

N0

∫

R2

∫ π

−πξ∂p

∂tdθ dx = − s2

N0

∫

R2

∫ π

−πξ (ξ.∇

xp) dθ dx

− sλ

N0

∫

R2

∫ π

−πξ p dθ dx

+sλ

N0

∫

R2

∫ π

−π

∫ π

−πξ T (θ, θ′) p(θ′) dθ′ dθ dx.


The first term in the right hand side of the above is zero due to the divergence theorem

(3.27). Recalling that ξ = (cos θ, sin θ) and substituting for V(t) from (3.17) and for

T (θ, θ′) from (3.12), gives

dV

dt= −λV +

sλ

N0

∫

R2

∫ π

−π

∫ π

−π(cos θ, sin θ) (h(θ − θ′) + k(θ)) p(θ′) dθ′ dθ dx.

The order of the integral in the above can be changed (changing dθ and dθ′) to give

dV

dt= − λV +

sλ

N0

∫

R2

∫ π

−π

∫ π

−π(cos θ, sin θ) h(θ − θ′) p(θ′) dθ dθ′ dx

+sλ

N0

∫

R2

∫ π

−π

∫ π

−π(cos θ, sin θ) k(θ)) p(θ′) dθ dθ′ dx.

The distribution k(θ) is symmetric about θ = 0, so∫ π

−πsin θ k(θ) dθ = 0.

We also use the definition of χ from (3.14) to give

dV

dt= − λV

+sλ

N0

∫

R2

∫ π

−π

∫ π

−π(cos θ, sin θ) h(θ − θ′) p(θ′) dθ dθ′ dx + λχsξ1.

Fixing θ′, we can make the substitution δ = θ−θ′, and hence dδ = dθ. Using the standard

trigonometric identities gives

cos θ = cos δ cos θ′ − sin δ sin θ′, sin θ = sin δ cos θ′ + cos δ sin θ′.

Substituting these identities back into the equation and noting that we now have to change

the limits of integration, gives

dV

dt= − λV + λχsξ1

+sλξ1

N0

∫

R2

∫ π

−π

∫ π−θ′

−π−θ′(cos δ cos θ′ − sin δ sin θ′) h(δ) p(θ′) dδ dθ′ dx

+sλξ2

N0

∫

R2

∫ π

−π

∫ π−θ′

−π−θ′(sin δ cos θ′ + cos δ sin θ′) h(δ) p(θ′) dδ dθ′ dx.

Now, because the function being integrated is 2π-periodic, the limits from −π−θ′ to π−θ′

are the same as the limits from −π to π. The distribution h(δ) is also symmetric about

δ = 0 so that∫ π

−πsin δ h(δ) dδ = 0.

Recalling the definition of ψd from (3.15), we get

dV

dt= − λV + λχsξ1

+sλ

N0

∫

R2

∫ π

−π(cos θ′, sin θ′) ψd p(θ

′) dθ′ dx.

This simplifies todV

dt= −λ0V + λχsξ1 (3.29)

where

λ0 = λ(1 − ψd).


3.3.2.5 Deriving equation for D2(t)

Multiply the original equation (3.11) by ‖x‖2, integrate over θ and x, and divide by N0,

to give

1

N0

∫

R2

∫ π

−π‖x‖2 ∂p

∂tdθ dx = − s

N0

∫

R2

∫ π

−π‖x‖2 (ξ.∇

xp) dθ dx

− λ

N0

∫

R2

∫ π

−π‖x‖2 p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−π‖x‖2 T (θ, θ′) p(θ′) dθ′ dθ dx.

In the final term, we use the fact that the order of integration of dθ and dθ′ can be

changed to give the inner integral

∫ π

−πT (θ, θ′) dθ = 1.

Substituting for D2(t) from (3.18) and using the divergence theorem (3.24) gives

dD2

dt=

s

N0

∫

R2

∫ π

−π(ξp).∇

x(‖x‖2) dθ dx

− λD2 +λ

N0

∫

R2

∫ π

−π‖x‖2 p(θ′) dθ′ dx.

The final two terms cancel, giving

dD2

dt=

s

N0

∫

R2

∫ π

−π(ξp).∇

x(‖x‖2) dθ dx.

This givesdD2

dt=

s

N0

∫

R2

∫ π

−π(ξp).(2x1, 2x2) dθ dx

which can be rewritten as

dD2

dt=

2s

N0

∫

R2

∫ π

−π(ξ.x)p dθ dx.

Finally, we can substitute for B(t) from (3.19) to give the final differential equation for

D2(t),dD2

dt= 2sB. (3.30)

3.3.2.6 Deriving equation for B(t)

Multiply the original equation (3.11) by (x.ξ), integrate over θ and x, and divide by N0,

to give

1

N0

∫

R2

∫ π

−π(x.ξ)

∂p

∂tdθ dx = − s

N0

∫

R2

∫ π

−π(x.ξ)(ξ.∇

xp) dθ dx

− λ

N0

∫

R2

∫ π

−π(x.ξ) p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−π(x.ξ) T (θ, θ′) p(θ′) dθ′ dθ dx.


Substituting for B(t) from (3.19) and for T (θ, θ′) from (3.12), and using the divergence

theorem (3.24) gives

dB

dt=

s

N0

∫

R2

∫ π

−πp dθ dx− λB

+λ

N0

∫

R2

∫ π

−π

∫ π

−π(x1 cos θ + x2 sin θ) (h(θ − θ′) + k(θ)) p(θ′) dθ′ dθ dx.

The order of the integration in the above can be changed (changing dθ and dθ′) to give

dB

dt= s− λB +

λ

N0

∫

R2

∫ π

−π

∫ π

−π(x1 cos θ + x2 sin θ) k(θ) p(θ′) dθ dθ′ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−π(x1 cos θ + x2 sin θ) (θ − θ′) p(θ′) dθ dθ′ dx

The distribution k(θ) is symmetric about θ = 0 so that∫ π

−πsin θ k(θ) dθ = 0.

We use the definition of χ from (3.14) to give

dB

dt= s− λB + λχx1 +

λ

N0

∫

R2

∫ π

−π(x1 cos θ + x2 sin θ) h(θ − θ′) p(θ′) dθdθ′ dx,

where

x1 =1

N0

∫

R2

∫ π

−πx1 p(θ

′) dθ′dx = H.ξ1. (3.31)

Fixing θ′, we can make the substitution δ = θ − θ′, and hence have dδ = dθ. Using the

standard trigonometric identities we have

cos θ = cos δ cos θ′ − sin δ sin θ′, sin θ = sin δ cos θ′ + cos δ sin θ′.

Substituting back into the equation and noting that we now have to change the limits of

integration, gives

dB

dt= s− λB + λχx1

+λ

N0

∫

R2

∫ π

−π

∫ π−θ′

−π−θ′(x1 cos δ cos θ′ − x1 sin δ sin θ′) h(δ) p(θ′) dδdθ′ dx

+λ

N0

∫

R2

∫ π

−π

∫ π−θ′

−π−θ′(x2 sin δ cos θ′ + x2 cos δ sin θ′) h(δ) p(θ′) dδdθ′ dx.

Now, because the function being integrated is 2π-periodic, the limits from −π−θ′ to π−θ′

are the same as the limits from −π to π. The distribution h(δ) is symmetric about δ = 0,

so∫ π

−πsin δ h(δ) dδ = 0.

Recalling the definition of ψd from (3.15), we get

dB

dt= s− λB + λχx1 +

λ

N0

∫

R2

∫ π

−π(x1 cos θ′ + x2 sin θ′) ψd p(θ

′) dδdθ′ dx.

This simplifies todB

dt= s− λ0B + λχx1. (3.32)


3.3.2.7 Deriving equation for σ2(t)

Rather than derive a differential equation for σ2(t), we derive an equation that relates it

to the other statistics defined earlier.

σ2(t) =1

N0

∫

R2

∫ π

−π‖x − H(t)‖2p(x, θ, t) dθdx

=1

N0

∫

R2

∫ π

−π(x − H(t)).(x − H(t))p(x, θ, t) dθdx

=1

N0

∫

R2

∫ π

−π(x.x− 2x.H(t) + H(t).H(t))p(x, θ, t) dθdx

The H(t) terms can be taken outside the integral (we are taking the average of an average),

and using the definition of D2(t), we have

σ2(t) = D2(t) − 2H(t)1

N0

∫

R2

∫ π

−πx p(x, θ, t) dθdx + H(t).H(t)

= D2(t) − 2‖H(t)‖2 + ‖H(t)‖2.

So finally we have

σ2(t) = D2(t) − ‖H(t)‖2. (3.33)

3.3.2.8 Final system of equations

The final system of equations that Othmer et al. derived is closed due to their choice of

probability distributions for the reorientation kernel T (θ, θ′). The system is

dH

dt= V, (3.34)

dV

dt= −λ0V + λχsξ1, (3.35)

dD2

dt= 2sB, (3.36)

anddB

dt= s− λ0B + λχx1, (3.37)

where

ξ1 = (1, 0) (the direction of the gradient),

x1 = H.ξ1,

and λ0 = λ(1 − ψd).

We also have the equation relating σ2 to H(t) and D2(t),

σ2(t) = D2(t) − ‖H(t)‖2. (3.38)

3.3.3 Solving equations for spatial statistics

The above system of differential equations is straightforward to solve as all the equations

are linear.


3.3.3.1 Solution for V(t)

From (3.35) and assuming that V(0) = 0, we get

V(t) = sλχ

λ0(1 − e−λ0t)ξ1.

This can be written in the form Othmer et al. give in their paper,

V(t) = sχ

1 − ψd(1 − e−λ0t)ξ1. (3.39)

As t tends to infinity, the mean velocity of movement becomes parallel to the direction of

the gradient and approaches the value

V∞ = sχ

1 − ψdξ1.

For this to be consistent we must have (1−ψd) ≥ χ. We cannot have an absolute velocity

that is greater than the speed of movement.

3.3.3.2 Solution for H(t)

From (3.34) and assuming that H(0) = 0, we get

H(t) = sχ

1 − ψd

(

t− 1

λ0(1 − e−λ0t)

)

ξ1. (3.40)

3.3.3.3 Solution for B(t)

From (3.37) and assuming that B(0) = 0, we get

B(t) =s

λ0

(

1 − e−λ0t + C2I

(

λ0t− 2(1 − e−λ0t) + λ0te−λ0t

))

, (3.41)

where C2I = λ2χ2/λ2

0.

3.3.3.4 Solution for D2(t)

From (3.36) and assuming that D2(0) = 0, we get

D2(t) =2s2

λ0

[

(1 − 2C2I )t− C2

I te−λ0t +

(3C2I − 1)

λ0(1 − e−λ0t) +

C2Iλ0t

2

2

]

. (3.42)

3.3.3.5 Solution for σ2(t)

Substituting (3.40) and (3.42) into (3.33) gives

σ2(t) =2s2

λ0

[

(1 − 2C2I )t− C2

I te−λ0t +

(3C2I − 1)

λ0(1 − e−λ0t) +

C2Iλ0t

2

2

]

−2s2

λ0

[ C2I

2λ0(λ2

0t2 + e−2λ0t + 1 + 2λ0te

−λ0t − 2λ0t− 2e−λ0t)]

.


This simplifies to

σ2(t) =2s2

λ0

(

(1 − C2I )t− 2C2

I te−λ0t +

(2C2I − 1)

λ0(1 − e−λ0t) +

C2I

2λ0(1 − e−2λ0t)

)

. (3.43)

Note that in the paper of Othmer et al. there is an error in their solution for σ2(t). They

incorrectly have (1 − e−λ0t)2 instead of (1 − e−2λ0t) in the final term in the equation.

As t tends to infinity the solution tends to

σ2(t) ∼ 2s2

λ0

[

(1 − C2I )t+

1

λ0(5

2C2I − 1)

]

. (3.44)

The quantity

CI =χ

1 − ψd

measures the net effect of the bias due to the taxis-inducing gradient on the motion in the

direction ξ1. For the solution to give a sensible answer we must have CI ≤ 1.

3.3.3.6 Solution plots

The following plots show the general behaviour of the solutions to the equations of Othmer

et al. For each plot we have fixed λ0 = 1 and s = 1, and then plotted solutions for CI = 0.2,

CI = 0.4, CI = 0.6, and CI = 0.8.

CI=0.2CI=0.4CI=0.6CI=0.8

0

0.2

0.4

0.6

0.8

V(t)

2 4 6 8 10

t

(a) Vx1(t).

CI=0.2CI=0.4CI=0.6CI=0.8

0

1

2

3

4

5

6

7

H(t)

2 4 6 8 10

t

(b) Hx1(t).

Figure 3.3: Plots of Vx1(t) and Hx1(t) for various values of CI .

Looking at the plots, one can see that the solutions very quickly tend to their asymptotic

limits. Even after a small time period (t = 10), the solutions have reached a constant

gradient (except for D2(t) as it is proportional to t2). The plots for V(t) and H(t) show


CI=0.2CI=0.4CI=0.6CI=0.8

0

5

10

15

20

25

30

35

D^2(t)

2 4 6 8 10

t

(a) D2(t).

CI=0.2CI=0.4CI=0.6CI=0.8

0

1

2

3

4

5

6

7

Sig^2(t)

2 4 6 8 10

t

(b) σ2(t).

Figure 3.4: Plots of D2(t) and σ2(t) for various values of CI .

the velocity and average position in the x1 direction only (the preferred direction), as both

statistics will be zero in the x2 direction.

From Figure 3.3, one can see that the solution for V(t) quickly tends to a limiting value

given by sCI , and the solution for H(t) tends to a solution that is linear in time with

gradient also given by sCI . Thus the limiting behaviour of V(t) and the absolute velocity

H(t)/t are the same.

From Figure 3.4, one can see that the solution for D2(t) behaves as t2 as t increases,

while σ2(t) is linear in time. The spread about the origin has behaviour characteristic

of a wave propagation process, while the spread about the mean position has behaviour

characteristic of a diffusive process (where σ2(t) ∼ t).

As CI increases we get motion that is less random and more like a straight line, up to

CI = 1 when we have straight line motion. We cannot have CI > 1, as the maximum

possible movement in the preferred direction is a straight line. If CI = 1 then it follows

that V(t) = s, H(t) ∼ st, D2(t) ∼ s2t2, and σ2 ∼ 0, and we no longer have diffusive-like

motion.

3.4 Conclusions

In this chapter we have presented the velocity jump process model of Othmer et al. (1998)

and demonstrated how using a generalized master ‘linear transport equation’ (3.5), one can

derive the telegraph equation in one dimension. In two dimensions, the general solution

is not specified but it is possible to use a method of calculating differential equations


for the moments of the underlying spatial distribution if the reorientation kernel T (θ, θ′)

is specified. To model bias in the system, Othmer et al. used two separate probability

distributions for the bias and the turning angle, k(θ) and h(δ) respectively (see Figure

3.1). Assuming the reorientation kernel T (θ, θ′) is a superposition of these probability

distributions, it is straightforward to derive differential equations for the average velocity

V(t), the average position H(t), the spread about the origin D2(t) and a further higher

moment B(t). These equations are all linear and simple to solve. Using the solutions

for H(t) and D2(t) it is possible to define a further statistic, the spread about the mean

position σ2(t), given by σ2(t) = D2(t) − ‖H(t)‖2. The solutions to these equations are

dependent on the moments of the angular distributions h(δ) and k(θ), as well as the fixed

speed s and the turning frequency λ. Unlike equations derived from a position jump

process, the solution equations for the spatial statistics from the velocity jump process of

Othmer et al. are valid for all time.

The main results from this chapter are summarised below:

• Othmer et al. derived a generalized model for velocity jump processes using a linear

transport equation.

• In one dimension this generalized equation can be shown to simplify to the telegraph

equation.

• In two dimensions a general solution is not specified but by using a method of taking

the moments of the linear transport equation, differential equations for the higher

moments of the underlying spatial distribution can be derived.

• To model bias in the system, Othmer et al. suggested a reorientation probability

distribution that is a superposition of two separate probability distributions, one

a symmetric turn angle distribution h(δ), and one a symmetric bias distribution

k(θ). Using this superposition of two separate probability distributions to model the

reorientation, Othmer et al. derive a closed system of differential equations for the

moments of the underlying spatial distribution.

• These differential equations are easily solved to give final solution equations for the

spatial statistics of interest: the average position H(t), the average velocity V(t), the

spread about the origin D2(t) and the spread about the mean position σ2(t). These

solution equations are valid for all time, unlike equations derived from a position

jump process.

• The solution equations for the spatial statistics of interest are dependent on the

speed of movement s, the turning frequency λ, the mean cosine of the turning angle

distribution ψd and the taxis coefficient χ. The latter two parameters being moments

of the two angular distributions h(δ) and k(θ) used to model the reorientation.

Chapter 4

Velocity jump processes using

sinusoidal reorientation

4.1 Introduction

In the previous chapter we presented the generalised velocity jump process model of Oth-

mer et al. (1988). The two-dimensional example they looked at used two separate and

independent probability distributions k(θ) and h(θ − θ′) to account for the bias and cor-

relation in the reorientation respectively. The sum of these two distributions gives the

probability distribution for moving from angle θ′ to angle θ, denoted by T (θ, θ′). The

choice of these probability distributions results in a closed system for the moments of the

underlying spatial distribution.

After analysing data from experiments on algae, Hill & Hader (1997) suggested that the

mean turning angle is dependent on the absolute direction, and showed how this results

in two models for reorientation — sinusoidal and linear (see Section 1.5). Sinusoidal

reorientation has been observed in algae such as C. nivalis that are gyrotactic due to

being bottom heavy and thus subject to a gravitational torque, see for example Kessler

(1986), Pedley & Kessler (1990), Kessler et al. (1992).

In this chapter we develop the velocity jump process model and method of calculating

the moments used by Othmer et al. (1988) to derive equations for the moments of the

underlying spatial distribution of a population moving with a turning angle distribution,

T (θ, θ′), that has a mean turning angle given by Hill & Hader’s sinusoidal reorientation

model.

63

CHAPTER 4: Velocity jump processes using sinusoidal reorientation 64

4.2 Reorientation model

4.2.1 Hill & Hader’s general reorientation model

As discussed in the introductory chapter (see Section 1.5), when looking at the trajectories

of swimming micro-organisms such as C. nivalis, Hill & Hader derived an equation for

the probability distribution of the long-time orientation of the cells, from which certain

parameters can be calculated. After analysing experimental results they showed that

the mean turning angle is dependent on the direction of movement and for gyrotactic

movement is given by

µδ = −B−1τ sin(θ − θ0) (−π ≤ θ, θ0 < π), (4.1)

from (1.87) and (1.99), where τ is the average time between turns, B is the average

reorientation time and θ0 is the preferred direction.

Without loss of generality, we shall assume that θ0 = 0 in the subsequent analysis.

4.2.1.1 Mean turning angle

When sampling the trajectories of swimming cells, Hill & Hader showed that the amplitude

of the mean turning angle was dependent on the sampling time step used, τ , and the

reorientation time, B. We define the dimensionless parameter dτ as

dτ = B−1τ. (4.2)

From (1.104), the smallest average time step between turns that Hill & Hader measured

is τ = 0.08 s and we will use this value as an estimate of the actual time between turns in

our model, τ . However, it should be noted that this value for τ was due to experimental

constraints and not necessarily due to the actual turning frequency of the algae. The

turning frequency is given by

λ =1

τ. (4.3)

From Table 1.1 we have 0.19 ≤ B−1 ≤ 0.80, and so for values of the parameters that

correspond to experimental data we have dτ ≪ 1, a fact that we shall exploit in later

analysis.

We assume that the cells are negatively gyrotactic and hence individuals prefer to move

upwards against gravity, and that the amplitude of the reorientation coefficient B−1 is

fixed and not dependent on the spatial position.

4.2.1.2 Variance of the turning angle

Hill & Hader also showed that the variance of the turning angle σ2δ is dependent on the

sampling time step and a constant parameter σ20 (the variance per unit time step), and is

given by

σ2δ = σ2

0τ. (4.4)


From (1.64) we define the concentration parameter that corresponds to σ20 as

κ0 = A−11

(

e−σ2

0/2)

, (4.5)

where A−11 (z) is the inverse of A1(z) = I1(z)

I0(z), and In(z) is the modified Bessel function of

order n.

Similarly, the concentration parameter that corresponds to σ2δ is

κ = A−11

(

e−σ2

0τ/2)

, (4.6)

Combining (4.5) and (4.6) gives

A1(κ) = [A1(κ0)]τ . (4.7)

From now on in our model, when we specify κ we mean the κ defined in (4.6) that is

related to the concentration parameter for unit time (κ0) by (4.7).

4.2.2 The reorientation kernel T (θ, θ′)

In the velocity jump process model of Othmer et al. the reorientation kernel T (θ, θ′) is

not restricted to any particular distribution, except for the condition that

∫ π

−πT (θ, θ′) dθ = 1, where T (θ, θ′) ≥ 0. (4.8)

Hill & Hader did not specify a particular probability distribution for the turning angle,

but for their reorientation models showed that the mean turning angle, µδ, was dependent

on the absolute angle, θ, with an amplitude dependent on the time step used (see (1.87)

and (1.99)). They also specified that the variance of the turning angle was independent

of the absolute angle and dependent on the time step used.

For our model we assume a von Mises distribution for the turning angle (or reorientation)

distribution. This is the simplest symmetric and unimodal circular probability distribu-

tion, and it takes the form

f(δ) =1

2πI0(κ)eκ cos(δ−µδ)

where µδ is the mean turning angle and

∫ π

−πf(δ) dδ = 1,

see Section 1.2.3.2.

The von Mises distribution has been used to model correlated random walks by, for exam-

ple, Siniff & Jessen (1969), Okubo (1980), Batschelet (1981), Bovet & Behhamou (1988)

etc. By setting the mean turning angle to be zero, the von Mises distribution is peaked

around θ = 0, and a walker is likely to carry on moving in the same direction as previously

— a realistic model for animal movement. By increasing or decreasing the concentration


parameter κ, one can increase or decrease how sharply peaked the turning angle distri-

bution is, and thus how correlated the random walk will be. As the mean turning angle

is always zero these models do not include bias and the population average position will

always be the origin (or start point) and there is no average drift.

By using Hill & Hader’s mean turning angle that is dependent on the absolute angle

(rather than always being zero) we can introduce bias into the distribution so that at each

turn there is a balance between the walker trying to continue in the same direction as

previously and trying to turn back to the preferred direction.

4.2.3 Sinusoidal reorientation model

To model the sinusoidal reorientation observed in gyrotaxis using the velocity jump process

model of Othmer et al. we substitute for Hill & Hader’s µδ = −dτ sin θ (assuming that

θ0 = 0 for convenience) and recognise that if we define δ = θ−θ′, the von Mises distribution

f(δ) can be used as the reorientation kernel T (θ, θ′). This gives

T (θ, θ′) =1

2πI0(κ)eκ cos(θ−θ′+dτ sin θ′) (4.9)

where (4.9) satisfies the conditions in (4.8). Thus we now have a probability distribution

for reorientation that implicitly includes bias.

The velocity jump process of Othmer et al. (1988), presented in the previous chapter, is

such that the turning process is discrete in time — a walker will move for a certain random

time, then turn through a random angle, then move for a certain random time etc. Hill &

Hader assumed a continuous time turning model as the time step between turns tended

to zero, but these two models are not incompatible.

From (4.2) and (4.7), one can see that as τ → 0, the concentration parameter κ→ ∞ and

dτ → 0. If these values for the reorientation parameters are substituted into (4.9), the

reorientation distribution becomes ever more sharply peaked about the previous direction

of movement as τ → 0. This is exactly the behaviour that Hill & Hader proposed in their

continuous random walk model.

In reality, most measurements of animal movement patterns are discretized in either time

or space and continuous random walks are not observed unless the sampling length is

very small. In the case of a movement process with a discretized observed time step with

average value of τ (such as Hill & Hader’s experimental data) our model is appropriate as

an approximation to the underlying continuous random walk. Our model is also suitable for

random walks where the underlying motion may not actually be continuous, for example

butterflies moving in discrete jumps from site to site as modelled by Kareiva & Shigesada

(1983).


4.2.4 The biological relevance of the turning angle distribution param-

eters

The turning or reorientation parameters dτ and κ affect the reorientation kernel given by

(4.9) in different ways. From (4.9) one can see that the reorientation distribution will

be centred on (and take its maximum value at) θ = θ′ − dτ sin θ′, since κ cos(α) has its

maximum when α = 0. Thus, the new mean absolute angle of movement θ will be balanced

between carrying on in the same direction (θ = θ′) and moving back towards the preferred

direction (the dτ sin θ′ term). The larger the value of dτ , the more the mean absolute angle

of movement is shifted back towards the preferred direction. However, it should be made

clear that it is just the mean direction of movement that is shifted back to the preferred

direction — the distribution will still be spread around this value with a corresponding

concentration parameter κ.

The concentration parameter κ gives a measure of how peaked the reorientation distribu-

tion, (4.9), will be peaked about the ‘shifted mean’ described above. If κ is small then

the distribution will be quite flat and there will be a lot of randomness in the choice of

direction at each step. Conversely, if κ is large then the distribution is sharply peaked

about the shifted mean and the walker is likely to move in a direction very close to the

shifted mean. So κ can be thought of as the ability of a random walker to overcome the

inherent randomness in its movement through the environment (which could be due to

turbulence and other external factors, or internal mechanisms of movement).

For these reasons we can think of dτ as a sensing ability and κ as a swimming or orientating

ability.

We now extend the velocity jump process model of Othmer at al. (1988) using our prob-

ability distribution for reorientation that implicitly includes bias (4.9).

4.3 Defining statistics of interest

The statistics of interest are the mean location of cells H(t), their mean squared displace-

ment D2(t), and their mean velocity V(t), which are defined as follows

H(t) =1

N0

∫

R2

∫ π

−πx p(x, θ, t) dθ dx, (4.10)

V(t) =s

N0

∫

R2

∫ π

−πξ p(x, θ, t) dθ dx, (4.11)

D2(t) =1

N0

∫

R2

∫ π

−π‖x‖2 p(x, θ, t) dθ dx, (4.12)

D2x1(t) =

1

N0

∫

R2

∫ π

−πx2

1 p(x, θ, t) dθ dx, (4.13)

and D2x2(t) =

1

N0

∫

R2

∫ π

−πx2

2 p(x, θ, t) dθ dx. (4.14)


In addition, we introduce the following higher order moments

Fn(t) =1

N0

∫

R2

∫ π

−πcosnθ p(x, θ, t) dθ dx, (4.15)

Gn(t) =1

N0

∫

R2

∫ π

−πx1 cosnθ p(x, θ, t) dθ dx, (4.16)

Yn(t) =1

N0

∫

R2

∫ π

−πsinnθ p(x, θ, t) dθ dx, (4.17)

and Zn(t) =1

N0

∫

R2

∫ π

−πx2 sinnθ p(x, θ, t) dθ dx. (4.18)

Note that using these definitions V(t) = s(F1(t), Y1(t)) and D2(t) = D2x1(t) +D2

x2(t).

4.4 Results and assumptions to be used in analysis

4.4.1 Integrals of the von Mises distribution

For the subsequent analysis we need to be able to calculate certain integrals of the modified

von Mises distribution. Mardia & Jupp (1999) give the following standard integrals for

the modified von Mises distribution:

1

2πI0(κ)

∫ π

−πcos(pθ − pµ) eκ cos(θ−µ) dθ =

Ip(κ)

I0(κ), (4.19)

1

2πI0(κ)

∫ π

−πsin(pθ − pµ) eκ cos(θ−µ) dθ = 0, (4.20)

where Ip(κ) is the modified Bessel function of order p.

We now derive further standard integrals that we will need later. Working with equation

(4.19), we can expand the cosine term using the standard trigonometric identities

Ip(κ)

I0(κ)=

1

2πI0(κ)

∫ π

−π(cos pθ cos pµ+ sin pθ sin pµ) eκ cos(θ−µ) dθ.

Rearranging and taking µ-dependent terms outside of the integral gives

Ip(κ)

I0(κ)=

cos pµ

2πI0(κ)

∫ π

−πcos pθ eκ cos(θ−µ) dθ +

sin pµ

2πI0(κ)

∫ π

−πsin pθ eκ cos(θ−µ) dθ. (4.21)

Expanding the sine term in equation (4.20), and rearranging the terms results in a similar

equation

0 =cos pµ

2πI0(κ)

∫ π

−πsin pθ eκ cos(θ−µ) dθ − sin pµ

2πI0(κ)

∫ π

−πcos pθ eκ cos(θ−µ) dθ. (4.22)

If we multiply equation (4.21) by cos pµ and equation (4.22) by sin pµ, and then take the

latter from the former (i.e. cos pµ (4.21) − sin pµ (4.22)), we get

1

2πI0(κ)

∫ π

−πcos pθ eκ cos(θ−µ) dθ =

Ip(κ)

I0(κ)cos pµ. (4.23)

A similar calculation using (4.20) gives

1

2πI0(κ)

∫ π

−πsin pθ eκ cos(θ−µ) dθ =

Ip(κ)

I0(κ)sin pµ. (4.24)


Substituting for µ = θ′ − dτ sin θ′ from Hill & Hader’s model for reorientation, we get the

following integrals

1

2πI0(κ)

∫ π

−πcos pθ eκ cos(θ−θ′+dτ sin θ′) dθ =

Ip(κ)

I0(κ)cos(pθ′ − pdτ sin θ′), (4.25)

1

2πI0(κ)

∫ π

−πsin pθ eκ cos(θ−θ′+dτ sin θ′) dθ =

Ip(κ)

I0(κ)sin(pθ′ − pdτ sin θ′). (4.26)

These will be used in the subsequent analysis.

4.4.2 Asymptotic expansions of the trigonometric functions

We assume that 0 ≤ dτ ≪ 1, and seek an asymptotic expansion in powers of dτ for the

trigonometric functions.

The standard Taylor Series expansions for the trigonometric functions that will be needed

in the later analysis are:

cos(nθ − ndτ sin θ) = cosnθ + ndτ sin θ sinnθ − n2d2τ

2sin2 θ cosnθ +O(d3

τ ) (4.27)

sin(nθ − ndτ sin θ) = sinnθ − ndτ sin θ cosnθ − n2d2τ

2sin2 θ sinnθ +O(d3

τ ). (4.28)

Using the standard trigonometric identities, (4.27) and (4.28) reduce for n = 1 and n = 2

to

cos(θ − dτ sin θ) =dτ2

+

(

1 − d2τ

8

)

cos θ − dτ2

cos 2θ +d2τ

8cos 3θ +O(d3

τ ), (4.29)

cos(2θ − 2dτ sin θ) =d2τ

2+ dτ cos θ +

(

1 − d2τ

)

cos 2θ − dτ cos 3θ +O(d3τ ), (4.30)

sin(θ − dτ sin θ) =

(

1 − 3d2τ

8

)

sin θ − dτ2

sin 2θ +d2τ

8sin 3θ +O(d3

τ ), (4.31)

sin(2θ − 2dτ sin θ) = dτ sin θ +(

1 − d2τ

)

sin 2θ − dτ sin 3θ +O(d3τ ). (4.32)

4.4.3 Previous results

• As in the previous chapter (see Section 3.3.2.1), we assume that we can change the

order of integration between dx, dθ and dθ′.

• As in the previous chapter (see Section 3.3.2.2), we use the divergence theorem to

show that the following integrals hold

∫

R2

−a(ξ.∇xp) dx =

∫

R2

pξ.(∇xa) dx, (4.33)

for scalar functions a, and

∫

R2

−v(ξ.∇xp) dx =

∫

R2

(p(ξ.∇x)v) dx, (4.34)

for vector functions v.


4.4.4 Other assumptions

• We assume that all the population starts at (0, 0) at t = 0, and assume that at

t = 0, the directions of movement of the population are spread equally around the

unit circle. Hence all the statistics and higher moments previously defined will be

zero at t = 0.

• Also, for convenience in the subsequent calculations, we shall write p(θ) or p in place

of p(θ,x, t).

4.5 Differential equations for the spatial statistics and higher

order moments

Using a similar method to Othmer et al., differential equations for the statistics of interest

and higher moments can be derived using the linear transport equation given in (3.11).


4.5.1.1 Differential equation for H(t)

In the previous chapter, the differential equation derived for H(t) for the reorientation

model used by Othmer et al. (3.28), was independent of the choice of T (θ, θ′), and so the

equation for H(t) for our model that uses sinusoidal re-orientation will be the same, i.e.

dH

dt= V. (4.35)

4.5.1.2 Differential equation for V(t)

From the definitions of F1(t) and Y1(t) in (4.15) and (4.17) respectively, we have

V(t) = s(F1(t), Y1(t)), (4.36)

and thus it is not necessary to derive a differential equation for V(t) directly if we are able

to find solutions for F1(t) and Y1(t).

4.5.1.3 Differential equation for D2(t)

From the definitions of D2x1(t) and D2

x2(t) in (4.13) and (4.14) respectively, we have

D2(t) = D2x1(t) +D2

x2(t), (4.37)

and thus it is not necessary to derive a differential equation for D2(t) directly if we are

able to find solutions for D2x1(t) and D2

x2(t).


4.5.1.4 Differential equation for D2x1(t)

Multiply the linear transport equation (3.11) by x21, integrate over θ and x, and divide by

N0, to give

1

N0

∫

R2

∫ π

−πx2

1

∂p

∂tdθ dx = − s

N0

∫

R2

∫ π

−πx2

1(ξ.∇xp) dθ dx

− λ

N0

∫

R2

∫ π

−πx2

1 p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−πx2

1 T (θ, θ′) p(θ′) dθ′ dθ dx.

Substituting for D2x1(t) from (4.13) and using the divergence theorem (see Section 3.3.2.2),

gives

dD2x1

dt=

s

N0

∫

R2

∫ π

−π(ξp).(∇

xx2

1) dθ dx− λ

N0

∫

R2

∫ π

−πx2

1p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−πx2

1 T (θ, θ′) p(θ′) dθ′ dθ dx.

The last two terms in the above cancel and we get

dD2x1

dt=

s

N0

∫

R2

∫ π

−π(ξp).(2x1, 0) dθ dx,

which can be written as

dD2x1

dt=

2s

N0

∫

R2

∫ π

−πpx1 cos θ dθ dx.

Recalling the definition of G1(t) from (4.16) gives

dD2x1

dt= 2sG1. (4.38)


Multiply the linear transport equation (3.11) by x22, integrate over θ and x, and divide by

N0, to give

1

N0

∫

R2

∫ π

−πx2

2

∂p

∂tdθ dx = − s

N0

∫

R2

∫ π

−πx2

2(ξ.∇xp) dθ dx

− λ

N0

∫

R2

∫ π

−πx2

2 p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−πx2

2 T (θ, θ′) p(θ′) dθ′ dθ dx.

Substituting for D2x2(t) from (4.14) and using the divergence theorem (see Section 3.3.2.2),

gives

dD2x2

dt=

s

N0

∫

R2

∫ π

−π(ξp).(∇

xx2

2) dθ dx− λ

N0

∫

R2

∫ π

−πx2

2p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−πx2

2 T (θ, θ′) p(θ′) dθ′ dθ dx.


The last two terms in the above cancel and we get

dD2x2

dt=

s

N0

∫

R2

∫ π

−π(ξp).(0, 2x2) dθ dx,

which can be written as

dD2x2

dt=

2s

N0

∫

R2

∫ π

−πpx2 sin θ dθ dx.

Recalling the definition of Z1(t) from (4.18) gives

dD2x2

dt= 2sZ1. (4.39)

4.5.2 Deriving equations for the higher order moments

4.5.2.1 Differential equation for F1(t)

Multiply the linear transport equation (3.11) by cos θ, integrate over θ and x, and divide

by N0, to give

1

N0

∫

R2

∫ π

−πcos θ

∂p

∂tdθ dx = − s

N0

∫

R2

∫ π

−πcos θ (ξ.∇

xp) dθ dx

− λ

N0

∫

R2

∫ π

−πcos θ p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−πcos θ T (θ, θ′) p(θ′) dθ′ dθ dx.

We can substitute for F1(t) from (4.15) and substitute for T (θ, θ′) from (4.9), giving

dF1

dt= − s

N0

∫

R2

∫ π

−π∇

x.(cos θξp) dθ dx

−λF1 +λ

N0

∫

R2

∫ π

−π

∫ π

−π

cos θ

2πI0(κ)eκ cos(θ−θ′+dτ sin θ) p(θ′) dθ′ dθ dx.


(see Section 3.3.2.2). Also, the order of the integration in the above can be changed (see

Section 3.3.2.1), and then using the von Mises integral (4.25), we get

dF1

dt= −λF1 +

λI1(κ)

N0I0(κ)

∫

R2

∫ π

−πcos(θ′ − dτ sin θ′) p(θ′) dθ′ dx.

We can expand the above using the Taylor Series expansion of the trigonometric functions

(4.29) and then substitute using the definition of Fn(t) in (4.15), to give

dF1

dt= −λ11F1 + as1 − as1F2 + as2F3 +O(d3

τ ), (4.40)

where

λ11 = λ

(

1 − (1 − d2τ

8)A1(κ)

)

, as1 =λdτ2A1(κ), as2 =

λd2τ

8A1(κ), (4.41)

and A1(κ) = I1(κ)I0(κ) .



Multiply the linear transport equation (3.11) by cos 2θ, integrate over θ and x, and divide

by N0, to give

1

N0

∫

R2

∫ π

−πcos 2θ

∂p

∂tdθ dx = − s

N0

∫

R2

∫ π

−πcos 2θ (ξ.∇

xp) dθ dx

− λ

N0

∫

R2

∫ π

−πcos 2θ p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−πcos 2θ T (θ, θ′) p(θ′) dθ′ dθ dx.

We can substitute for F2(t) from (4.15) and substitute for T (θ, θ′) from (4.9), giving

dF2

dt= − s

N0

∫

R2

∫ π

−π∇

x.(cos 2θξp) dθ dx− λF2

+λ

N0

∫

R2

∫ π

−π

∫ π

−π

cos 2θ



(see Section 3.3.2.2). Also the order of the integration in the above can be changed (see

Section 3.3.2.1)), and then using the von Mises integral (4.25), we get

dF2

dt= −λF2 +

λI2(κ)

N0I0(κ)

∫

R2

∫ π

−πcos(2θ′ − 2dτ sin θ′) p(θ′) dθ′ dx.


(4.30) and then substitute using the definition of Fn(t) in (4.15), to give

dF2

dt= −λ2F2 + bs1F1 − bs1F3 + bs2 + bs2F4 +O(d3

τ ), (4.42)

where

λ2 = λ(

1 − (1 − d2τ )A2(κ)

)

, bs1 = λdτA2(κ), bs2 =λd2

τ

2A2(κ, ) (4.43)

and A2(κ) = I2(κ)I0(κ) .

4.5.2.3 Differential equation for Y1(t)

Multiply the linear transport equation (3.11) by sin θ, integrate over θ and x, and divide

by N0, to give

1

N0

∫

R2

∫ π

−πsin θ

∂p

∂tdθ dx = − s

N0

∫

R2

∫ π

−πsin θ (ξ.∇

xp) dθ dx

− λ

N0

∫

R2

∫ π

−πsin θ p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−πsin θ T (θ, θ′) p(θ′) dθ′ dθ dx.


We can substitute for Y1(t) from (4.17) and substitute for T (θ, θ′) from (4.9), giving

dY1

dt= − s

N0

∫

R2

∫ π

−π∇

x.(sin θξp) dθ dx

−λY1 +λ

N0

∫

R2

∫ π

−π

∫ π

−π

sin θ



(see Section 3.3.2.2). Also, the order of the integration in the above can be changed (see

Section 3.3.2.1), and then using the von Mises integral (4.26), we get

dY1

dt= −λY1 +

λI1(κ)

N0I0(κ)

∫

R2

∫ π

−πsin(θ′ − dτ sin θ′) p(θ′) dθ′ dx.


(4.31) and then substitute using the definition of Yn(t) in (4.17), to give

dY1

dt= −λ12Y1 − as1Y2 + as2Y3 +O(d3

τ ), (4.44)

where

λ12 = λ

(

1 − (1 − 3d2τ

8)A1(κ)

)

, (4.45)

and as1 and as2 are as defined in (4.41).


Multiply the linear transport equation (3.11) by sin 2θ, integrate over θ and x, and divide

by N0, to give

1

N0

∫

R2

∫ π

−πsin 2θ

∂p

∂tdθ dx = − s

N0

∫

R2

∫ π

−πsin 2θ (ξ.∇

xp) dθ dx

− λ

N0

∫

R2

∫ π

−πsin 2θ p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−πsin 2θ T (θ, θ′) p(θ′) dθ′ dθ dx.

We can substitute for Y2(t) from (4.17) and substitute for T (θ, θ′) from (4.9), giving

dY2

dt= − s

N0

∫

R2

∫ π

−π∇

x.(sin 2θξp) dθ dx − λY2

+λ

N0

∫

R2

∫ π

−π

∫ π

−π

sin 2θ



(see Section 3.3.2.2). Also the order of the integration in the above can be changed (see

Section 3.3.2.1)), and then using the von Mises integral (4.26), we get

dY2

dt= −λY2 +

λI2(κ)

N0I0(κ)

∫

R2

∫ π

−πsin(2θ′ − 2dτ sin θ′) p(θ′) dθ′ dx.



(4.32) and then substitute using the definition of Yn(t) in (4.17), to give

dY2

dt= −λ2Y2 + bs1Y1 − bs1Y3 + bs2Y4 +O(d3

τ ), (4.46)

where λ2, bs1 and bs2 are as defined in (4.43).

4.5.2.5 Differential equation for G1(t)

Multiply the linear transport equation (3.11) by x1 cos θ, integrate over θ and x, and divide

by N0, to give

1

N0

∫

R2

∫ π

−πx1 cos θ

∂p

∂tdθ dx = − s

N0

∫

R2

∫ π

−πx1 cos θ (ξ.∇

xp) dθ dx

−λN0

∫

R2

∫ π

−πx1 cos θ p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−πx1 cos θ T (θ, θ′) p(θ′) dθ′ dθ dx.

We can substitute for G1(t) from (4.16) and substitute for T (θ, θ′) from (4.9), giving

dG1

dt= − s

N0

∫

R2

∫ π

−π(ξp).(∇

xx1 cos θ) dθ dx − λG1

+λ

N0

∫

R2

∫ π

−π

∫ π

−π

x1 cos θ


The order of the integration in the above can be changed (see Section 3.3.2.1), and then

using the von Mises integral (4.25) we get

dG1

dt=

s

N0

∫

R2

∫ π

−πp cos2 θ dθ dx − λG1

+λI1(κ)

N0I0(κ)

∫

R2

∫ π

−πx1 cos(θ′ − dτ sin θ′) p(θ′) dθ′ dx.

Expanding the above using the Taylor Series expansion of the trigonometric functions

(4.29) and the identity cos2 θ = 12(1 + cos 2θ), and then using the definition of Fn(t) and

Gn(t) in (4.15) and (4.16) respectively, gives

dG1

dt=s

2+s

2F2 − λ11G1 − as1G2 + as2G3 + as1x1 + h.o.t., (4.47)

where λ11, as1 and as2 are as defined in (4.41), and

x1 =

∫

R2

∫ π

−πx1p(θ

′) dθ′dx = H.ξ1. (4.48)

The higher order terms (h.o.t.) that have been rounded off may be dependent on x1, which

is itself time dependent.



Multiply the linear transport equation (3.11) by x1 cos 2θ, integrate over θ and x, and

divide by N0, to give

1

N0

∫

R2

∫ π

−πx1 cos 2θ

∂p

∂tdθ dx = − s

N0

∫

R2

∫ π

−πx1 cos 2θ (ξ.∇

xp) dθ dx

−λN0

∫

R2

∫ π

−πx1 cos 2θ p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−πx1 cos 2θ T (θ, θ′) p(θ′) dθ′ dθ dx.

We can substitute for G2(t) from (4.16) and substitute for T (θ, θ′) from (4.9), giving

dG2

dt= − s

N0

∫

R2

∫ π

−π(ξp).(∇

xx1 cos 2θ) dθ dx − λG2

+λ

N0

∫

R2

∫ π

−π

∫ π

−π

x1 cos 2θ




dG2

dt=

s

N0

∫

R2

∫ π

−πp cos θ cos 2θ dθ dx− λG2

+λI2(κ)

N0I0(κ)

∫

R2

∫ π

−πx1 cos(2θ′ − 2dτ sin θ′) p(θ′) dθ′ dx.


(4.30) and the identity cos θ cos 2θ = 12(cos θ + cos 3θ), and then using the definition of

Fn(t) and Gn(t) in (4.15) and (4.16) respectively, gives

dG2

dt=s

2F1 +

s

2F3 − λ2G2 + bs1G1 − bs1G3 + bs2G4 + bs2x1 + h.o.t., (4.49)

where λ2, bs1 and bs2 are as defined in (4.43), and x1 is as defined in (4.48).

The higher order terms (h.o.t.) that have been rounded off may be dependent on x1 which

is time dependent itself.

4.5.2.7 Differential equation for Z1(t)

Multiply the linear transport equation (3.11) by x2 sin θ, integrate over θ and x, and divide

by N0, to give

1

N0

∫

R2

∫ π

−πx2 sin θ

∂p

∂tdθ dx = − s

N0

∫

R2

∫ π

−πx2 sin θ (ξ.∇

xp) dθ dx

−λN0

∫

R2

∫ π

−πx2 sin θ p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−πx2 sin θ T (θ, θ′) p(θ′) dθ′ dθ dx.


We can substitute for Z1(t) from (4.18) and substitute for T (θ, θ′) from (4.9), giving

dZ1

dt= − s

N0

∫

R2

∫ π

−π(ξp).(∇

xx2 sin θ) dθ dx − λZ1

+λ

N0

∫

R2

∫ π

−π

∫ π

−π

x2 sin θ




dZ1

dt=

s

N0

∫

R2

∫ π

−πp sin2 θ dθ dx− λZ1

+λI1(κ)

N0I0(κ)

∫

R2

∫ π

−πx2 sin(θ′ − dτ sin θ′) p(θ′) dθ′ dx.


(4.31) and the identity sin2 θ = 12(1 − cos 2θ), and then using the definition of Fn(t) and

Zn(t) in (4.15) and (4.18) respectively, gives

dZ1

dt=s

2− s

2F2 − λ12Z1 − as1Z2 + as2Z3 + h.o.t., (4.50)

where λ12 is as defined in (4.45 and as1 and as2 are as defined in (4.41).


Multiply the linear transport equation (3.11) by x2 sin 2θ, integrate over θ and x, and


1

N0

∫

R2

∫ π

−πx2 sin 2θ

∂p

∂tdθ dx = − s

N0

∫

R2

∫ π

−πx2 sin 2θ (ξ.∇

xp) dθ dx

−λN0

∫

R2

∫ π

−πx2 sin 2θ p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−πx2 sin 2θ T (θ, θ′) p(θ′) dθ′ dθ dx.

We can substitute for Z2(t) from (4.18) and substitute for T (θ, θ′) from (4.9), giving

dZ2

dt= − s

N0

∫

R2

∫ π

−π(ξp).(∇

xx2 sin 2θ) dθ dx − λZ2

+λ

N0

∫

R2

∫ π

−π

∫ π

−π

x2 sin 2θ




dZ2

dt=

s

N0

∫

R2

∫ π

−πp sin θ sin 2θ dθ dx − λZ2

+λI2(κ)

N0I0(κ)

∫

R2

∫ π

−πx2 sin(2θ′ − 2dτ sin θ′) p(θ′) dθ′ dx.



(4.31) and the identity sin θ sin 2θ = 12(cos θ − cos 3θ), and then using the definition of

Fn(t) and Zn(t) in (4.15) and (4.18) respectively, gives

dZ2

dt=s

2F1 −

s

2F3 − λ2Z2 + bs1Z1 − bs1Z3 + bs2Z4 + h.o.t., (4.51)

where λ2, bs1 and bs2 are as defined in (4.43).

4.6 Closing and solving the system of equations for H(t),

V(t), Fn(t) and Yn(t)

The system of differential equations that we have derived is not closed. We have dif-

ferential equations for the higher order moments, Fn(t) and Yn(t), that are dependent

on successively higher order moments. We could carry on deriving differential equations

for these successively higher order moments until we had an extremely large ‘cascade’ of

equations. In principle this could be solved to any order as all the equations are linear,

but in practice this is likely to be very complicated.

The orientation distribution, p(θ, t), will on average be symmetric about the preferred

direction, θ = 0, and thus we would expect the angular moments Fn(t) and Yn(t) to

get smaller as n increases since they are integrals of a smooth function multiplied by

oscillating functions of decreasing wavelength and zero average (we shall demonstrate this

result in Chapter 7). We shall assume we can close the systems of equations up to O(d3τ )

by approximating the higher order moments, Fn(t), Yn(t) for n ≥ 3, as time independent

constants using the equation for the expected long-time angular distribution from Hill &

Hader (1997), see (1.100). This proves to be a reasonable approach as long as the higher

moments Fn(t) and Yn(t) for n ≥ 3, are not large. The assumption will no longer be valid

if the random walk is highly correlated (i.e. if the parameter κ is very large), see results

in Chapter 7.

4.6.1 Approximating the higher order moments

From (1.100), the expected long-time angular distribution for a population moving with

sinusoidal reorientation is given by

f(θ) =1

2πI0(z)e(z cos θ), (4.52)

where

z =2B−1

σ20

= − dτln(A1(κ))

, (4.53)

and A1(κ) = I1(κ)/I0(κ). Also, from (4.15) and (4.17) we have


Fn(t) =1

N0

∫

R2

∫ π

−πcosnθ p(x, θ, t) dθ dx,

Yn(t) =1

N0

∫

R2

∫ π

−πsinnθ p(x, θ, t) dθ dx.

To close the system we now make an assumption about the form of the underlying spatial

distribution p(x, θ, t). In general, it is not the case that the x and θ components of p(x, θ, t)

are independent, but we assume that when averaging over all positions x for these higher

moments we can approximate p(x, θ, t) by

p(x, θ, t) = f(θ)p(x, t), (4.54)

where f(θ) is the long-time angular distribution given by (4.52), and

∫ π

−πf(θ) dθ = 1, and

∫

R2

p(x, t) dx = 1. (4.55)

Thus the higher moments Fn(t) and Yn(t) for n ≥ 3 are given by

Fn(t) =1

N0

∫

R2

(∫ π

−π

cosnθ

2πI0(z)e(z cos θ) dθ

)

p(x, t) dx,

Yn(t) =1

N0

∫

R2

(∫ π

−π

sinnθ


)

p(x, t) dx,

and evaluating these using (4.19) and (4.20) gives

Fn(t) =In(z)

I0(z)= An(z), (4.56)

Yn(t) = 0. (4.57)

To close the systems of equations in (4.40)—(4.46), we use the time independent approx-

imations F3(t) = A3(z), F4(t) = A4(z), and Y3(t) = Y4(t) = 0, where z = −dτ/ ln(A1(κ)).

The differential equations for F1(t) and F2(t), (4.40) and (4.42), can now be written as

dF1

dt= −λ11F1 − as1F2 + as1 + as2A3(z) +O(d3

τ ), (4.58)

dF2

dt= bs1F1 − λ2F2 − bs1A3(z) + bs2 + bs2A4(z) +O(d3

τ ), (4.59)

and similarly for Y1(t) and Y2(t), (4.44) and (4.46) become

dY1

dt= −λ12Y1 − as1Y2 +O(d3

τ ), (4.60)

dY2

dt= bs1Y1 − λ2Y2 +O(d3

τ ). (4.61)

4.6.2 The general solution to a linear system of differential equations

If x is a vector with n components, A is an n × n matrix, and b(t) is a vector with n

components which may or may not be dependent on t, then the linear differential equation

x = Ax + b, (4.62)


is straightforward to solve by rearranging first.

If the n eigenvalues of A are given by ψi = ψ1, . . . , ψn, then we define D to be the n × n

diagonal matrix of eigenvalues (where the i-th diagonal entry is given by ψi and all other

entries are zero) and P to be the corresponding n × n matrix of eigenvectors. It is a

standard result in linear algebra (Lipschutz, 1987) that

D = P−1AP. (4.63)

Left-multiplying (4.62) by the inverse of P and using the fact that P−1P = I, where I is

the identity matrix, gives

P−1x = P−1APP−1x + P−1b, (4.64)

and using (4.63) this becomes

P−1x = DP−1x + P−1b, (4.65)

We now define y = P−1x to give the final equation

y = Dy + P−1b. (4.66)

The solution of (4.66) is given by

y =(

e−Dt)

−1∫

e−DtP−1b dt+(

e−Dt)

−1c, (4.67)

where c is a constant vector and e−Dt is the n×n diagonal matrix where the i-th diagonal

entry is given by e−ψit and all other entries are zero. The solution in (4.67) is then left-

multiplied by P to give x = Py.

4.6.3 Solving the final system of equations for H(t), V(t), Fn(t) and Yn(t)

Now that we have closed the original system of equations up to O(d3τ ), it is straightforward

to solve.

4.6.3.1 Solving equations for F1(t) and F2(t)

The differential equations in (4.58) and (4.59) form a linear coupled system. Together

with the initial conditions F1(0) = F2(0) = 0, this is solved using the method described in

Section 4.6.2. The solutions to leading order are

F1(t) = Af1

(

1 − e−φ1t)

+Bf1

(

1 − e−φ2t)

, (4.68)

and

F2(t) = Af2

(

1 − e−φ1t)

+Bf2

(

1 − e−φ2t)

, (4.69)

where

φ1 =1

2

(

λ11 + λ2 −√

(λ11 − λ2)2 − 4as1bs1

)

, (4.70)


φ2 =1

2

(

λ11 + λ2 +√

(λ11 − λ2)2 − 4as1bs1

)

, (4.71)

Af1 =λ2 − φ1

bs1φ1(φ2 − φ1)

(

bs1Cf1 − (λ2 − φ2)Cf2

)

, (4.72)

Bf1 =−(λ2 − φ2)

bs1φ2(φ2 − φ1)

(

bs1Cf1 − (λ2 − φ1)Cf2

)

, (4.73)

Af2 =1

φ1(φ2 − φ1)

(

bs1Cf1 − (λ2 − φ2)Cf2

)

, (4.74)

Bf2 =−1

φ2(φ2 − φ1)

(

bs1Cf1 − (λ2 − φ1)Cf2

)

, (4.75)

where the terms Cf1 and Cf2 correspond to the constant terms in the revised differential

equations (4.58) and (4.59) and are given by

Cf1 = (as1 + as2A3(z)) (4.76)

Cf2 =(

bs2 + bs2A4(z) − bs1A3(z))

(4.77)

and all other constant terms are as defined in (4.41) or (4.43).

Note that in the solutions (4.68) and (4.69), φ1 > 0 and φ2 > 0, and thus as t → ∞ the

long-time limiting solutions are

F1(∞) = Af1 +Bf1, (4.78)

F2(∞) = Af2 +Bf2, (4.79)

and thus the solutions tend to constant positive values that must be ≤ 1 as we shall see

later.

Solution for complex eigenvalues

If (λ11 −λ2)2 − 4as1bs1 < 0 then the eigenvalues in (4.68) and (4.69) are complex and take

the form φ1 = a + bi and φ2 = a − bi (where i =√−1). Recalling the definitions of the

constant terms from (4.41) and (4.43), one can see that this is likely to occur only if the

reorientation parameters κ and dτ are both large. Although we shall show later that these

complex-eigenvalue solutions are still valid in the sense that they give real-valued solutions

that are a reasonable match to simulation results, it seems likely that this behaviour occurs

because of the breakdown of the assumption that dτ is small.

The equations (4.68) and (4.69) give real valued solutions that are straightforward to

rearrange into the following form

F1C(t) = Af1C

(

1 − e−at cos(bt))

+Bf1Ce−at sin(bt), (4.80)

and

F2C(t) = Af2C

(


+Bf1Ce−at sin(bt), (4.81)


where

Af1C =Cf1bs1λ2 + Cf2(2aλ2 − a2 − b2 − λ2

2)

bs1(a2 + b2), (4.82)

Bf1C =Cf1bs1(a

2 − aλ2 + b2) + Cf2(a3 − 2a2λ2 + aλ2

2 + ab2)

bs1b(a2 + b2), (4.83)

Af2C =Cf1bs1 − Cf2(λ2 − 2a)

a2 + b2, (4.84)

Bf2C =−Cf1abs1 + Cf2(b

2 − a2 + aλ2)

b(a2 + b2). (4.85)

Since a > 0 is the real part of the eigenvalues, the solutions (4.80) and (4.81) will tend to

long-time limiting solutions given by

F1C(∞) = Af1C , (4.86)

F2C(∞) = Af2C , (4.87)

which must both be ≤ 1.

4.6.3.2 Solving equations for Y1(t) and Y2(t)


with the initial conditions Y1(0) = Y2(0) = 0, this gives the trivial solutions

Y1(t) = Y2(t) = 0. (4.88)

This is not unexpected as the Yn(t) terms are the angular moments associated with move-

ment in the direction perpendicular to the preferred direction, which we would expect to

be zero on average.

4.6.3.3 Solution for V(t)

From the definitions in (4.11), (4.15) and (4.17), one can see that if all Yn terms are zero

then

V(t) = sF1(t)ξ1, (4.89)

and substituting for F1(t) from (4.68) gives the leading order solution

V(t) = s(

Af1

(

1 − e−φ1t)

+Bf1

(

1 − e−φ2t))

ξ1, (4.90)

where the constants Af1 and Bf1 are as defined in (4.72) and (4.73).

If we have complex eigenvalues then from (4.80) we can write (4.90) as

VC(t) = s(

Af1C

(


+Bf1Ce−at sin(bt)

)

ξ1, (4.91)

where the constants Af1C and Bf1C are as defined in (4.82) and (4.83).


Thus, as t→ ∞ we have

V(∞) = s(Af1 +Bf1)ξ1 or sAf1Cξ1, (4.92)

and therefore to avoid a nonsensical solution, we must have Af1 +Bf1 ≤ 1 and Af1C ≤ 1,

or else the average velocity will be greater than the speed of movement.

4.6.3.4 Solution for H(t)

From (4.35) we havedH

dt= V.

Substituting our solution for V(t) from (4.90) gives

dH

dt= s

(

Af1

(

1 − e−φ1t)

+Bf1

(

1 − e−φ2t))

ξ1.

Integrating this and using the initial condition that H(0) = 0, gives the leading order

solution

H(t) = s

(

(Af1 +Bf1)t−Af1

φ1

(

1 − e−φ1t)

− Bf1

φ2

(

1 − e−φ2t)

)

ξ1, (4.93)

where Af1 and Bf1 are as given in (4.72) and (4.73).

If we have complex eigenvalues then (4.93) can be written as

H(t) = s

(

Af1Ct−(aAf1C − bBf1C)

a2 + b2

(


)

ξ1

−s(

(bAf1C + aBf1C)

a2 + b2e−at sin(bt)

)

ξ1, (4.94)

where Af1C and Bf1C are as given in (4.82) and (4.83).

As t→ ∞ the average position H(t) behaves as

H(∞) ∼ s(Af1 +Bf1)t ξ1 or sAf1Ctξ1, (4.95)

and thus the average absolute velocity, H(t)/t, will have the same limiting behaviour as

V(t).

If dτ = 0 and there is no bias, then (4.93) has solution H(t) = 0, and all the walkers will

simply diffuse away from the origin — there is no drift.

4.7 Closing and solving the system of equations for D2(t),

Gn(t) and Zn(t)

As with the system of differential equations for H(t) and associated moments, the system

of differential equations that we have derived is not closed. We have differential equations

for the higher order moments, Gn(t) and Zn(t), that are dependent on successively higher


order moments. We could carry on deriving differential equations for these successively

higher order moments until we had an extremely large ‘cascade’ of equations. As with

the previous system, in theory this would be possible to solve to a high order as all the

equations are linear, but in practice this is likely to be very complicated.

As with the angular moments, Fn(t) and Yn(t), we would expect the moments Gn(t) and

Zn(t) to get smaller as n increases. Unlike the previous system however, the moments are

dependent on both the angle and position. For n = 1, it is clear that we cannot assume

that the spatial distribution p(x, θ, t) can be split into f(θ)p(x, t), as for example in the

extreme case when the x1 value is very large and negative, it is much more likely that

cos θ is also negative (since in order to reach a point where x1 is large and negative the

walker will have had to move almost directly along an angle close to θ = ±π and hence

cos θ is likely to be negative). For the higher moments (n > 1), this ‘correlation’ between

the angle and spatial position will become weaker and we make a similar approximation

as for the previous system.

4.7.1 Approximating the higher order moments

In a similar way to Section 4.6.1 we assume that for n ≥ 3 we can approximate p(x, θ, t)

by

p(x, θ, t) = f(θ)p(x, t), (4.96)

where f(θ) is the long-time angular distribution given by (4.52), and∫ π

−πf(θ) dθ = 1, and

∫

R2

p(x, t) dx = 1. (4.97)

Thus from (4.16) and (4.18), the higher moments Gn(t) and Zn(t) for n ≥ 3 are given by

Gn(t) =1

N0

∫

R2

(∫ π

−π

cosnθ


)

x1p(x, t) dx,

Zn(t) =1

N0

∫

R2

(∫ π

−π

sinnθ


)

x2p(x, t) dx,

and evaluating these using (4.19) and (4.20) gives

Gn(t) =In(z)

I0(z)x1 = An(z)x1, (4.98)

Zn(t) = 0, (4.99)

where x1 = H.ξ1 from (4.48).

To close the systems of equations in (4.47)—(4.51), we use the approximations F3(t) =

A3(z), G3(t) = A3(z)x1, G4(t) = A4(z)x1, and Z3(t) = Z4(t) = 0, where z = −dτ/ ln(A1(κ)).

The differential equations for G1(t) and G2(t), (4.47) and (4.49), can now be written as

dG1

dt=

s

2+s

2F2 − λ11G1 − as1G2 + (as1 + as2A3(z)) x1 + h.o.t., (4.100)

dG2

dt=

s

2F1 +

s

2A3(z) − λ2G2 + bs1G1 +

(

bs2 + bs2A4(z) − bs1A3(z))

x1

+h.o.t., (4.101)


and similarly for Z1(t) and Z2(t), (4.50) and (4.51) become

dZ1

dt=

s

2− s

2F2 − λ12Z1 − as1Z2 + h.o.t., (4.102)

dZ2

dt=

s

2F1 −

s

2A3(z) − λ2Z2 + bs1Z1 + h.o.t. (4.103)

4.7.2 Solving the final system of equations for D2(t), Gn(t) and Zn(t)

Now that we have closed the original system of equations up to leading order it is straight-

forward to solve.

4.7.2.1 Solving equations for G1(t) and G2(t)


with the initial conditions G1(0) = G2(0) = 0 and the solutions from (4.68), (4.69) and

(4.93), this is straightforward to solve using the method described in Section 4.6.2. The

solution for G1(t) to leading order is

G1(t) = s(

(Af1 +Bf1)2t+Ag1

(

1 − e−φ1t)

+Bg1(

1 − e−φ2t)

+ Cg1te−φ1t

+Dg1te−φ2t + Eg1

(

e−φ1t − e−φ2t))

, (4.104)

where

Ag1 = −2A2

f1

φ1−Af1Bf1

(

1

φ1+

1

φ2

)

+(λ2 − φ1)

2φ1(φ2 − φ1)(1 +Af2 +Bf2)

−(λ2 − φ1)(λ2 − φ2)

2bs1φ1(φ2 − φ1)(Af1 +Bf1 +A3(z)) , (4.105)

Bg1 = −2B2

f1

φ2−Af1Bf1

(

1

φ1+

1

φ2

)

− (λ2 − φ2)

2φ2(φ2 − φ1)(1 +Af2 +Bf2)

+(λ2 − φ1)(λ2 − φ2)

2bs1φ2(φ2 − φ1)(Af1 +Bf1 +A3(z)) , (4.106)

Cg1 = A2f1 +

(λ2 − φ1)

2bs1(φ2 − φ1)

(

Af1(λ2 − φ2) −Af2bs1)

, (4.107)

Dg1 = B2f1 −

(λ2 − φ2)

2bs1(φ2 − φ1)

(

Bf1(λ2 − φ1) −Bf2bs1)

, (4.108)

Eg1 = −(λ2 − φ1)

(

bs1(Bf2 − 2Cf1Bf1/φ2) − (λ2 − φ2)(Bf1 − 2Cf2Bf1/φ2))

2bs1(φ2 − φ1)2

+(λ2 − φ2)

(

bs1(Af2 − 2Cf1Af1/φ1) − (λ2 − φ1)(Af1 − 2Cf2Af1/φ1))

2bs1(φ2 − φ1)2,

(4.109)

and φ1, φ2, Af1, Bf1, Af2, and Bf2 are as defined in (4.70) — (4.75).

The solution for G2(t) is of a similar form to (4.104) but is omitted as it is not required

when solving the subsequent differential equations for the spread.

Equation (4.104) gives real-valued solutions if the eigenvalues φi are complex conjugates.


4.7.2.2 Solving equations for Z1(t) and Z2(t)


with the initial conditions Z1(0) = Z2(0) = 0 and the solutions from (4.68) and (4.69),

this is straightforward to solve using the method described in Section 4.6.2. The solution

for Z1(t) to leading order is

Z1(t) = s(

Az1(

1 − e−φ3t)

+Bz1(

1 − e−φ4t)

+ Cz1(

e−φ1t − e−φ3t)

+ Dz1

(


+ Ez1(


+Fz1(

e−φ2t − e−φ4t))

, (4.110)

where

φ3 =1

2

(

λ12 + λ2 −√

(λ12 − λ2)2 − 4as1bs1

)

, (4.111)

φ4 =1

2

(

λ12 + λ2 +√

(λ12 − λ2)2 − 4as1bs1

)

, (4.112)

Az1 =(λ2 − φ3)

2bs1φ3(φ4 − φ3)

(

bs1(1 −Af2 −Bf2) − (λ2 − φ4)(Af1 +Bf1 −A3(z)))

, (4.113)

Bz1 = − (λ2 − φ4)

2bs1φ4(φ4 − φ3)

(

bs1(1 −Af2 −Bf2) − (λ2 − φ3)(Af1 +Bf1 −A3(z)))

, (4.114)

Cz1 =(λ2 − φ3)

2bs1(φ3 − φ1)(φ4 − φ3)

(

bs1Af2 + (λ2 − φ4)Af1

)

, (4.115)

Dz1 =(λ2 − φ3)

2bs1(φ3 − φ2)(φ4 − φ3)

(

bs1Bf2 + (λ2 − φ4)Bf1

)

, (4.116)

Ez1 = − (λ2 − φ4)

2bs1(φ4 − φ1)(φ4 − φ3)

(

bs1Af2 + (λ2 − φ3)Af1

)

, (4.117)

Fz1 = − (λ2 − φ4)

2bs1(φ4 − φ2)(φ4 − φ3)

(

bs1Bf2 + (λ2 − φ3)Bf1

)

, (4.118)

and φ1, φ2, Af1, Bf1, Af2, and Bf2 are as defined in (4.70) — (4.75).

The solution for Z2(t) is of a similar form but is omitted as it is not required when solving

the subsequent differential equations for the spread.

Equation (4.110) gives real-valued solutions if the eigenvalues φi are complex conjugates.

4.7.2.3 Solving equation for D2x1(t)

We havedD2

x1

dt= 2sG1. (4.119)

Substituting for G1(t) from (4.104), integrating, and recalling the initial condition that

D2x1(0) = 0, gives

D2x1(t) = 2s2

(

(Af1 +Bf1)2

2t2 +Ag1

(

t− 1

φ1(1 − e−φ1t)

)

+Bg1

(

t− 1

φ2(1 − e−φ2t)

)


−Cg1φ1

(

te−φ1t − 1

φ1(1 − e−φ1t)

)

− Dg1

φ2

(

te−φ2t − 1

φ2(1 − e−φ2t)

)

+Eg1

(

1

φ1(1 − e−φ1t) − 1

φ2(1 − e−φ2t)

))

, (4.120)

where the constant terms are as given in (4.105) — (4.109), and φ1, φ2, Af1, Bf1, Af2,

and Bf2 are as defined in (4.70) — (4.75).

As t→ ∞ the spread about the origin in the preferred direction, D2x1(t), behaves as

D2x1(t) ∼ s2(Af1 +Bf1)

2t2, (4.121)

and thus the limiting behaviour of the solution is such that D2x1(t) ∼ t2.

4.7.2.4 Solving equation for D2x2(t)

We havedD2

x2

dt= 2sZ1. (4.122)

Substituting for Z1(t) from (4.110), integrating, and recalling the initial condition that

D2x2(0) = 0, gives

D2x2(t) = 2s2

(

(Az1 +Bz1)t−1

φ3(Az1 + Cz1 +Dz1)

(

1 − e−φ3t)

− 1

φ4(Bz1 +Ez1 + Fz1)

(

1 − e−φ4t)

+1

φ1(Cz1 + Ez1)

(

1 − e−φ1t)

+1

φ2(Dz1 + Fz1)

(

1 − e−φ2t)

)

, (4.123)

where the constant terms are as given in (4.113) — (4.118).

As t→ ∞ the spread about the origin in the non-preferred direction, D2x2(t), behaves as

D2x2(t) ∼ 2s2(Az1 +Bz1)t, (4.124)

and thus the limiting behaviour of the solution is such that D2x2(t) ∼ t — the characteristic

behaviour of a diffusive process.

4.7.2.5 Solving equation for D2(t)

From the definitions in (4.12) — (4.14), we have

D2(t) = D2x1(t) +D2

x2(t), (4.125)

so that the total spread D2(t) is found by adding the equations for the spread in each

direction (4.120) and (4.123).

If dτ = 0 and there is no bias in the system, then the original differential equations for

G1(t) and Z1(t), (4.47) and (4.50), reduce to

dG1

dt=s

2− λ0G1, (4.126)

dZ1

dt=s

2− λ0Z1, (4.127)


where λ0 = λ(

1 − I1(κ)I0(κ)

)

. This gives the solutions

D2x1(t) = D2

x2(t) =s2

λ0

(

t− 1

λ0(1 − e−λ0t)

)

, (4.128)

and hence the spread about the origin is the same in each direction, and our solution for

D2(t) is the same as in the model of Othmer et al. (1988) with no bias (3.42).

In the absence of bias the spread about the origin in both directions will increase as the

random walk becomes more correlated (κ→ ∞), up to the limiting value of D2(t) = (st)2.

This corresponds to straight line motion directly from the origin along the initial direction

of facing. If bias is present, then this will no longer be true in general as the spread about

the origin will be greater in the preferred direction and if we increase κ to a large enough

value then the spread in the non-preferred direction will start to decrease.

4.7.3 Equations for the spread about the mean position

Now that we have solutions for H(t), D2x1(t) and D2

x2(t) we can solve the equation for

σ2(t). From the definition of σ2x1 and σ2

x2 as the spread about the mean position in the

preferred and non-preferred directions respectively, we have

σ2x1(t) = D2

x1(t) − ‖Hx1(t)‖2, (4.129)

σ2x2(t) = D2

x2(t) − ‖Hx2(t)‖2, (4.130)

where the total spread about the mean position is given by

σ2(t) = σ2x1(t) + σ2

x2(t). (4.131)

4.7.3.1 Equation for σ2x1

The solution for σ2x1 is found by substituting for (4.120) and (4.93) into (4.129). Although

we omit the full solution here, it should be noted that the t2 terms in D2x1(t) and ‖Hx1(t)‖2

cancel so that σ2x1 ∼ t, and the spread about the mean position in the preferred direction

has the characteristic behaviour of a diffusive process (unlike the spread about the origin

in the preferred direction D2x1(t)).

The equation for σ2x1(t) is likely to be highly sensitive to errors in our other equations. For

example, a small relative error in D2x1(t) (which is large) and a small error in H(t) which

is then squared, can result in a large relative error in σ2x1(t). This will occur when the

assumptions we have made about the reorientation parameters being small are no longer

valid.

4.7.3.2 Equation for σ2x2

The mean position in the non-preferred direction has been shown to be zero, Hx2(t) = 0

for all t. Thus the spread about the mean position in the non-preferred direction is the


same as the spread about the origin in the non-preferred direction,

σ2x2(t) = D2

x2(t). (4.132)

If there is no bias in the system, then the average position H(t) is zero in both directions

and σ2x1 = σ2

x2 = D2x1(t) = D2

x2(t) as given in (4.128).

4.8 Solution plots

Our final solutions for the statistics of interest are of a similar form as the statistic equa-

tions in the model of Othmer et al. (see (3.39)-(3.43)), but with different constant coeffi-

cients and different exponentially decaying terms as we have included higher moments in

our system. The limiting behaviour of our solutions and those of Othmer et al. have the

same form.

The following plots show the general behaviour of the solutions to the equations for the

statistics of interest that we have derived. For each plot we have fixed λ = 1 and s = 1,

and either dτ = 0.1 or dτ = 0.3, and then plotted solutions for κ = 0.1, κ = 1, κ = 2,

κ = 4, and κ = 8 in order to illustrate a wide range of parameter values.

To compare the statistics for the spread in different directions D2x1 and D2

x2, and σ2x1 and

σ2x2, we have plotted solutions for dτ = 0.3 only, as for smaller values of dτ there is less

difference between the spread in each direction.

The plots showing V(t) and H(t) show the expected values for the statistics in the preferred

direction only (x1). The expected values of these statistics in the x2-direction are zero.

4.8.1 Comment on solutions

In general all the plots show sensible behaviour except for the extreme parameter values

in our range — dτ = 0.3 and κ = 8. In this case we start to get nonsensical results —

our assumption that we can round off and/or approximate higher order terms is no longer

valid.

From Figures 4.1 and 4.2, it is clear that the solution for V(t) quickly tends to a fixed

limiting value and H(t) quickly tends to a linear time dependent solution with a fixed

gradient. Both the fixed value for V(t) and the fixed gradient for H(t) are given by

s(Af1 +Bf1) which has a maximum value of 1.

From Figures 4.3 and 4.4, one can see that the solution for D2(t) increases as both κ and

dτ increase, but the solution for σ2(t) decreases as dτ increases, and for dτ = 0.3 starts to

decrease for larger values of κ also. If bias is present, then more of the motion is used to

move in the preferred direction and there is less diffusion away from the mean position.


k=0.1k=1k=2k=4k=8

0

0.1

0.2

0.3

0.4

0.5

V(t)

20 40 60 80 100

t

(a) dτ = 0.1.

k=0.1k=1k=2k=4k=8

0

0.2

0.4

0.6

0.8

V(t)

20 40 60 80 100

t

(b) dτ = 0.3.

Figure 4.1: Plot of V(t) for dτ = 0.1 and dτ = 0.3 and various values of κ. (The scale of

each plot is different)

k=0.1k=1k=2k=4k=8

0

10

20

30

40

50

H(t)

20 40 60 80 100

t

(a) dτ = 0.1.

k=0.1k=1k=2k=4k=8

0

20

40

60

80

H(t)

20 40 60 80 100

t

(b) dτ = 0.3.

Figure 4.2: Plot of H(t) for dτ = 0.1 and dτ = 0.3 and various values of κ. (The scale of



k=0.1k=1k=2k=4k=8

0

1000

2000

3000

4000

20 40 60 80 100

t

(a) dτ = 0.1.

k=0.1k=1k=2k=4k=8

0

1000

2000

3000

4000

5000

6000

7000

20 40 60 80 100

t

(b) dτ = 0.3.

Figure 4.3: Plot of D2(t) for dτ = 0.1 and dτ = 0.3 and various values of κ. (The scale of


k=0.1k=1k=2k=4k=8

0

200

400

600

800

1000

1200

1400

20 40 60 80 100

t

(a) dτ = 0.1.

k=0.1k=1k=2k=4k=8

0

100

200

300

400

500

20 40 60 80 100

t

(b) dτ = 0.3.

Figure 4.4: Plot of σ2(t) for dτ = 0.1 and dτ = 0.3 and various values of κ. (The scale of



k=0.1k=1k=2k=4k=8

0

1000

2000

3000

4000

5000

6000

7000

20 40 60 80 100

t

(a) D2

x1(t).

k=0.1k=1k=2k=4k=8

0

50

100

150

200

250

300

20 40 60 80 100

t

(b) D2

x2(t).

Figure 4.5: Plot of D2x1(t) and D2



k=0.1k=1k=2k=4k=8

–50

0

50

100

150

200

250

20 40 60 80 100

t

(a) σ2

x1(t).

k=0.1k=1k=2k=4k=8

0

50

100

150

200

250

300

20 40 60 80 100

t

(b) σ2

x2(t).

Figure 4.6: Plot of σ2x1(t) and σ2




From Figures 4.5 and 4.6, our equations show that the spread will increase as κ increases

for small κ, but for larger values the spread will start to decrease if κ is increased further.

The plot for σ2x1(t) clearly shows that when κ = 8 our model breaks down as, although

we expect a small spread when we have large values for the parameters and ‘straight-line’

motion, it is nonsensical to have ‘negative’ spread. We have already discussed how the

equation for σ2x1(t) is likely to be highly sensitive to errors in the other statistic equations

even if these errors are of a small relative size.

More detailed studies of how accurate these equations are and the effect of the parameters

on the solutions are given in Chapter 6.

4.9 Working with the equations for the statistics of interest

4.9.1 Limitations of the model and solutions

Although we have found some useful general results and we have equations to find the

statistics of interest we should be aware of the assumptions we have made during the

modelling process and how this introduces limitations to the model.

• We have assumed 0 ≤ dτ ≪ 1 so that we ignored some higher order dnτ terms in the

asymptotic expansion. This may make our solutions less accurate as these higher

order terms are not taken into account in the final equations. We would expect our

solutions to become less accurate as the assumption that dτ ≪ 1 becomes less valid

(i.e. if we make dτ large).

• To close the systems of differential equations in (4.40) — (4.50) we first assumed that

the underlying spatial distribution p(x, θ, t) could be approximated by p(x, t)f(θ)

and then approximated the higher moments using the equation for the expected long-

time angular distribution f(θ) from Hill & Hader (1.100). These approximations are

likely to become less valid as the reorientation parameters κ and dτ increase.

• It is clear from solution plots that our solution equations break down for large values

of the parameters κ and dτ due to either or both of the above assumptions. The

equation for σ2x1(t) is highly sensitive to errors in the other equations and we get

nonsensical results for this equation with large values of the parameters.

4.9.2 Rescaling the equations

The equations for the statistics of interest that were derived in the previous section for the

statistics of interest are dependent on time t, and four parameters — the turning frequency

λ, the fixed speed of movement s, and the non-dimensional reorientation parameters κ and


dτ . The first two parameters can be scaled out of the equations to make them simpler.

The equations will then only be dependent on time and the parameters κ and dτ .

From (4.70), (4.71), (4.111) and (4.112), one can see that the eigenvalues φi are all multi-

ples of λ. It can be seen from (4.72) — (4.75), (4.105) — (4.109) and (4.113) — (4.118)

that the constant terms Af1, Bf1, Af2 and Bf2 are all non-dimensional, while the constant

terms Ag1, Az1 etc are all multiples of 1/λ.

Let us now rewrite φi in the form

φi = λψi for i = 1, 2, 3, 4, (4.133)

and let Ag1 = λAg1, Az1 = λAz1 etc.

We now use the following rescalings

t = λt so t =t

λ,

V(t) = V(t)/s so V(t) = V(t)s,

H(t) = H(t)λ/s so H(t) = H(t)s/λ,

D2x1(t) = D2

x1(t)λ2/s2 so D2

x1(t) = D2x1(t)s

2/λ2,

D2x2(t) = D2

x2(t)λ2/s2 so D2

x2(t) = D2x2(t)s

2/λ2. (4.134)

Using these rescalings, the equation for the average velocity V(t), (4.90), becomes

V(t)s = s(

Af1

(

1 − e−λψ1 t/λ)

+Bf1

(

1 − e−λψ2 t/λ))

ξ1, (4.135)

which simplifies to

V(t) =(

Af1

(

1 − e−ψ1 t)

+Bf1

(

1 − e−ψ2t))

ξ1. (4.136)

In a similar way, the equation for the average position H(t), (4.93), rescales to

H(t) =

(

(Af1 +Bf1)t−Af1

ψ1

(

1 − e−ψ1 t)

− Bf1

ψ2

(

1 − e−ψ2 t)

)

ξ1, (4.137)

and the equations for the spread about the origin in each direction (4.120) and (4.123)

rescale to

D2x1(t) = 2

(

(Af1 +Bf1)2

2t2 + Ag1

(

t− 1

ψ1(1 − e−ψ1 t)

)

+ Bg1

(

t− 1

ψ2(1 − e−ψ2 t)

)

− Cg1ψ1

(

te−ψ1 t − 1

ψ1(1 − e−ψ1 t)

)

− Dg1

ψ2

(

te−ψ2 t − 1

ψ2(1 − e−ψ2 t)

)

+Eg1

(

1

ψ1(1 − e−ψ1 t) − 1

ψ2(1 − e−ψ2 t)

))

, (4.138)

and

D2x2(t) = 2

(

(Az1 + Bz1)t−1

ψ3(Az1 + Cz1 + Dz1)

(

1 − e−ψ3 t)

− 1

ψ4(Bz1 + Ez1 + Fz1)

(

1 − e−ψ4 t)

+1

ψ1(Cz1 + Ez1)

(

1 − e−ψ1 t)

+1

ψ2(Dz1 + Fz1)

(

1 − e−ψ2 t)

)

. (4.139)


We also define the non-dimensionalised equation for the spread about the mean position

to be

σ2x1(t) = D2

x1(t) − ‖Hx1(t)‖2, (4.140)

σ2x2(t) = D2

x2(t) − ‖Hx2(t)‖2. (4.141)

The rescaled equations can also be obtained by setting the speed to be s = 1 and the

turning frequency to be λ = 1 in the original equations (4.90), (4.93), (4.120) and (4.123),

and changing the time and distance units accordingly.

The non-dimensionalised equations are now dependent on only two parameters — κ and

dτ .

4.9.3 Limits on the parameters

We have observed two possibly unrealistic properties of our solution equations that occur

with large values of the reorientation parameters.

4.9.3.1 Complex eigenvalues

The equations for the spatial statistics derived previously allowed for complex eigenvalues.

The solutions, (4.91) and (4.94), are real valued and there is no reason to assume that they

are not valid. However, it is not clear whether this behaviour is part of the underlying

behaviour of the system, or as seems more likely, is due to the rounding assumptions made

during the derivation of the equations.

From (4.70) and (4.71), the function inside the square root that gives rise to the complex

eigenvalues is given by

ζ1 = (λ11 − λ2)2 − 4as1bs1, (4.142)

where from (4.41) and (4.43) all the constant terms are dependent on the reorientation

parameters κ and dτ . The function ζ1 is of the above form because we have made the

rounding assumptions that dτ is small to enable an asymptotic expansion, and that we

can approximate higher moments as fixed constants using Hill & Hader’s equation for the

long-time angular distribution. With these assumptions we have a closed coupled system

of two differential equations.

Figure 4.7 shows how the non-dimensionalised (s = 1, λ = 1) function ζ1 changes and

becomes negative (resulting in complex eigenvalues), as κ increases for dτ = 0, 0.1, 0.2, 0.3.

Using (4.111) and (4.112), a very similar plot is obtained for φ3 and φ4, where the λ11

terms in (4.142) are replaced by λ12.

From Figure 4.7 it is clear that the larger values of dτ result in complex eigenvalues for

smaller values of κ, while if dτ = 0 the eigenvalues are never complex and ζ1 → 0 as

κ→ ∞.


d=0d=0.1d=0.2d=0.3

–0.1

–0.05

0

0.05

0.1

0.15

2 4 6 8 10 12 14 16 18 20

k

Figure 4.7: Plot of ζ1 against κ for dτ = 0, 0.1, 0.2, 0.3.

As κ gets larger our assumption that we can approximate the higher order moments such

as Fn(t) for n > 2 will become less valid. All the higher moments Fn → 1 ∀n as κ → ∞.

As dτ gets larger our assumption that dτ ≪ 1 so that we can complete an asymptotic

expansion to a leading order will become less valid.

Thus, it seems likely that the complex eigenvalues that are obtained when we have larger

values of the reorientation parameters are due to the two rounding assumptions made and

are not part of the underlying behaviour of the system, although we have not proved this

to be true.

4.9.3.2 Negative spread

There is no doubt that our solution equation for σ2x1(t) breaks down for larger values of the

reorientation parameters as can be seen in Figure 4.6. Since σ2x1(t) = D2

x1(t) − ‖H(t)‖2,

a small relative error in the equations for H(t) and D2x1 can result in a large error in

σ2x1(t). The errors in the equations are introduced because of the two rounding assumptions

introduced to close the system.

The solution for σ2x1(t) is linear in t and from (4.93) and (4.120), the leading order term

is given by ζ2t where

ζ2 = Ag1 +Bg1 − (Af1 +Bf1)

(

Af1

φ1+Bf1

φ2

)

, (4.143)

and the constants are as defined earlier in (4.72) etc. If ζ2 becomes negative then the long-

time solution for σ2x1(t) will become negative resulting in ‘negative spread’ as discussed

previously and seen in Figure 4.6.


Figure 4.8 shows how the non-dimensionalised (s = 1, λ = 1) function ζ2 changes and

becomes negative as κ increases for dτ = 0.1, 0.2, 0.3.

d=0.1d=0.2d=0.3

–0.5

0

0.5

1

1.5

2

2.5

3

2 4 6 8 10 12 14 16 18 20

k

Figure 4.8: Plot of ζ2 against κ for dτ = 0.1, 0.2, 0.3.

From Figure 4.8 it is clear that the larger values of dτ result in ‘negative spread’ for

smaller values of κ. Although the exact values of the reorientation parameters do not quite

correspond, there seems to be a parallel between the behaviour shown in Figures 4.7 and

4.8 — some of our equations become unrealistic or nonsensical if we choose reorientation

parameters that are too large.

4.10 Conclusions

We have shown in this chapter that it is possible to extend the velocity jump process

model and method of calculating moments used by Othmer et al. (1988). We have devel-

oped their model by using a reorientation probability distribution based on the von Mises

distribution with mean turning angle given by the sinusoidal reorientation model of Hill &

Hader (1997) that implicitly includes bias. The original probability distributions used by

Othmer et al. were chosen so that a closed system of differential equations was derived for

the moments of the underlying spatial distribution (see Chapter 3). When using a more re-

alistic reorientation probability distribution that includes sinusoidal reorientation, we have

shown in our analysis that the system of differential equations derived for the moments

of the spatial distribution is not closed, even when working to leading order by assuming

that the reorientation parameter dτ is small. We are able to close the system of differen-

tial equations by approximating the higher order moments using Hill & Hader’s equation

for the expected long-time angular distribution (1.100), and making an assumption about


the underlying spatial distribution. This results in a system of equations for the spatial

statistics (V(t), H(t), D2(t) and σ2(t)) of the population of random walkers that are valid

for all time and not just long-time approximations as in the diffusion model. Othmer et

al. did not derive equations for the spread in each direction, but we have extended the

analysis to include equations for D2x1(t), D

2x2(t), σ

2x1(t) and σ2

x2(t). The equations for the

spatial statistics are dependent on the turning frequency λ, the fixed speed of movement

s, and the reorientation parameters κ and dτ . Using suitable rescalings the equations can

be non-dimensionalised to be dependent on the reorientation parameters only. Because of

the various assumptions made during the analysis, the equations are not valid over all the

parameter space for κ and dτ , and if both are ‘large’ then the equations underestimate

the spread about the mean position, or give the nonsensical result of ‘negative spread.’

The sinusoidal reorientation model is motivated by the gyrotactic movement of algae such

as C. nivalis under a gravitational torque, but the model and solution equations presented

in this chapter could be used to describe the movement of any random walker moving in

a homogenous environment with sinusoidal reorientation.


• We have developed the velocity jump process of Othmer et al. to include a realistic

reorientation probability distribution that implicitly introduces bias to the movement

by including the dependence of the mean turning angle on the sine of the absolute

angle (sinusoidal reorientation model) from Hill & Hader’s experimental results.

• We have derived equations for the statistics of the underlying spatial distribution

that are valid for all time, these statistics being the mean velocity V(t), the mean

position H(t), the spread about the origin D2(t) and the spread about the mean

position σ2(t).

• We also have equations that are valid for all time for the spread in each of the

preferred and non-preferred directions, D2x1(t), D

2x2(t), σ

2x1(t) and σ2

x2(t).

• To close and solve the system of differential equations we had to make certain as-

sumptions about the reorientation parameters and the properties of the higher mo-

ments. We assumed that the parameter dτ was small to allow Taylor expansions of

the trigonometric functions and so all the equations are only leading order approxi-

mations. We also approximated the higher moments using Hill & Hader’s equation

for the expected long-time angular distribution in order to close the final system of

differential equations.

• Because of these assumptions, the equations for the spatial statistics are not valid

for large values of the reorientation parameters to avoid nonsensical results.

Chapter 5

Velocity jump processes using

linear reorientation

5.1 Introduction

In the previous chapter we have shown how it is possible to extend the velocity jump

process and method of calculating moments used by Othmer et al. (1988) to include a

reorientation distribution, T (θ, θ′), that has a mean turning angle given by the sinusoidal

reorientation model of Hill & Hader (1997). After analysing data from experiments on

algae, Hill & Hader suggested that the mean turning angle is dependent on the absolute

direction, and showed how this results in two models for reorientation — sinusoidal and

linear (Section 1.5). Linear reorientation has been observed in algae such as C. nivalis or

P. gatunese that move towards or away from light, a process known as phototaxis, see for

example Hill & Vincent (1993) and Vincent & Hill (1996). In this chapter we develop the

velocity jump process model and method of calculating the moments used by Othmer et

al. (1988) to derive equations for the moments of the underlying spatial distribution of

a population moving with a reorientation distribution, T (θ, θ′), that has a mean turning

angle given by Hill & Hader’s linear reorientation model.

5.2 Results and assumptions to be used in analysis

5.2.1 Reorientation model

As discussed in the introductory chapter, when looking at the trajectories of swimming

micro-organisms such as C. nivalis, Hill & Hader (1997) derived a probability distribution

function for the orientation of the cells, from which certain parameters can be calculated.

After analysing experimental results they showed that the mean turning angle is dependent

99

CHAPTER 5: Velocity jump processes using linear reorientation 100

on the direction of movement and for phototaxis is given by

µδ(θ) =

−B−1τθ, −π < θ < π,

0, θ = ±π,(5.1)

from (1.87) and (1.101), where τ is the average time between turns, B is the average

reorientation time and θ0 = 0 is the preferred direction. We define dτ and κ in the same

way as in the sinusoidal model (see (4.2) and (4.6)). We assume that the taxis is positive

(i.e. individuals prefer to move towards the light source), and that the amplitude of the

reorientation coefficient B−1 is fixed and not dependent on the spatial position. As in

the sinusoidal model, the symmetric probability distribution we use for the reorientation

distribution is the von Mises distribution (see Section 1.2.3.2).

To model the linear reorientation observed in phototaxis using the velocity jump process

model of Othmer et al. we substitute for Hill & Hader’s µδ = −dτθ (assuming that θ0 = 0

for convenience) and recognise that if we define δ = θ− θ′, the von Mises distribution f(δ)

can be used as the reorientation kernel T (θ, θ′). This gives

T (θ, θ′) =1

2πI0(κ)eκ cos(θ−θ′+dτ θ′) (5.2)

where∫ π

−πT (θ, θ′) dθ = 1.

Thus we have a probability distribution for reorientation that implicitly includes bias. We

now extend the velocity jump process model of Othmer at al. (1988) using our probability

distribution for reorientation that implicitly includes bias.

5.2.2 Defining higher order moments

The statistics of interest are the mean location of cells H(t), their mean squared displace-

ment D2(t), and their mean velocity V(t), which are defined as in (4.10)—(4.12). We also

derive equations for the statistics of the spread in each direction D2x1(t) and D2

x2(t) which

are defined in (4.13) and (4.14). In addition we recall the definition of σ2(t) (3.20).

We introduce the following extra auxiliary functions

Fn(t) =1

N0

∫

R2

∫ π

−πcosnθ p(x, θ, t) dθ dx, (5.3)

Yn(t) =1

N0

∫

R2

∫ π

−πsinnθ p(x, θ, t) dθ dx, (5.4)

Kn(t) =1

N0

∫

R2

∫ π

−πθ sinnθ p(x, θ, t) dθ dx, (5.5)

Ln(t) =1

N0

∫

R2

∫ π

−πθ cosnθ p(x, θ, t) dθ dx, (5.6)

Mn(t) =1

N0

∫

R2

∫ π

−πθ2 cosnθ p(x, θ, t) dθ dx, (5.7)

Nn(t) =1

N0

∫

R2

∫ π

−πθ2 sinnθ p(x, θ, t) dθ dx, (5.8)


Gn(t) =1

N0

∫

R2

∫ π

−πx1 cosnθ p(x, θ, t) dθ dx, (5.9)

Zn(t) =1

N0

∫

R2

∫ π

−πx2 sinnθ p(x, θ, t) dθ dx, (5.10)

Pn(t) =1

N0

∫

R2

∫ π

−πx1θ sinnθ p(x, θ, t) dθ dx, (5.11)

Qn(t) =1

N0

∫

R2

∫ π

−πx2θ cosnθ p(x, θ, t) dθ dx, (5.12)

Rn(t) =1

N0

∫

R2

∫ π

−πx1θ

2 cosnθ p(x, θ, t) dθ dx, (5.13)

Sn(t) =1

N0

∫

R2

∫ π

−πx2θ

2 sinnθ p(x, θ, t) dθ dx, (5.14)

which will be used in the subsequent analysis. The higher moments Fn(t), Gn(t), Yn(t)

and Zn(t) were used in the sinusoidal model and have been discussed previously. The

other moments are dependent on the functions θ sinnθ, θ cosnθ, θ2 cosnθ and θ2 sinnθ.

As in the sinusoidal model, we assume that all the population starts at the origin (0, 0)

and that the initial directions are equally distributed around the unit circle. Using this

assumption

Fn(0) = Gn(0) = Yn(0) = Zn(0) = 0,

and

Pn(0) = Qn(0) = Rn(0) = Sn(0) = 0.

The initial conditions for Kn(0), Ln(0), Mn(0) and Nn(0) are straightforward to derive.

If we start at the origin with an equal spread of directions then p(0, θ, 0) = 1/2π, and so

we have

K1(0) =1

N0

∫

R2

∫ π

−π

θ sin θ

2πdθ dx = 1,

L1(0) =1

N0

∫

R2

∫ π

−π

θ cos θ

2πdθ dx = 0,

M1(0) =1

N0

∫

R2

∫ π

−π

θ2 cos θ

2πdθ dx = −2.

N1(0) =1

N0

∫

R2

∫ π

−π

θ2 sin θ

2πdθ dx = 0,

K2(0) =1

N0

∫

R2

∫ π

−π

θ sin 2θ

2πdθ dx = −1

2,

L2(0) =1

N0

∫

R2

∫ π

−π

θ cos 2θ

2πdθ dx = 0,

M2(0) =1

N0

∫

R2

∫ π

−π

θ2 cos 2θ

2πdθ dx =

1

2,

and N2(0) =1

N0

∫

R2

∫ π

−π

θ2 sin 2θ

2πdθ dx = 0.

All other statistics we are interested in are assumed to be 0 at t = 0.


5.2.3 Integrals of the von Mises distribution

From (4.23) and (4.24), we use the following identities for the integrals of the modified

von Mises distribution∫ π

−π

cos pθ

2πI0(κ)eκ cos(θ−θ′+dτ θ′) dθ =

Ip(κ)

I0(κ)cos(pθ′ − pdτθ

′) (5.15)

∫ π

−π

sin pθ

2πI0(κ)eκ cos(θ−θ′+dτ θ′) dθ =

Ip(κ)

I0(κ)sin(pθ′ − pdτθ

′). (5.16)

5.2.4 Asymptotic expansions of the trigonometric functions

We assume that 0 ≤ dτ ≪ 1, and seek an asymptotic expansion in powers of dτ for the

trigonometric functions.

The standard Taylor Series expansions for the trigonometric functions that will be needed

in the later analysis are:

cos(nθ − ndτθ) = cosnθ + ndτθ sinnθ − n2d2τ

2θ2 cosnθ +O(d3

τ ) (5.17)

sin(nθ − ndτθ) = sinnθ − ndτθ cosnθ − n2d2τ

2θ2 sinnθ +O(d3

τ ). (5.18)

Using the standard trigonometric identities, (5.17) and (5.18) reduce for n = 1 and n = 2

to

cos(θ − dτθ) = cos θ + dτθ sin θ − d2τ

2θ2 cos θ +O(d3

τ ), (5.19)

cos(2θ − 2dτθ) = cos 2θ + 2dτθ sin 2θ − 2d2τθ

2 cos 2θ +O(d3τ ), (5.20)

sin(θ − dτθ) = sin θ − dτθ cos θ − d2τ

2θ2 sin θ +O(d3

τ ), (5.21)

sin(2θ − 2dτθ) = sin 2θ − 2dτθ cos 2θ − 2d2τθ

2 sin 2θ +O(d3τ ). (5.22)

5.2.5 Previous results

• As in the previous chapters (see Section 3.3.2.1), we assume that we can change the

order of integration between dx, dθ and dθ′.

• As in the previous chapters (see Section 3.3.2.2), we use the divergence theorem to

show that the following integrals hold∫

R2

−a(ξ.∇xp) dx =

∫

R2

pξ.(∇xa) dx, (5.23)

for scalar functions a, and∫

R2

−v(ξ.∇xp) dx =

∫

R2

(p(ξ.∇x)v) dx, (5.24)

for vector functions v.

• Also, for convenience in the subsequent calculations, we shall write p(θ) or p in place

of p(θ,x, t).


5.3 Differential equations for the spatial statistics and higher

order moments

Using a similar method to Othmer et al. (1988), differential equations for the statistics

of interest and higher moments can be derived using the evolution equation given earlier

(3.11).


5.3.1.1 Differential equation for H(t)

In the previous chapters the differential equation derived for H(t) (see (3.28)), was in-

dependent of the choice of T (θ, θ′), and so the equation for H(t) for the case of linear

re-orientation will be the same, i.e.dH

dt= V. (5.25)

5.3.1.2 Differential equation for V(t)

From the definitions of F1(t) and Y1(t) in (5.3) and (5.4) respectively, we have

V(t) = s(F1(t), Y1(t)), (5.26)

and thus it is not necessary to derive a differential equation for V(t) directly if we are able

to find solutions for F1(t) and Y1(t).

5.3.1.3 Differential equation for D2(t)

From the definitions of D2x1(t) and D2

x2(t) in (4.13) and (4.14) respectively, we have

D2(t) = D2x1(t) +D2

x2(t), (5.27)

and thus it is not necessary to derive a differential equation for D2(t) directly if we are

able to find solutions for D2x1(t) and D2

x2(t).


In the previous chapter the differential equation derived for D2x1(t) (see (4.38)), was in-

dependent of the choice of T (θ, θ′), and so the equation for D2x1(t) for the case of linear

re-orientation will be the samedD2

x1

dt= 2sG1, (5.28)

where G1 is defined in (5.9).



In the previous chapter the differential equation derived for D2x2(t) (see (4.39)), was in-

dependent of the choice of T (θ, θ′), and so the equation for D2x2(t) for the case of linear

re-orientation will be the samedD2

x2

dt= 2sZ1, (5.29)

where Z1 is defined in (5.10).

5.3.1.6 Equation for σ2(t)

We also have the equation derived earlier for σ2(t) (see (3.43)),

σ2(t) = D2(t) − ‖H(t)‖2, (5.30)

and the corresponding equations for the spread in each direction

σ2x1(t) = D2

x1(t) − ‖Hx1(t)‖2, (5.31)

and σ2x2(t) = D2

x2(t) − ‖Hx2(t)‖2. (5.32)

5.3.2 Deriving equations for the higher order moments

5.3.2.1 Estimating the integrals

To derive differential equations for all the following higher order moments it is necessary

to solve the following integrals

kn(µ, κ) =

∫ π

−πθ sinnθ eκ cos(θ−µ) dθ, (5.33)

ln(µ, κ) =

∫ π

−πθ cosnθ eκ cos(θ−µ) dθ, (5.34)

mn(µ, κ) =

∫ π

−πθ2 cosnθ eκ cos(θ−µ) dθ , (5.35)

and

nn(µ, κ) =

∫ π

−πθ2 sinnθ eκ cos(θ−µ) dθ , (5.36)

for n = 1 and 2, and where µ = θ′ − dτθ′.

These integrals are then multiplied by p(θ′) and integrated over dθ′ and dx to give differ-

ential equations for the moments of the population distribution.

While it is possible to solve the integrals numerically for fixed parameter values, we are

unable to solve these integrals in an analytic way. It should be noted that even if an

analytic solution exists, the final solutions will still only be leading order approximations

due to the assumption we have made that dτ ≪ 1. As we are able to evaluate the integrals

numerically, we fit simple known functions (dependent on µ and κ), to allow us to make

further analytical progress.


5.3.2.2 Fitting a function to integral (5.33) with n = 1

By inspection we fitted functions of Bessel functions to plots of the integral that had been

evaluated for fixed values of the parameter κ. We found that as κ increases, increasingly

complex terms involving higher order Bessel functions are needed to obtain a good fit. To

avoid over-complication, the function

k1(µ, κ) = 2πI0(κ) − πI1(κ) cos µ− πI2(κ) cos 2µ, (5.37)

was used to approximate (5.33) with n = 1. Plots comparing the function k1(µ, κ) to the

exact integral for |µ| ≤ π, are shown for various values of κ in Figure 5.1.

Note that when κ = 0, both the exact integral and the test function k1(µ, κ) equal 2π.

This function is a good fit for small values of the parameter κ but becomes increasingly

less of a fit as κ gets larger.

5.3.2.3 Fitting functions to the further integrals

Similarly, we fitted the following functions to the integrals given in (5.33) — (5.36) re-

spectively, for n = 1 and n = 2.

l1(µ, κ) = −I1(κ)π sinµ+ 2I2(κ)π sin 2µ, (5.38)

m1(µ, κ) = −4πI0(κ) + 8πI1(κ) cos µ− 4πI2(κ) cos 2µ, (5.39)

n1(µ, κ) = 6πI1(κ) sinµ, (5.40)

k2(µ, κ) = −πI0(κ) + 3πI1(κ) cos µ− 3

2πI2(κ) cos 2µ, (5.41)

l2(µ, κ) = −3

2I1(κ)π sinµ− I2(κ)π sin 2µ, (5.42)

m2(µ, κ) = πI0(κ) − 4πI1(κ) cos µ+ 6πI2(κ) cos 2µ, (5.43)

n2(µ, κ) = −4πI1(κ) sinµ+ 8πI2(κ) sin 2µ. (5.44)

Plots comparing the functions defined above, (5.38) — (5.44) to the exact integrals for

|µ| ≤ π, are shown for various values of κ in Figures 5.2 — 5.8.

5.3.2.4 Comment on the fitted functions

The functions (5.37)—(5.44) are used to approximate the integrals (5.33)—(5.36) with

n = 1 and 2, in the subsequent analysis.

The method of estimating the exact integrals (5.33)—(5.36) by fitting simple functions that

approximate the solution could be considered quite crude, and more sophisticated ways

may possibly be found to calculate better approximations or even the exact solutions.

However, it should be noted that even if we had exact solutions to these integrals our final

differential equations will still only be approximations themselves due to the assumptions

we make in working to a leading order and assuming O(d3τ ) terms are negligible. As

we are aware that our final differential equations and solutions will only be leading order


Test function k1Exact integral

Legend

5.8

6

6.2

6.4

6.6

6.8

7

7.2

7.4

–3 –2 –1 0 1 2 3µ

(a) κ = 0.5.


Legend

6

6.5

7

7.5

8

8.5

9

–3 –2 –1 0 1 2 3µ

(b) κ = 1.


Legend

8

10

12

14

16

18

–3 –2 –1 0 1 2 3µ

(c) κ = 2.


Legend

20

40

60

80

100

–3 –2 –1 0 1 2 3µ

(d) κ = 4.

Figure 5.1: Plots comparing k1(µ, κ) to the exact integral for various values of κ. (The

scale of each plot is different).


Test function l1Exact integral

Legend

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

–3 –2 –1 1 2 3µ

(a) κ = 0.5.


Legend

–2

–1

0

1

2

–3 –2 –1 1 2 3µ

(b) κ = 1.


Legend

–10

–5

5

10

–3 –2 –1 1 2 3µ

(c) κ = 2.


Legend

–80

–60

–40

–20

0

20

40

60

80

–3 –2 –1 1 2 3µ

(d) κ = 4.

Figure 5.2: Plots comparing l1(µ, κ) to the exact integral for various values of κ. (The



Test function m1Exact integral

Legend

–20

–18

–16

–14

–12

–10

–8

–3 –2 –1 0 1 2 3µ

(a) κ = 0.5.


Legend

–30

–25

–20

–15

–10

–5

–3 –2 –1 0 1 2 3µ

(b) κ = 1.


Legend

–60

–40

–20

0–3 –2 –1 1 2 3µ

(c) κ = 2.


Legend

–400

–300

–200

–100

–3 –2 –1 1 2 3µ

(d) κ = 4.

Figure 5.3: Plots comparing m1(µ, κ) to the exact integral for various values of κ. (The



Test function n1Exact integral

Legend

–4

–2

0

2

4

–3 –2 –1 1 2 3µ

(a) κ = 0.5.


Legend

–10

–5

0

5

10

–3 –2 –1 1 2 3µ

(b) κ = 1.


Legend

–30

–20

–10

10

20

30

–3 –2 –1 1 2 3µ

(c) κ = 2.


Legend

–200

–100

0

100

200

–3 –2 –1 1 2 3µ

(d) κ = 4.

Figure 5.4: Plots comparing n1(µ, κ) to the exact integral for various values of κ. (The




Legend

–5

–4

–3

–2

–1

–3 –2 –1 0 1 2 3µ

(a) κ = 0.5.


Legend

–10

–8

–6

–4

–2

–3 –2 –1 1 2 3µ

(b) κ = 1.


Legend

–25

–20

–15

–10

–5

0

5

–3 –2 –1 1 2 3µ

(c) κ = 2.


Legend

–160

–140

–120

–100

–80

–60

–40

–20

20

–3 –2 –1 1 2 3µ

(d) κ = 4.

Figure 5.5: Plots comparing k2(µ, κ) to the exact integral for various values of κ. (The




Legend

–1

–0.5

0

0.5

1

–3 –2 –1 1 2 3µ

(a) κ = 0.5.


Legend

–2

–1

0

1

2

–3 –2 –1 1 2 3µ

(b) κ = 1.


Legend

–8

–6

–4

–2

0

2

4

6

8

–3 –2 –1 1 2 3µ

(c) κ = 2.


Legend

–60

–40

–20

0

20

40

60

–3 –2 –1 1 2 3µ

(d) κ = 4.

Figure 5.6: Plots comparing l2(µ, κ) to the exact integral for various values of κ. (The




Legend

1

2

3

4

5

6

7

–3 –2 –1 0 1 2 3µ

(a) κ = 0.5.


Legend

2

4

6

8

10

12

14

–3 –2 –1 1 2 3µ

(b) κ = 1.


Legend

–10

0

10

20

30

40

–3 –2 –1 1 2 3µ

(c) κ = 2.


Legend

–100

0

100

200

300

–3 –2 –1 1 2 3µ

(d) κ = 4.

Figure 5.7: Plots comparing m2(µ, κ) to the exact integral for various values of κ. (The




Legend

–3

–2

–1

0

1

2

3

–3 –2 –1 1 2 3µ

(a) κ = 0.5.


Legend

–8

–6

–4

–2

0

2

4

6

8

–3 –2 –1 1 2 3µ

(b) κ = 1.


Legend

–30

–20

–10

10

20

30

–3 –2 –1 1 2 3µ

(c) κ = 2.


Legend

–200

–100

100

200

–3 –2 –1 1 2 3µ

(d) κ = 4.

Figure 5.8: Plots comparing n2(µ, κ) to the exact integral for various values of κ. (The



approximations to the true solutions, this method of estimating the exact integrals with the

fitted functions is reasonable. The final solutions are validated by comparing to numerical

simulations in Chapter 6.


Multiply the original linear transport equation (3.11) by cos θ, integrate over θ and x, and


1

N0

∫

R2

∫ π

−πcos θ

∂p

∂tdθ dx =

s

N0

∫

R2

∫ π

−πcos θ (ξ.∇

xp) dθ dx

− λ

N0

∫

R2

∫ π

−πcos θ p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−πcos θ T (θ, θ′) p(θ′) dθ′ dθ dx.


(see Section 3.3.2.2). Substituting for F1(t) from (5.3) and for T (θ, θ′) from (5.2), gives

dF1

dt= −λF1 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πcos θ eκ cos(θ−µ) p(θ′) dθ′ dθ dx,

where µ = θ′ − dτθ′.

The order of the integral in the above can be changed (see Section 3.3.2.1), and then using

(5.15) we get

dF1

dt= −λF1 +

λI1(κ)

I0(κ)N0

∫

R2

∫ π

−πcos(θ′ − dτθ

′) p(θ′) dθ′ dx.

Expanding this using the Taylor Series expansion of the trigonometric functions (5.19)

and then substituting using the definitions of the higher moments given in (5.3) — (5.14),

givesdF1

dt= −λ

(

1 − I1(κ)

I0(κ)

)

F1 +dτλI1(κ)

I0(κ)K1 −

d2τλI1(κ)

2I0(κ)M1 +O(d3

τ ). (5.45)

5.3.2.6 Differential equation for K1(t)

Recalling the definition of K1(t) from (5.5) and using the same method as in Section

5.3.2.5, we get

dK1

dt= −λK1 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πθ sin θ eκ cos(θ−µ) p(θ′) dθ′ dθ dx,



(5.37) and substituting for µ we get

dK1

dt= −λK1 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π(2πI0(κ) − πI1(κ) cos(θ′ − dτθ

′)) p(θ′) dθ′ dx

− λ

2πI0(κ)N0

∫

R2

∫ π

−π(πI2(κ) cos(2θ′ − 2dτθ

′)) p(θ′) dθ′ dx,



and (5.20), and then substituting using the definitions of the higher moments given in

(5.3) — (5.14), gives

dK1

dt= −λ

(

1 +dτI1(κ)

2I0(κ)

)

K1 + λ− λI1(κ)

2I0(κ)F1

+λI1(κ)d

2τ

4I0(κ)M1 −

λI2(κ)

2I0(κ)F2 −

λI2(κ)dτI0(κ)

K2 +λI2(κ)d

2τ

I0(κ)M2 +O(d3

τ ). (5.46)

5.3.2.7 Differential equation for M1(t)

Recalling the definition of M1(t) from (5.7) and using the same method as in Section

5.3.2.5, we get

dM1

dt= −λM1 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πθ2 cos θ eκ cos(θ−µ) p(θ′) dθ′ dθ dx,




dM1

dt= −λM1 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π(−4πI0(κ) + 8πI1(κ) cos(θ′ − dτθ

′)) p(θ′) dθ′ dx

− λ

2πI0(κ)N0

∫

R2

∫ π

−π(4πI2(κ) cos(2θ′ − 2dτθ

′)) p(θ′) dθ′ dx,



(5.3) — (5.14), gives

dM1

dt= −λ

(

1 +2d2τ I1(κ)

I0(κ)

)

M1 − 2λ+4λI1(κ)

I0(κ)F1 +

4λI1(κ)dτI0(κ)

K1

−2λI2(κ)

I0(κ)F2 −

4λI2(κ)dτI0(κ)

K2 +4λI2(κ)d

2τ

I0(κ)M2 +O(d3

τ ). (5.47)


Recalling the definition of F2(t) from (5.3) and using the same method as in Section 5.3.2.5,

we get

dF2

dt= −λF2 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πcos 2θ eκ cos(θ−µ) p(θ′) dθ′ dθ dx,




dF2

dt= −λF2 +

λI2(κ)

I0(κ)N0

∫

R2

∫ π

−π

∫ π

−πcos(2θ′ − 2dτθ

′)) p(θ′) dθ′ dθ dx,




gives

dF2

dt= −λ

(

1 − I2(κ)

I0(κ)

)

F2 +2λI2(κ)dτI0(κ)

K2 −2λI2(κ)d

2τ

I0(κ)M2 +O(d3

τ ). (5.48)

5.3.2.9 Differential equation for K2(t)

Recalling the definition of K2(t) from (5.5) and using the same method as in Section

5.3.2.5, we get

dK2

dt= −λK2 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πθ sin 2θ eκ cos(θ−µ) p(θ′) dθ′ dθ dx,




dK2

dt= −λK2 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π(−πI0(κ) + 3πI1(κ) cos(θ′ − dτθ

′)) p(θ′) dθ′ dθ dx

− λ

2πI0(κ)N0

∫

R2

∫ π

−π(3

2πI2(κ) cos(2θ′ − 2dτθ




(5.3) — (5.14), gives

dK2

dt= −λ

(

1 +3dτ I2(κ)

2I0(κ)

)

K2 −λ

2+

3λI1(κ)

2I0(κ)F1 +

3λI1(κ)dτ2I0(κ)

K1

−3λI1(κ)d2τ

4I0(κ)M1 −

3λI2(κ)

4I0(κ)F2 +

3λI2(κ)d2τ

2I0(κ)M2 +O(d3

τ ). (5.49)

5.3.2.10 Differential equation for M2(t)

Recalling the definition of M2(t) from (5.7) and using the same method as in Section

5.3.2.5, we get

dM2

dt= −λM2 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πθ2 cos 2θ eκ cos(θ−µ) p(θ′) dθ′ dθ dx,




dM2

dt= −λM2 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π(πI0(κ) − 4πI1(κ) cos(θ′ − dτθ


+λ

2πI0(κ)N0

∫

R2

∫ π

−π(6πI2(κ) cos(2θ′ − 2dτθ





(5.3) — (5.14), gives

dM2

dt= −λ

(

1 +6d2τ I2(κ)

I0(κ)

)

M2 +λ

2− 2λI1(κ)

I0(κ)F1 −

2λI1(κ)dτI0(κ)

K1

+λI1(κ)d

2τ

I0(κ)M1 +

3λI2(κ)

I0(κ)F2 +

6λI2(κ)dτI0(κ)

K2 +O(d3τ ). (5.50)


Recalling the definition of Y1(t) from (5.4) and using the same method as in Section 5.3.2.5,

we get

dY1

dt= −λY1 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πsin θ eκ cos(θ−µ) p(θ′) dθ′ dθ dx,




dY1

dt= −λY1 +

λI1(κ)

I0(κ)N0

∫

R2

∫ π

−π

∫ π

−πsin(θ′ − dτθ




givesdY1

dt= −λ

(

1 − I1(κ)

I0(κ)

)

Y1 −λI1(κ)dτI0(κ)

L1 −λI1(κ)d

2τ

2I0(κ)N1 +O(d3

τ ). (5.51)

5.3.2.12 Differential equation for L1(t)

Recalling the definition of L1(t) from (5.6) and using the same method as in Section

5.3.2.5, we get

dL1

dt= −λL1 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πθ cos θ eκ cos(θ−µ) p(θ′) dθ′ dθ dx,




dL1

dt= −λL1 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π(−πI1(κ) sin(θ′ − dτθ


+λ

2πI0(κ)N0

∫

R2

∫ π

−π(2πI2(κ) sin(2θ′ − 2dτθ





(5.3) — (5.14), gives

dL1

dt= −λ

(

1 − dτ I1(κ)

2I0(κ)

)

L1 −λI1(κ)

2I0(κ)Y1 +

λI1(κ)d2τ

4I0(κ)N1 +

λI2(κ)

I0(κ)Y2

−2λI2(κ)dτI0(κ)

L2 −2λI2(κ)d

2τ

I0(κ)N2 +O(d3

τ ). (5.52)

5.3.2.13 Differential equation for N1(t)

Recalling the definition of N1(t) from (5.8) and using the same method as in Section

5.3.2.5, we get

dN1

dt= −λN1 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πθ2 sin θ eκ cos(θ−µ) p(θ′) dθ′ dθ dx,




dN1

dt= −λN1 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−π(6πI1(κ) sin(θ′ − dτθ




gives

dN1

dt= −λ

(

1 +3d2τ I1(κ)

2I0(κ)

)

N1 +3λI1(κ)

I0(κ)Y1 −

3λI1(κ)dτI0(κ)

L1 +O(d3τ ). (5.53)


Recalling the definition of Y2(t) from (5.4) and using the same method as in Section 5.3.2.5,

we get

dY2

dt= −λY2 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πsin 2θ eκ cos(θ−µ) p(θ′) dθ′ dθ dx,




dY2

dt= −λY2 +

λI2(κ)

I0(κ)N0

∫

R2

∫ π

−π

∫ π

−πsin(2θ′ − 2dτθ




gives

dY2

dt= −λ

(

1 − I2(κ)

I0(κ)

)

Y2 −2λI2(κ)dτI0(κ)

L2 −2λI2(κ)d

2τ

I0(κ)N2 +O(d3

τ ). (5.54)


5.3.2.15 Differential equation for L2(t)

Recalling the definition of L2(t) from (5.6) and using the same method as in Section

5.3.2.5, we get

dL2

dt= −λL2 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πθ cos 2θ eκ cos(θ−µ) p(θ′) dθ′ dθ dx,




dL2

dt= −λL2 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π(−3

2πI1(κ) sin(θ′ − dτθ


− λ

2πI0(κ)N0

∫

R2

∫ π

−π(πI2(κ) sin(2θ′ − 2dτθ




(5.3) — (5.14), gives

dL2

dt= −λ

(

1 − dτ I2(κ)

I0(κ)

)

L2 −3λI1(κ)

4I0(κ)Y1 +

3λI1(κ)dτ4I0(κ)

L1 +3λI1(κ)d

2τ

8I0(κ)N1

−λI2(κ)2I0(κ)

Y2 +λI2(κ)d

2τ

I0(κ)N2 +O(d3

τ ). (5.55)

5.3.2.16 Differential equation for N2(t)

Recalling the definition of N2(t) from (5.8) and using the same method as in Section

5.3.2.5, we get

dN2

dt= −λN1 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πθ2 sin 2θ eκ cos(θ−µ) p(θ′) dθ′ dθ dx,




dN2

dt= −λN2 +

λ

2πI0(κ)N0

∫

R2

∫ π

−π(−4πI1(κ) sin(θ′ − dτθ


+λ

2πI0(κ)N0

∫

R2

∫ π

−π(8πI2(κ) sin(2θ′ − 2dτθ




(5.3) — (5.14), gives

dN2

dt= −λ

(

1 +8d2τ I2(κ)

I0(κ)

)

N2 −2λI1(κ)

I0(κ)Y1 +

2λI1(κ)dτI0(κ)

L1 +λI1(κ)d

2τ

I0(κ)N1

+4λI2(κ)

I0(κ)Y2 −

8λI2(κ)dτI0(κ)

L2 +O(d3τ ). (5.56)



Multiply the original equation (3.11) by x1 cos θ, integrate over θ and x, and divide by

N0, to give

1

N0

∫

R2

∫ π

−πx1 cos θ

∂p

∂tdθ dx =

s

N0

∫

R2

∫ π

−πx1 cos θ (ξ.∇

xp) dθ dx

− λ

N0

∫

R2

∫ π

−πx1 cos θ p dθ dx

+λ

N0

∫

R2

∫ π

−π

∫ π

−πx1 cos θ T (θ, θ′) p(θ′) dθ′ dθ dx.

The first term in the right hand side can be rewritten using the divergence theorem (see

Section 3.3.2.2) and then substituting for G1(t) from (5.9) and for T (θ, θ′) from (5.2), gives

dG1

dt=

s

N0

∫

R2

∫ π

−πpξ.(∇

xx1 cos θ) dθ dx − λG1

+λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πx1 cos θ eκ cos(θ−θ′+dτ θ′) p(θ′) dθ′ dθ dx.


using the von Mises integral (5.15), we get

dG1

dt=

s

N0

∫

R2

∫ π

−πcos2 θp dθ dx − λG1

+λI1(κ)

N0I0(κ)

∫

R2

∫ π

−πx1 cos(θ′ − dτθ

′) p(θ′) dθ′ dx.



gives

dG1

dt=s

2+s

2F2 − λ

(

1 − I1(κ)

I0(κ)

)

G1 +λdτI1(κ)

I0(κ)P1 −

λd2τI1(κ)

2I0(κ)R1 + h.o.t., (5.57)

where h.o.t. represents higher order terms that are assumed small compared to the leading

order terms in the equation.

5.3.2.18 Differential equation for P1(t)

Recalling the definition of P1(t) from (5.11) and using the same method as in Section

5.3.2.17, we get

dP1

dt=

s

N0

∫

R2

∫ π

−πpξ.(∇

xx1θ sin θ) dθ dx − λP1

+λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πx1θ sin θ eκ cos(θ−θ′+dτ θ′) p(θ′) dθ′ dθ dx.


using (5.37) and substituting for µ we get

dP1

dt=

s

N0

∫

R2

∫ π

−πθ sin θ cos θp dθ dx− λP1


+λ

2πI0(κ)N0

∫

R2

∫ π

−πx1(2πI0(κ) − πI1(κ) cos(θ′ − dτθ

′)) p(θ′) dθ′ dx

− λ

2πI0(κ)N0

∫

R2

∫ π

−π(πI2(κ)x1 cos(2θ′ − 2dτθ

′) p(θ′) dθ′ dx.



(5.3) — (5.14), gives

dP1

dt=

s

2K2 − λ

(

1 +dτI1(κ)

2I0(κ)

)

P1 + λx1 −λI1(κ)

2I0(κ)G1 +

d2τλI1(κ)

4I0(κ)R1

−λI2(κ)2I0(κ)

G2 −dτλI2(κ)

I0(κ)P2 +

d2τλI2(κ)

I0(κ)R2 + h.o.t. (5.58)

5.3.2.19 Differential equation for R1(t)

Recalling the definition of R1(t) from (5.13) and using the same method as in Section

5.3.2.17, we get

dR1

dt=

s

N0

∫

R2

∫ π

−πpξ.(∇

xx1θ

2 cos θ) dθ dx

−λR1 +λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πx1θ

2 cos θ eκ cos(θ−θ′+dτ θ′) p(θ′) dθ′ dθ dx.



dR1

dt=

s

N0

∫

R2

∫ π

−πθ2 cos2 θp dθ dx− λR1

+λ

2πI0(κ)N0

∫

R2

∫ π

−πx1(−4πI0(κ) + 8πI1(κ) cos(θ′ − dτθ

′)) p(θ′) dθ′ dx

− λ

2πI0(κ)N0

∫

R2

∫ π

−πx1(4πI2(κ) cos(2θ′ − 2dτθ

′)) p(θ′) dθ′ dx.



(5.3) — (5.14), gives

dR1

dt=

s

2M0 +

s

2M2 − λ

(

1 +2d2τ I1(κ)

2I0(κ)

)

R1 − 2λx1 +4λI1(κ)

I0(κ)G1

+4dτλI1(κ)

I0(κ)P1 −

2λI2(κ)

I0(κ)G2 −

4dτλI2(κ)

I0(κ)P2 +

4d2τλI2(κ)

I0(κ)R2

+h.o.t. (5.59)


Recalling the definition of G2(t) from (5.9) and using the same method as in Section

5.3.2.17, we get

dG2

dt=

s

N0

∫

R2

∫ π

−πpξ.(∇

xx1 cos 2θ) dθ dx− λG2

+λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πx1 cos 2θ eκ cos(θ−θ′+dτ θ′) p(θ′) dθ′ dθ dx.




dG2

dt=

s

N0

∫

R2

∫ π

−πcos 2θ cos θp dθ dx

−λG2 +λI2(κ)

N0I0(κ)

∫

R2

∫ π

−πx1 cos(2θ′ − 2dτθ

′) p(θ′) dθ′ dx.



gives

dG2

dt=

s

2F1 +

s

2F3 − λ

(

1 − I2(κ)

I0(κ)

)

G2 +2dτλI2(κ)

I0(κ)P2 −

2d2τλI2(κ)

I0(κ)R2

+h.o.t. (5.60)

5.3.2.21 Differential equation for P2(t)

Recalling the definition of P2(t) from (5.11) and using the same method as in Section

5.3.2.17, we get

dP2

dt=

s

N0

∫

R2

∫ π

−πpξ.(∇

xx1θ sin 2θ) dθ dx− λP2

+λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πx1θ sin 2θ eκ cos(θ−θ′+dτ θ′) p(θ′) dθ′ dθ dx.



dP2

dt=

s

N0

∫

R2

∫ π

−πθ cos θ sin 2θp dθ dx

−λP2 +λ

2πI0(κ)N0

∫

R2

∫ π

−πx1(−πI0(κ) + 3πI1(κ) cos(θ′ − dτθ

′)) p(θ′) dθ′ dx

− λ

2πI0(κ)N0

∫

R2

∫ π

−πx1(

3

2πI2(κ) cos(2θ′ − 2dτθ

′)) p(θ′) dθ′ dx.



(5.3) — (5.14), gives

dP2

dt=

s

2K1 +

s

2K3 − λ

(

1 +3dτ I2(κ)

2I0(κ)

)

P2 −λ

2x1 +

3λI1(κ)

2I0(κ)G1

+3dτλI1(κ)

2I0(κ)P1 −

3d2τλI1(κ)

4I0(κ)R1 −

3λI2(κ)

4I0(κ)G2 +

3d2τλI2(κ)

2I0(κ)R2

+h.o.t. (5.61)

5.3.2.22 Differential equation for R2(t)

Recalling the definition of R2(t) from (5.13) and using the same method as in Section

5.3.2.17, we get

dR2

dt=

s

N0

∫

R2

∫ π

−πpξ.(∇

xx1θ

2 cos 2θ) dθ dx


−λR2 +λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πx1θ

2 cos 2θ eκ cos(θ−θ′+dτ θ′) p(θ′) dθ′ dθ dx.



dR2

dt=

s

N0

∫

R2

∫ π

−πθ2 cos θ cos 2θp dθ dx

−λR2 +λ

2πI0(κ)N0

∫

R2

∫ π

−πx1(πI0(κ) − 4πI1(κ) cos(θ′ − dτθ

′)) p(θ′) dθ′ dx

+λ

2πI0(κ)N0

∫

R2

∫ π

−πx1(6πI2(κ) cos(2θ′ − 2dτθ

′)) p(θ′) dθ′ dx.



(5.3) — (5.14), gives

dR2

dt=

s

2M1 +

s

2M3 − λ

(

1 +6d2τ I2(κ)

I0(κ)

)

R2 +λ

2x1 −

2λI1(κ)

I0(κ)G1

−2dτλI1(κ)

I0(κ)P1 +

d2τλI1(κ)

I0(κ)R1 +

3λI2(κ)

I0(κ)G2 +

6dτλI2(κ)

I0(κ)P2

+h.o.t. (5.62)


Recalling the definition of Z1(t) from (5.10) and using the same method as in Section

5.3.2.17, we get

dZ1

dt=

s

N0

∫

R2

∫ π

−πpξ.(∇

xx2 sin θ) dθ dx− λZ1

+λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πx2 sin θ eκ cos(θ−θ′+dτ θ′) p(θ′) dθ′ dθ dx.


using the von Mises integral (5.16), we get

dZ1

dt=

s

N0

∫

R2

∫ π

−πsin2 θp dθ dx− λZ1

+λI1(κ)

N0I0(κ)

∫

R2

∫ π

−πx2 sin(θ′ − dτθ

′) p(θ′) dθ′ dx. (5.63)



gives

dZ1

dt=s

2− s

2F2 − λ

(

1 − I1(κ)

I0(κ)

)

Z1 −λdτ I1(κ)

I0(κ)Q1 −

λd2τI1(κ)

2I0(κ)S1 + h.o.t. (5.64)


5.3.2.24 Differential equation for Q1(t)

Recalling the definition of Q1(t) from (5.12) and using the same method as in Section

5.3.2.17, we get

dQ1

dt=

s

N0

∫

R2

∫ π

−πpξ.(∇

xx2θ cos θ) dθ dx− λQ1

+λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πx2θ cos θ eκ cos(θ−θ′+dτ θ′) p(θ′) dθ′ dθ dx.



dQ1

dt=

s

N0

∫

R2

∫ π

−πθ sin θ cos θp dθ dx

−λQ1 +λ

2πI0(κ)N0

∫

R2

∫ π

−πx2(−πI1(κ) sin(θ′ − dτθ

′)) p(θ′) dθ′ dx

+λ

2πI0(κ)N0

∫

R2

∫ π

−πx2(2πI2(κ) sin(2θ′ − 2dτθ

′)) p(θ′) dθ′ dx.



(5.3) — (5.14), gives

dQ1

dt=

s

2K2 − λ

(

1 − dτI1(κ)

2I0(κ)

)

Q1 −λI1(κ)

2I0(κ)Z1 +

d2τλI1(κ)

4I0(κ)S1

+λI2(κ)

I0(κ)Z2 −

2dτλI2(κ)

I0(κ)Q2 −

2d2τλI2(κ)

I0(κ)S2 + h.o.t. (5.65)

5.3.2.25 Differential equation for S1(t)

Recalling the definition of S1(t) from (5.14) and using the same method as in Section

5.3.2.17, we get

dS1

dt=

s

N0

∫

R2

∫ π

−πpξ.(∇

xx2θ

2 sin θ) dθ dx − λS1

+λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πx2θ

2 sin θ eκ cos(θ−θ′+dτ θ′) p(θ′) dθ′ dθ dx.



dS1

dt=

s

N0

∫

R2

∫ π

−πθ2 sin2 θp dθ dx− λS1

+λ

2πI0(κ)N0

∫

R2

∫ π

−πx2(6πI1(κ) sin(θ′ − dτθ

′)) p(θ′) dθ′ dx.



gives

dS1

dt=

s

2M0 −

s

2M2 − λ

(

1 +3d2τ I1(κ)

2I0(κ)

)

S1 +3λI1(κ)

I0(κ)Z1 −

3dτλI1(κ)

I0(κ)Q1

+h.o.t. (5.66)



Recalling the definition of Z2(t) from (5.9) and using the same method as in Section

5.3.2.17, we get

dZ2

dt=

s

N0

∫

R2

∫ π

−πpξ.(∇

xx2 sin 2θ) dθ dx − λZ2

+λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πx2 sin 2θ eκ cos(θ−θ′+dτ θ′) p(θ′) dθ′ dθ dx.



dZ2

dt=

s

N0

∫

R2

∫ π

−πsin θ sin 2θp dθ dx

−λZ2 +λI2(κ)

N0I0(κ)

∫

R2

∫ π

−πx2 sin(2θ′ − 2dτθ

′) p(θ′) dθ′ dx.



gives

dZ2

dt=

s

2F1 −

s

2F3 − λ

(

1 − I2(κ)

I0(κ)

)

Z2 −2dτλI2(κ)

I0(κ)Q2 −

2d2τλI2(κ)

I0(κ)S2

+h.o.t. (5.67)

5.3.2.27 Differential equation for Q2(t)

Recalling the definition of Q2(t) from (5.12) and using the same method as in Section

5.3.2.17, we get

dQ2

dt=

s

N0

∫

R2

∫ π

−πpξ.(∇

xx2θ cos 2θ) dθ dx − λQ2

+λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πx2θ cos 2θ eκ cos(θ−θ′+dτ θ′) p(θ′) dθ′ dθ dx.



dQ2

dt=

s

N0

∫

R2

∫ π

−πθ sin θ cos 2θp dθ dx

−λQ2 −λ

2πI0(κ)N0

∫

R2

∫ π

−πx2(

3

2πI1(κ) sin(θ′ − dτθ

′)) p(θ′) dθ′ dx

− λ

2πI0(κ)N0

∫

R2

∫ π

−πx2πI2(κ) sin(2θ′ − 2dτθ

′)) p(θ′) dθ′ dx.



(5.3) — (5.14), gives

dQ2

dt=

s

2K3 −

s

2K1 − λ

(

1 − dτI2(κ)

I0(κ)

)

Q2 −3λI1(κ)

4I0(κ)Z1 +

3dτλI1(κ)

4I0(κ)Q1

+3d2τλI1(κ)

8I0(κ)S1 −

λI2(κ)

2I0(κ)Z2 +

d2τλI2(κ)

I0(κ)S2 + h.o.t. (5.68)


5.3.2.28 Differential equation for S2(t)

Recalling the definition of S2(t) from (5.14) and using the same method as in Section

5.3.2.17, we get

dS2

dt=

s

N0

∫

R2

∫ π

−πpξ.(∇

xx2θ

2 sin 2θ) dθ dx − λS2

+λ

2πI0(κ)N0

∫

R2

∫ π

−π

∫ π

−πx2θ

2 sin 2θ eκ cos(θ−θ′+dτ θ′) p(θ′) dθ′ dθ dx.



dS2

dt=

s

N0

∫

R2

∫ π

−πθ2 sin θ sin 2θp dθ dx

−λS2 +λ

2πI0(κ)N0

∫

R2

∫ π

−πx2(−4πI1(κ) sin(θ′ − dτθ

′)) p(θ′) dθ′ dx

+λ

2πI0(κ)N0

∫

R2

∫ π

−πx2(8πI2(κ) sin(2θ′ − 2dτθ

′)) p(θ′) dθ′ dx.



(5.3) — (5.14), gives

dS2

dt=

s

2M1 −

s

2M3 − λ

(

1 +8d2τ I2(κ)

I0(κ)

)

S2 −2λI1(κ)

I0(κ)Z1 +

2dτλI1(κ)

I0(κ)Q1

+d2τλI1(κ)

I0(κ)S1 +

4λI2(κ)

I0(κ)Z2 −

8dτλI2(κ)

I0(κ)Q2 + h.o.t. (5.69)

5.3.3 System of equations for non-spatial moments

The differential equations derived above can be split into four separate systems of equa-

tions, two for the non-spatial higher moments and two for the spatial higher moments.

The non-spatial higher moments can themselves be split into a system arising from the

even higher moments (Fn(t), Kn(t) and Mn(t)) and a system arising from the odd higher

moments (Yn(t), Ln(t) and Nn(t)).

For simplicity, we can write the first system as

F1

K1

M1

F2

K2

M2

= λ

a11 a12 a13 0 0 0

a21 a22 a23 a24 a25 a26

a31 a32 a33 a34 a35 a36

0 0 0 a44 a45 a46

a51 a52 a53 a54 a55 a56

a61 a62 a63 a64 a65 a66

F1

K1

M1

F2

K2

M2

+ λ

0

1

−2

0

−12

12

+O(d3τ ), (5.70)

or

F = AFF + BF , (5.71)


where the aij correspond to the constant coefficients (dependent on dτ and κ only) from

the differential equations given previously.

We also have the initial conditions

F1(0)

K1(0)

M1(0)

F2(0)

K2(0)

M2(0)

=

0

1

−2

0

−12

12

. (5.72)

Similarly, we can write the second system as

Y1

L1

N1

Y2

L2

N2

= λ

b11 b12 b13 0 0 0

b21 b22 b23 b24 b25 b26

b31 b32 b33 0 0 0

0 0 0 b44 b45 b46

b51 b52 b53 b54 b55 b56

b61 b62 b63 b64 b65 b66

Y1

L1

N1

Y2

L2

N2

+O(d3τ ), (5.73)

or

Y = AYY, (5.74)

where the bij correspond to the constant coefficients (dependent on dτ and κ only) from


We also have the initial conditions that Y1(0) = L1(0) = N1(0) = Y2(0) = L2(0) =

N2(0) = 0.

5.3.4 System of equations for spatial moments

The spatial higher moments can themselves be split into a system arising from the even

higher moments (Gn(t), Pn(t) and Rn(t)) and a system arising from the odd higher mo-

ments (Zn(t), Qn(t) and Sn(t)).

For simplicity, we can write the first system as

G1

P1

R1

G2

P2

R2

= λ

c11 c12 c13 0 0 0

c21 c22 c23 c24 c25 c26

c31 c32 c33 c34 c35 c36

0 0 0 c44 c45 c46

c51 c52 c53 c54 c55 c56

c61 c62 c63 c64 c65 c66

G1

P1

R1

G2

P2

R2

+ λ

0

1

−2

0

−12

12

x1


+s

2

1 + F2

K2

M0 +M2

F1 + F3

K1 +K3

M1 +M3

+ h.o.t., (5.75)

or

G = AFG + BG(t), (5.76)

where the cij correspond to the constant coefficients (dependent on dτ and κ only) from


We also have the initial conditions that G1(0) = P1(0) = R1(0) = G2(0) = P2(0) =

R2(0) = 0.

Similarly, we can write the second system as

Z1

Q1

S1

Z2

Q2

S2

= λ

d11 d12 d13 0 0 0

d21 d22 d23 d24 d25 d26

d31 d32 d33 0 0 0

0 0 0 d44 d45 d46

d51 d52 d53 d54 d55 d56

d61 d62 d63 d64 d65 d66

Z1

Q1

S1

Z2

Q2

S2

+s

2

1 − F2

K2

M0 −M2

F1 − F3

K3 −K1

M1 −M3

+ h.o.t, (5.77)

or

Z = AY Z + BZ(t), (5.78)

where the dij correspond to the constant coefficients (dependent on dτ and κ only) from


We also have the initial conditions that Z1(0) = Q1(0) = S1(0) = Z2(0) = Q2(0) =

S2(0) = 0.

5.3.4.1 Higher moments

The systems of equations in (5.75) and (5.77) are not closed. We have the terms M0,

F3, K3 and M3. We know from the definitions of Fn, Kn and Mn (5.3), (5.5) and (5.7)

respectively, that these higher moments are only dependent on the angular distribution of

p(θ,x, t) and not on the spatial distribution. We can close the two systems of equations


in (5.75) and (5.77) by approximating these higher moments with the expected long-time

angular distribution of p(θ,x, t) as in Section 4.6.1. From Hill & Hader’s results, the

expected long-time angular distribution for a population moving with linear reorientation

is given by


, λ = B−1/σ20 , (5.79)

where B(λ) is the normalization function defined in (1.103). In a similar way to the higher

moment approximations in Section 4.6.1 for the sinusoidal model, we estimate the higher

moments as follows

F3 =1

N0

∫

R2

∫ π

−πcos 3θ B(λ)e−λθ

2

dθ dx, (5.80)

K3 =1

N0

∫

R2

∫ π

−πθ sin 3θ B(λ)e−λθ

2

dθ dx, (5.81)

M0 =1

N0

∫

R2

∫ π

−πθ2B(λ)e−λθ

2

dθ dx, (5.82)

M3 =1

N0

∫

R2

∫ π

−πθ2 cos 3θ B(λ)e−λθ

2

dθ dx, (5.83)

where all the moments evaluate to a constant independent of t. Note that, although M0(t)

is not a higher moment as such we approximate it in a similar way to the higher moments.

Simulation results given in Section 7.2.1 show that M0(t) reaches the steady state solution

very quickly and thus this approximation is reasonable.

Under this assumption the systems in (5.75) and (5.77) are closed and can be solved.

5.4 Solving the systems of equations

The systems of differential equations in (5.70)—(5.77) are all linear and are readily solved

in principle. We use the method as described in Section 4.6.2, to solve a general system

of linear differential equations.

5.4.1 Solving for the non-spatial higher order moments

We have a closed system of differential equations for V(t), H(t) and the associated higher

order moments Fn(t), Kn(t), Mn(t), Yn(t), Ln(t) and Nn(t), see (5.25), (5.70) and (5.73).

The equations for the higher order moments form two coupled systems and these must be

solved first.

(5.70) and (5.73) are systems of linear differential equations and are readily solved in

principle. However, because of the complicated coefficients that are dependent on the

parameters dτ , κ and λ, the solution becomes algebraically cumbersome. It is easier to

leave the solutions in their most general form and then solve the system numerically for

fixed values of the parameters. The numerical solutions were found using Maple.


In general, the eigenvalues of the matrix in (5.70) are −λφ1,−λφ2,−λ,−λ,−λ,−λ, where

φ1 = Ψ − 1

4

√Ω, (5.84)

φ2 = Ψ +1

4

√Ω, (5.85)

and

Ψ = 1 −A1(κ)

(

1

2− dτ

4− d2

τ

)

−A2(κ)

(

1

2− 3dτ

4− 3d2

τ

)

, (5.86)

Ω = A1(κ)(

4 − 4dτ − 15d2τ + 8d3

τ + 16d4τ

)

+A2(κ)(

4 − 12dτ − 39d2τ + 72d3

τ + 144d4τ

)

−A1(κ)A2(κ)(

8 − 16dτ − 34d2τ + 32d3

τ + 32d4τ

)

, (5.87)

and An(κ) = In(κ)I0(κ) , such that the real part of φi > 0 for all parameter values.

To leading order, the system has general solutions

Fn(t) = Afne−λφ1t +Bfne

−λφ2t + Cfn,

Kn(t) = Akne−λφ1t +Bkne

−λφ2t + Ckn,

Mn(t) = Amne−λφ1t +Bmne

−λφ2t + Cmn, (5.88)

where Afn, Bfn, Cfn etc are all O(1) constants.

When solving the system numerically, the constants multiplying e−λt terms in the solution

are found to be O(10−m), where m is the number of digits used in the numerical solutions.

This suggests these terms are in fact zero, and it may be possible to show algebraically

that this is true but we do not do so here.

The solution to (5.73) is more straightforward. Although the matrices in (5.70) and (5.73)

are similar, the fact that all the initial conditions in (5.73) are zero, results in the following

trivial solution

Yn(t) = Ln(t) = Nn(t) = 0 ∀ t. (5.89)

The higher moments that correspond to the non-preferred direction are zero for all time.

5.4.2 Solving for V(t) and H(t)

5.4.2.1 Equation for V(t)

From the definitions in (4.11), (5.3) and (5.4), one can see that if all Yn terms are zero

then

V(t) = sF1(t)ξ1, (5.90)

and substituting for F1(t) from (5.88) gives the leading order solution

V(t) = s(

Af1e−λφ1t +Bf1e

−λφ2t +Cf1

)

ξ1, (5.91)

where Af1, Bf1 and Cf1 are all constants such that Cf1 = −(Af1 + Bf1) and V(0) = 0.

(These constant terms Af1 etc, are not the same as those defined in the previous chapter

for the sinusoidal model).


The long-time solution for V∞ is given by

V∞ ∼ sCf1ξ1 (5.92)

Thus we would expect similar long-time behaviour as in the solutions for the sinusoidal

model and the solutions of Othmer et al. which have the same form. For the solution to

be self consistent Cf1 < 1, as the velocity V must be smaller than the speed of movement

s.

5.4.2.2 Equation for H(t)

From (5.25), we have the differential equation for H(t),

dH

dt= V,

Integrating (5.91) and recalling the initial condition H(0) = 0 gives

H(t) = s

(

Cf1t+Af1

φ1+Bf1

φ2+EH(t)

)

ξ1 (5.93)

where the term EH(t) takes into account all the decaying exponential terms dependent on

the eigenvalues φi and the constants multiplying them. Thus EH(t) → 0 as t→ ∞, and

H∞ ∼ sCf1tξ1. (5.94)

5.4.3 Solving for the spatial higher order moments

We have a closed system of differential equations for D2(t) and the other associated higher

moments Gn(t), Pn(t), Rn(t), Zn(t), Qn(t) and Sn(t). The systems of equations in (5.75)

and (5.77) have the leading order solutions

Gn(t) = Agn +Bgnt+ Egn(t), (5.95)

Pn(t) = Apn +Bpnt+ Epn(t), (5.96)

Rn(t) = Arn +Brnt+ Ern(t), (5.97)

Zn(t) = Azn +Ezn(t), (5.98)

Qn(t) = Aqn + Eqn(t), (5.99)

Sn(t) = Asn +Esn(t), (5.100)

where the constants Agn, Bgn, etc can be calculated numerically for particular values of the

parameters κ and dτ . The Egn(t) and similar terms correspond to decaying exponential

functions that are dependent on the eigenvalues of the relevant matrices in (5.75) and

(5.77) respectively.


5.4.4 Solving for D2(t) and σ2(t)

From (5.28) and (5.29) we have

dD2x1

dt= 2sG1, and

dD2x2

dt= 2sZ1.

Using (5.95) and (5.98), the leading order solutions for D2x1(t) and D2

x2(t) are

D2x1(t) = ADx1 +BDx1t+ CDx1t

2 + EDx1(t), (5.101)

D2x2(t) = ADx2 +BDx2t+ EDx2(t), (5.102)

where ADx1, BDx1 etc are constants and EDx1 and EDx2 are decaying exponential terms.

Since, D2(t) = D2x1(t) +D2

x2(t), the leading order solution for D2(t) is

D2(t) = AD +BDt+ CDt2 + ED(t). (5.103)

In a similar way, using (5.30)—(5.32) the leading order solutions for σ2(t) are given by

σ2x1(t) = Aσx1 +Bσx1t+ Eσx1(t), (5.104)

σ2x2(t) = Aσx2 +Bσx2t+ Eσx2(t), (5.105)

σ2(t) = Aσ +Bσt+ Eσ(t), (5.106)

where the Aσx1 etc are constants and the Eσx1 etc correspond to decaying exponential

functions.

As t→ ∞ the E(t) terms will decay away to zero and the dominant terms in the equations

will either be of O(t2) (for D2x1(t), or O(t) for the other statistics. As Hx2(t) = 0 for all t,

it follows that D2x2(t) = σ2

x2(t).

If there is no bias and dτ = 0,

D2(t) = σ2(t) =2s2

λ1

(

t− 1

λ1(1 − e−λ1t)

)

. (5.107)

Thus when dτ = 0 we have the exact long-time solutions

D2∞

= σ2∞

=2s2t

λ1− 1

λ1. (5.108)

This is obviously the same result as the sinusoidal model with zero bias, and gives the

characteristic diffusive behaviour, D2(t) ∼ t, which is obtained with an unbiased random

walk.

5.5 Final system of solutions

We have not presented full analytic solutions for the system of equations as we did for the

sinusoidal case in the previous chapter. There are many complicated coefficients and as

such the system is easier to work with by solving numerically for particular values of the

parameters. We show the solutions as numerical values that correspond to the long-time

limit solutions and also as solution plots for a short time range (t = 0 to 100).


5.5.1 Numerical solutions

As discussed previously the solutions that we have found numerically contain exponential

terms that quickly decay away to zero. We present here the numerical values for the

dominant terms that are present in the long-time solutions. We have fixed the parameters

s = 1 and λ = 1 (effectively non-dimensionalising our equations, see Section 4.9.2), and

then calculated the numerical solutions for various values of κ and dτ . All solutions are

shown are accurate to 4 decimal places.

5.5.1.1 Long-time numerical solutions for V(t)

We have shown in (5.92) that the long-time solution for V(t) is given by

V∞ = sCf1ξ1. (5.109)

The solution will tend to a fixed limiting value that is dependent on the parameters κ and

dτ . This limiting value has been calculated up to 4 decimal places for various values of the

parameters in Table (5.1). When there is no bias (dτ = 0), then V(t) = 0 for all values of

κ.

κ 0.1 0.5 1.0 2.0 4.0 8.0

Cf1 for dτ = 0.1 0.0058 0.0345 0.0842 0.2197 0.4732 0.7156

Cf1 for dτ = 0.2 0.0125 0.0727 0.1695 0.3895 0.6729 0.8477

Cf1 for dτ = 0.3 0.0202 0.1131 0.2487 0.5052 0.7593 0.8917

Table 5.1: Long-time numerical solutions for V(t) with linear reorientation

5.5.1.2 Long-time numerical solutions for H(t)

We have shown in (5.94) that the long-time solution for H(t) is given by

H∞ ∼ sCf1tξ1, (5.110)

The solution will tend to a time dependent solution with a fixed gradient Cf1 that is

dependent on the parameters κ and dτ . The value of Cf1 is given in Table 5.1 for particular

values of the reorientation parameters.

5.5.1.3 Long-time numerical solutions for D2(t) and σ2(t)

The long-time solutions for D2x1,∞, D2

x2,∞ and σ2x1,∞ have a dominant term of O(t2) or

O(t) that is dependent on the parameters κ and dτ . The dominant terms in the long-time

solutions have been calculated up to 4 decimal places for various values of the parameters

in Table (5.2), Table (5.3), and Table (5.4) respectively. The long-time solutions for σ2x2,∞


are the same as D2x2,∞ and are omitted. The dominant terms in the long-time solutions

for D2(t) and σ2(t) can be found by adding the appropriate long-time solutions for D2x1,∞

and D2x2,∞ or σ2

x1,∞ and σ2x2,∞ respectively and these solutions are not displayed.

κ 0.1 0.5 1.0

D2x1,∞ for dτ = 0.1 0.0000t2 + 1.0485t 0.0012t2 + 1.2815t 0.0071t2 + 1.6434t

D2x1,∞ for dτ = 0.2 0.0002t2 + 1.0424t 0.0053t2 + 1.2188t 0.0287t2 + 1.3808t

D2x1,∞ for dτ = 0.3 0.0004t2 + 1.0345t 0.0128t2 + 1.1376t 0.0618t2 + 1.0868t

κ 2.0 4.0 8.0

D2x1,∞ for dτ = 0.1 0.0483t2 + 2.2220t 0.2239t2 + 0.6225t 0.5212t2 − 5.3196t

D2x1,∞ for dτ = 0.2 0.1517t2 + 0.9347t 0.4528t2 − 1.9376t 0.7187t2 − 4.8707t

D2x1,∞ for dτ = 0.3 0.2552t2 + 0.1228t 0.5765t2 − 2.1426t 0.7952t2 − 3.7264t

Table 5.2: Long-time numerical solutions for D2x1(t) with linear reorientation

κ 0.1 0.5 1.0 2.0 4.0 8.0

D2x2,∞ for dτ = 0.1 1.0547t 1.3385t 1.8648t 3.4903t 6.7977t 7.4193t

D2x2,∞ for dτ = 0.2 1.0556t 1.3431t 1.8519t 3.0751t 3.6194t 1.9337t

D2x2,∞ for dτ = 0.3 1.0556t 1.3371t 1.7792t 2.4467t 2.0504t 1.1349t

Table 5.3: Long-time numerical solutions for D2x2(t) with linear reorientation

κ 0.1 0.5 1.0 2.0 4.0 8.0

σ2x1,∞ for dτ = 0.1 1.0486t 1.2846t 1.6677t 2.4948t 2.7496t 1.0650t

σ2x1,∞ for dτ = 0.2 1.0427t 1.2321t 1.4713t 1.6242t 0.8118t −0.0498t

σ2x1,∞ for dτ = 0.3 1.0353t 1.1681t 1.2625t 1.0476t 0.3150t −0.1440t

Table 5.4: Long-time numerical solutions for σ2x1(t) with linear reorientation

5.5.2 Solution plots

Although we have left our solutions in a general form it is straightforward to calculate

the values of the various constants and the eigenvalues numerically. The following plots

show the general behaviour of the solutions to the equations for the statistics of interest

that we have derived. For each plot we have fixed λ = 1 and s = 1 (effectively non-

dimensionalising our equations, see Section 4.9.2), and either dτ = 0.1 or dτ = 0.3, and

then plotted solutions for κ = 0.5, 1, 2, 4, 8. Note that the solution plots show the full

solution, they are not just the long-time solutions as calculated previously. To compare


the spread in different directions D2x1 and D2

x2, and σ2x1 and σ2

x2, we have plotted solutions

for dτ = 0.3 only, as for smaller values of dτ there is less difference between the values of

the two statistics.

k=0.5k=1k=2k=4k=8

0

0.2

0.4

0.6

0.8

1

V(t)

20 40 60 80 100

t

(a) dτ = 0.1.

k=0.5k=1k=2k=4k=8

0

0.2

0.4

0.6

0.8

1

V(t)

20 40 60 80 100

t

(b) dτ = 0.3.

Figure 5.9: Plot of V(t) for dτ = 0.1 and dτ = 0.3 and various values of κ.

k=0.5k=1k=2k=4k=8

0

20

40

60

80

100

H(t)

20 40 60 80 100

t

(a) dτ = 0.1.

k=0.5k=1k=2k=4k=8

0

20

40

60

80

100

H(t)

20 40 60 80 100

t

(b) dτ = 0.3.

Figure 5.10: Plot of H(t) for dτ = 0.1 and dτ = 0.3 and various values of κ.


k=0.5k=1k=2k=4k=8

0

1000

2000

3000

4000

5000

20 40 60 80 100

t

(a) dτ = 0.1.

k=0.5k=1k=2k=4k=8

0

2000

4000

6000

20 40 60 80 100

t

(b) dτ = 0.3.

Figure 5.11: Plot of D2(t) for dτ = 0.1 and dτ = 0.3 and various values of κ. (The scale

of each plot is different)

k=0.5k=1k=2k=4k=8

0

200

400

600

800

20 40 60 80 100

t

(a) dτ = 0.1.

k=0.5k=1k=2k=4k=8

0

50

100

150

200

250

300

350

20 40 60 80 100

t

(b) dτ = 0.3.

Figure 5.12: Plot of σ2(t) for dτ = 0.1 and dτ = 0.3 and various values of κ. (The scale of



k=0.5k=1k=2k=4k=8

0

1000

2000

3000

4000

5000

6000

7000

20 40 60 80 100

t

(a) D2

x1(t).

k=0.5k=1k=2k=4k=8

0

50

100

150

200

20 40 60 80 100

t

(b) D2

x2(t).

Figure 5.13: Plot of D2x1(t) and D2



k=0.5k=1k=2k=4k=8

0

20

40

60

80

100

120

20 40 60 80 100

t

(a) σ2

x1(t).

k=0.5k=1k=2k=4k=8

0

50

100

150

200

20 40 60 80 100

t

(b) σ2

x2(t).

Figure 5.14: Plot of σ2x1(t) and σ2




5.5.3 Comment on solutions

Looking at both the long-time and full solutions we can see a few general results.

• The average velocity V(t), and hence the average position H(t), get larger as each

or either of the parameters (κ and dτ ) gets larger.

• The parameter dτ seems to have little effect on the absolute velocity and average

position if κ is small, whereas when κ is large changing dτ affects the solution much

more. If we think of dτ as the ‘sensing ability’ parameter and κ as the ‘swimming

ability’ parameter this makes sense — it is of no use being good at sensing (large

dτ ) if one cannot swim (small κ), see Section 4.2.4.

• The spread about the mean σ2(t) increases as the parameters get larger but reaches a

maximum and starts to decrease as one looks at the more extreme parameter values.

For extreme values of the parameters, σ2x1(t) becomes negative and our solutions are

not valid. We have the same problem of ‘negative spread’ as encountered with the

sinusoidal model.

A more detailed study of how well our theoretical equations fit real and simulated data,

and the effect of the parameter values on the solutions is given in Chapter 6.

5.5.4 Limitations of the model and solutions

Although we have found some useful general results and we have equations to find the

statistics of interest we should be aware of the assumptions we have made during the

modelling process and how this introduces limitations to the model.

• We have assumed dτ ≪ 1 and rounded off higher order dnτ terms. This will make

our solutions less accurate, as these higher order terms aren’t taken into account in

the final solutions.

• We have estimated the complicated integrals discussed previously with fitted func-

tions. These functions are quite good fits for small values of κ but become increas-

ingly poorer fits as κ increases. We have effectively not included higher order Bessel

functions that would improve the fit but make the solution unworkably complicated.

• Because we have rounded off higher order terms with the two assumptions given

above, the solutions may not be valid for all parameter ranges. When the reorienta-

tion parameters are both large the solution for σ2x1(t) becomes negative and is not

valid — we have the nonsensical result of ‘negative spread’ as encountered in the

sinusoidal model.


5.6 Comparing solutions of the sinusoidal and linear models

From the results presented in this chapter and the previous chapter, we have time-

dependent solutions for the statistics of interest for both the sinusoidal and linear reorien-

tation models. The non-dimensionalised solutions are dependent only on the parameters

of the respective reorientation distributions.

We can compare the behaviour of the statistics of interest between the two reorientation

models for fixed parameter values. The most useful statistics to compare in a biased

random walk are the mean position H(t) and the total spread about the mean position

σ2(t).

In Table (5.5) and Table (5.6) we compare the long-time numerical solutions for the linear

and sinusoidal models for the statistics H(t) and σ2(t) respectively. The linear solutions

have been found numerically as described earlier in this chapter, while the sinusoidal

solutions have been found by substituting the relevant parameter values into the non-

dimensionalised long-time general solutions, (4.95) etc, derived in the previous chapter.

5.6.1 Comparing solutions for H(t)

κ 0.1 0.5 1.0 2.0 4.0 8.0

H∞ (lin, dτ = 0.1) 0.0058 t 0.0345 t 0.0842 t 0.2197 t 0.4732 t 0.7156 t

H∞ (sin, dτ = 0.1) 0.0026 t 0.0160 t 0.0402 t 0.1143 t 0.2999 t 0.5793 t

H∞ (lin, dτ = 0.3) 0.0202 t 0.1131 t 0.2487 t 0.5052 t 0.7593 t 0.8917 t

H∞ (sin, dτ = 0.3) 0.0079 t 0.0478 t 0.1187 t 0.3185 t 0.6665 t 0.8875 t

Table 5.5: Comparing long-time numerical solutions for H(t)

The dominant term in the long-time solutions, for both the linear and sinusoidal models,

is the O(t) term, and it is this that we must look at to compare the different models.

Looking at the results in Table (5.5) one can see that the long-time average position,

H∞, is consistently larger for the linear model, for all parameter values. This result is

not unexpected when recalling the way the reorientation responses work. In the sinusoidal

model, if a walker is facing completely opposite to the preferred direction (θ = π) then there

is no average response as sin θ = 0. In the linear model the walker will have its maximum

average response in this case. Thus we would expect the linear model to move the walkers

back to the preferred direction quicker and hence have a larger absolute velocity, and this

is what is suggested by the results presented in Table (5.5).


κ 0.1 0.5 1.0 2.0 4.0 8.0

σ2∞

(lin, dτ = 0.1) 2.1033 t 2.6232 t 3.5325 t 5.9851 t 9.5474 t 8.4843 t

σ2∞

(sin, dτ = 0.1) 2.1048 t 2.6368 t 3.5942 t 6.4246 t 12.2222 t 13.938 t

σ2∞

(lin, dτ = 0.3) 2.0909 t 2.5052 t 3.0418 t 3.4944 t 2.3654 t 0.9909 t

σ2∞

(sin, dτ = 0.3) 2.1024 t 2.6096 t 3.4528 t 5.1640 t 4.1195 t 0.1842 t

Table 5.6: Comparing long-time numerical solutions for σ2(t)

5.6.2 Comparing solutions for σ2(t)

Looking at the results in Table (5.6) one can see that the long-time spread about the mean

position σ2∞

, is larger for the sinusoidal model except for κ = 8 and dτ = 0.3. The results

for κ = 8 and dτ = 0.3 are unreliable as we know that σ2x1(t) is not valid for either model

with these values of the parameters.

Excluding the latter result, the spread is always larger for the sinusoidal reorientation

model. This can be explained in a similar way to the result for the average position —

the linear response would seem to be a ‘better’ response in that walkers are redirected to

the preferred direction quicker. Once moving in the preferred direction they are then less

likely to move off in other directions and hence the overall spread about the mean position

will be less.

5.7 Conclusions

In a similar manner to the previous chapter that developed the sinusoidal model, we have

shown in this chapter that it is possible to extend the velocity jump process model and

method of calculating moments used by Othmer et al. (1988). We have extended their

model by using a reorientation probability distribution based on the von Mises distribution

with mean turning angle given by the linear reorientation model of Hill & Hader (1997)

that implicitly includes bias. The original probability distributions used by Othmer et

al. were chosen so that they resulted in a closed system of differential equations for the

moments of the underlying spatial distribution. By using a more realistic reorientation

probability distribution, we have shown that to derive a system of differential equations for

the moments, it is necessary to estimate certain intractable integrals with simple known

functions. Even when working to leading order, by assuming that the reorientation pa-

rameter dτ is small, the system of differential equations is not closed and we have had to

estimate further higher angular moments using Hill & Hader’s equation for the expected

long-time angular distribution (1.102). The final system of differential equations is linear

and readily solved in principle. However, the eigenvalues and other constants in the system

are algebraically cumbersome and general solutions dependent on only the reorientation


parameters κ and dτ are not presented. Instead, solutions have been found for particular

parameter values, although it is straightforward to find solution equations for any pa-

rameter values. For these particular parameter values we have solution equations for the

spatial statistics (V(t), H(t), D2(t) and σ2(t)) of the population of random walkers that

are valid for all time and not just long-time approximations as in the diffusion model. We

also have solution equations for the spread in each direction (D2x1(t), D

2x2(t), σ

2x1(t) and

σ2x2(t)). Because of the various assumptions made during the analysis, the equations may

be not valid over all the parameter space for κ and dτ , and if both are ‘large’ then the

solutions underestimate the spread about the mean position and can give the nonsensical

result of ‘negative spread.’

The linear reorientation model is motivated by the phototactic movement of algae such

as C. nivalis or P. gatunense moving towards a light source, but the model and solutions

presented in this chapter could be used to describe the movement of any population of

random walkers moving in a homogenous environment with linear reorientation.


• We have extended the velocity jump process of Othmer et al. to include a realistic

reorientation probability distribution that implicitly introduces bias to the movement

by including the dependence of the mean turning angle on the the absolute angle

(linear reorientation model) from Hill & Hader’s experimental results.

• We have derived solution equations for the statistics of the underlying spatial dis-

tribution that are valid for all time, these statistics being the mean velocity V(t),

the mean position H(t), the spread about the origin D2(t) and the spread about the

mean position σ2(t). The general form of the equations are algebraically cumbersome

so solutions for particular parameter values have been presented.

• We also have equations that are valid for all time for the spread in each of the

preferred and non-preferred directions, D2x1(t), D

2x2(t), σ

2x1(t) and σ2

x2(t).

• To close and solve the derived system of differential equations we have had to i)

assume that the parameter dτ is small to allow Taylor expansions of the trigonometric

functions, and hence all the solution equations are only leading order approximations;

ii) fit simple known functions to several intractable integrals that occur when deriving

equations for the higher moments, and estimate some further higher moments using

Hill & Hader’s equation for the expected long-time angular distribution.

• Because of these assumptions, the equations for the spatial statistics are only valid

for smaller values of the reorientation parameters to avoid nonsensical results.

• Comparisons between the equations for the average position for the linear and sinu-

soidal models shows that the linear model is ‘better’ in the sense that the distance

moved in the preferred direction is greater.

Chapter 6

Spatial statistics of simulated

random walks

6.1 Introduction

In the previous two chapters we have presented a new model that develops the velocity

jump process model of Othmer et al. (1988) to include the sinusoidal and linear reorien-

tation models of Hill & Hader (1997). During the relevant analysis in each chapter we

made a number of assumptions and the final asymptotic solutions for the statistics of the

underlying spatial distribution are themselves only leading order approximations.

In this chapter we set up and run simulations of biased and correlated random walks

for populations of walkers moving with the reorientation models specified in the previous

chapters (see Sections 4.2 and 5.2). The motivation for carrying out these simulations is

as follows:

i) to test the validity of the asymptotic solutions with respect to simulated random

walks using the same parameters — because of various assumptions made during

the derivation of the solution equations we do not know if they will be valid over all

parameter ranges; and to carry out a detailed study of the effect of the parameters

on the final asymptotic solutions.

ii) to study the effect of changing the sampling length on the angular statistics of a

random walk and to validate the theory of Hill & Hader (1997).

iii) to get information about the spatial statistics for parameter ranges when the theo-

retical model breaks down or for more complicated models that are not covered by

the theoretical model (e.g. spatial dependence of the reorientation parameters).

In this chapter we will carry out systematic studies to investigate point i) above, while ii)

is studied in Chapter 7 and iii) is studied in Chapter 8.

142

CHAPTER 6: Spatial statistics of simulated random walks 143

6.2 Computer simulations of random walks

6.2.1 Simulation of an individual random walk

1) Enter parameter values and initialconditions.

2) Produce uniform random deviates between 0 and 1.

3) Produce a random time step length.

4) Produce a random angle from the von Mises distribution.

5) Calculate new position using random angle and time step length.

6) Stop after required number of steps.

7) Calculate positions for each unit time.

Figure 6.1: Simple algorithm for an individual random walk.

A simple algorithm to simulate the random walk of an individual organism is straight-

forward to program (see Figure 6.1). In the basic algorithm we assume that the walker

moves in a stepwise fashion using the reorientation models described in the previous two

chapters. The walker moves forward with a fixed speed for a time step of random length,

the length of the time step being given by a Poisson process with turning frequency λ

(so the mean run length time is 1/λ). The walker then changes direction, with the new

direction (as an angle) given by a probability distribution with parameters κ and dτ that

are fixed at the start of the simulation. The probability distribution used is the von Mises

distribution as introduced in previous chapters. The mean of this probability distribution

depends on the direction the walker is currently moving in and so introduces bias. In


the basic model there are no other external effects on the movement of the organism, e.g.

environmental factors, interactions or flow.

6.2.1.1 Algorithm to simulate an individual random walk

Step 1

The parameters that are fixed at the start of the simulation are the speed s, the turning

rate λ and the reorientation parameters κ and dτ . The initial position (x0, y0) is assumed

to be (0, 0), while the initial direction θ is chosen at random (p(θ) = 1/2π).

Step 2

A simple loop is used to calculate the new position at each step. To produce a random

time step from a Poisson process and a random angle from the von Mises distribution, it is

first necessary to produce random deviates between 0 and 1. An algorithm that produces

uniform random deviates between 0 and 1 that have no correlation between successive

deviates (RAN1), was taken from Numerical Recipes in C (Press et al., 1992). It produces

an effectively infinite sequence with no correlation between successive values. However the

algorithm needs a seed number and produces the same sequence for the same seed. The

internal computer timer was used to produce different seeds for each simulation.

Step 3

To produce a random time step length as a result of a Poisson process of intensity λ,

another algorithm (GAMDEV), was taken from Numerical Recipes in C (Press et al.,

1992).

Step 4

To produce a random angle from the von Mises distribution, an algorithm by Fisher and

Best (1979) was used. This uses a wrapped Cauchy distribution as an envelope to give

an acceptance-rejection method to produce random angles from the required von Mises

distribution.

Step 5

Using the angle produced from step 4 it is merely simple trigonometry to calculate the

new position from the old position, given the fixed speed and the time step. The position

after each time step is then sent to a data file.


Step 6

A counter keeps track of the total time (the sum of all the random time steps) and when

the required maximum time is reached the simulation stops.

Step 7

The data file produced has the (x, y) position after each random time step. To compare

with the theoretical results, we now calculate the (x, y) position after each unit time

step (i.e. the position at t = 1, t = 2 etc), and save the results in a different data file.

This is simply a linear interpolation of the raw data, and will not have an effect on the

spatial statistics. The effect of changing the sampling length on the angular statistics is

investigated in Chapter 7.

6.2.1.2 Examples of simulated random walks of an individual walker

The following plots are examples of how changing the reorientation parameters (κ and dτ )

and introducing bias alters the movement pattern of the organism. All initial directions are

in the preferred direction θ = 0 (so that the x1-direction from our theory in the previous

chapters corresponds to the y-direction). Each random walk has 1000 steps and starts at

(x, y) = (0, 0), with the same turning rate and speed in each (λ = 1 and s = 1).

Figure 6.2: i) Random walk with κ = 0.1, dτ = 0. The random walk is close to being

completely random (Brownian) motion.

Figure 6.2 shows that when κ ≈ 0 and dτ = 0, the motion is close to random (Brownian

motion). Figure 6.3 shows that when κ is large but dτ = 0, the motion is more correlated


Figure 6.3: ii) Random walk with κ = 2, dτ = 0. The random walk appears more

correlated but there is no overall preferred direction.

Figure 6.4: iii) Random walk with κ = 0.5, dτ = 0.2. The random walk is less correlated

but there is a definite preferred direction (y-direction).


Figure 6.5: iv) Random walk with κ = 4, dτ = 0.3. The random walk is highly correlated

and the preferred direction is clear.

(tendency to move in the same direction as previously) but there is no overall preferred

direction. Figure 6.4 shows that when κ is small and dτ is non-zero, one starts to see a

preferred direction. Figure 6.5 shows that when κ is large and dτ is non-zero then there

is a definite preferred direction and correlation.

6.2.2 Collecting average statistics for a set of random walks

The analysis completed in the previous chapters resulted in equations for the average

statistics for the whole population and not just an individual organism. To be able to

compare the statistics calculated from our theory with statistics from simulations we need

to run simulations for a whole population of organisms and not just one. The simplest way

to achieve this is to simulate a number of random walks with the same given parameters

and then calculate the average statistics for all these random walks. These can then be

compared to the results one gets when entering the same parameters into the equations

derived in the previous chapters. This simple approach is valid provided we assume there

is no interaction between individuals — our basic model does assume this. The next step

is to set up an algorithm to simulate a set of random walks and then calculate the average

statistics for them.


1) Enter parameter values and initialconditions.

2) Simulate an individual random walkwith given parameters.

4) Stop after required number of randomwalks (ie. required population size).

3) Send position data to a results file.

5) Apply required sampling length toresults file.

6) Calculate average statistics for eachtime step from data.

7) Calculate confidence intervals forthe average staistics at each time step.

8) Calculate average statistics for eachtime step from equations.

9) Stop after required number of timesteps.

Figure 6.6: Algorithm used to calculate average statistics for a set of random walks.


6.2.2.1 Algorithm to simulate a population of random walkers

Step 1

As described in the previous section, certain parameters and initial conditions need to be

entered into the algorithm. The parameters are the same as those needed in simulating

an individual random walk. One also needs to enter the population size (number of

random walks to be simulated). All the random walks use the same parameters and initial

conditions. We assume that the population all start at the same point, (x0, y0) = (0, 0),

and initial orientations θ0 are chosen randomly from a uniform distribution on a unit circle

(so for a large population we would expect E(cos θ0) ≈ 0).

Steps 2 and 3

This part of the algorithm is simply the algorithm for an individual random walk repeated

as many times as necessary. This produces a data file with a set of ‘runs’ each of the same

length. Each ‘run’ consists of the (x, y) position at each unit time step.

Step 4

The algorithm includes a counter that counts the number of ‘runs’ completed so far. When

this reaches the required population size the algorithm moves on to the next step.

Step 5

The data file consists of a set of ‘runs’ each of which consists of a list of coordinates at each

time step. It is possible to produce a second data file from this first file where one looks

at the coordinates at every second time step (or third, or fourth etc). This is effectively

changing the sampling length used. This will be investigated further in Chapter 7 when

looking at angular statistics and for the results presented in this chapter the time step

remains unchanged and the data file is thus also unchanged.

Steps 6, 7 and 8

The algorithm now looks through the data file taking the coordinates for the first time step

in each ‘run’ and calculating average statistics from them. The algorithm also calculates

the values predicted by the equations derived in the previous chapter for the average

statistics. This part of the algorithm uses a program taken from Numerical Recipes (Press

et al., 1992) to calculate Bessel functions (BESSI). These steps form a loop in the program,

the algorithm moving onto the next time step in each run after every loop.


Step 9

A counter is again used to track where the algorithm is in the data file. When the end of

the data file is reached the algorithm stops and the results for each time step are presented

in a new data file.

6.3 Simulations to validate theoretical results

In the following systematic study a number of simulations were carried out using the

algorithm in Figure 6.6. The results of the simulation and the expected value given by our

theory are then compared and displayed as a set of plots. The turning rate and speed are

both fixed for all the simulations (λ = 1 and s = 1), this is effectively non-dimensionalising

with respect to time as discussed in Section 4.9.2. The only parameters that are changed

between each simulation run are κ and dτ . Each simulation run used a population of 1000

walkers (n = 1000) and produced statistics from t = 0 to t = 100.

All the subsequent plots are samples of the behaviour of the population. Each simulation

run looks at the movement of 1000 walkers, and by nature each individual run includes

a lot of ‘random noise.’ For this reason no two simulation runs will be exactly the same

and will give slightly different plots each time. What we have included is typical plots of

the average behaviour. When looking at a particular statistic, completely new runs were

simulated. This means for example, that the simulation run to calculate H(t) with κ = 2

and dτ = 0.1 is a different simulation run to the run to calculate σ2(t) with κ = 2 and

dτ = 0.1.

6.3.1 Mean position — H(t)

The mean position is only non-zero in the preferred direction (which we assume to be the

y-direction in our model, so in the notation of the previous chapters x1 = y and x2 = x).

The expected mean position in the non-preferred direction (x2 = x) is always 0. Hence,

in the following studies we look at the mean of the position in the y-direction (i.e. we

use Hy(t)). As there is a relatively large amount of random noise in the mean position

for smaller values of the parameters, even with n = 1000, the comparison between theory

and simulation has been done by comparing the theoretical value with a 95% confidence

interval calculated from the sample mean and variance.

6.3.1.1 Confidence intervals for the population mean

Clarke & Cooke (1992) give the following definitions for the confidence interval of a pop-

ulation mean. In sampling from a normal distribution with known variance σ2 and whose

mean takes some unknown value µ, the sample mean x has a N(µ, σ2/n) distribution.

Also, Z =√n(x− µ)/σ has a N(0, 1) distribution. Using a standard table of the normal


distribution, we can say that

Pr(−1.96 ≤ x− µ

σ/√n≤ +1.96) = 0.95.

This statement is equivalent to

Pr(x− 1.96σ/√n ≤ µ ≤ x+ 1.96σ/

√n) = 0.95. (6.1)

We call the interval in the brackets a 95% confidence interval for µ.

Using the Central Limit Theorem, in general the mean of a non-normal distribution will

tend to be normally distributed as the sample size increases. Let us now suppose that a

large sample is available, of n observations from a non-normal distribution. The statement

in (6.1) is now true with probability approximately 0.95, rather than exactly.

If we do not know the population variance σ2, then we must use the estimated standard

deviation s instead of σ. When the sample size is small the ratio

x− µ

s/√n

has a t-distribution, with (n − 1) degrees of freedom, rather than a normal distribution.

The appropriate 95% confidence interval is

x− s√nt(n−1,0.05) ≤ µ ≤ x+

s√nt(n−1,0.05).

As the sample size n increases, the t-distribution becomes closer to the normal distribution.

In fact, t(120,0.05) = 1.98 and as n increases t(n−1,0.05) approaches the value 1.96. In our

simulations we will be dealing with sample sizes of n = 1000 so an appropriate 95 %

confidence interval for the population mean in our simulations is

x− 1.96s√n≤ µ ≤ x+ 1.96

s√n

(6.2)

To calculate the confidence interval for Hy(t), we must use the sample variance in the

y-direction, σ2y(t), not the sample value of the total variance σ2(t).

6.3.1.2 Results for the sinusoidal reorientation model

Plots comparing the mean position Hy(t) from our theoretical equation with sinusoidal

reorientation (4.93) to simulated random walks are shown in Figures 6.7 — 6.9. When

dτ = 0 there is no bias and no overall drift so that Hy(t) = 0, and plots are omitted.

6.3.1.3 Comments on the sinusoidal reorientation model

The plots displayed in Figures 6.7-6.9 show for various values of κ and dτ , the theoretical

expected value for the mean y-position calculated from our equation for Hy(t) with a

sinusoidal reorientation model (4.93), and a 95% confidence interval calculated from the

simulation mean and variance.


(a) κ = 0.1, dτ = 0.1. (b) κ = 0.5, dτ = 0.1.

(c) κ = 1, dτ = 0.1. (d) κ = 2, dτ = 0.1.

(e) κ = 4, dτ = 0.1. (f) κ = 8, dτ = 0.1.

Figure 6.7: Plots showing theoretical Hy(t) (—), and 95% confidence interval from sim-

ulated (· · ·), against time for sinusoidal reorientation with dτ = 0.1. (The scale used for

each plot is different.)


(a) κ = 0.1, dτ = 0.2. (b) κ = 0.5, dτ = 0.2.

(c) κ = 1, dτ = 0.2. (d) κ = 2, dτ = 0.2.

(e) κ = 4, dτ = 0.2. (f) κ = 8, dτ = 0.2.





(a) κ = 0.1, dτ = 0.3. (b) κ = 0.5, dτ = 0.3.

(c) κ = 1, dτ = 0.3. (d) κ = 2, dτ = 0.3.

(e) κ = 4, dτ = 0.3. (f) κ = 8, dτ = 0.3.





Simulations with 0.1 ≤ dτ ≤ 0.2.

When dτ 6= 0 we have a bias towards the y-direction and there is an overall drift in this

direction. Looking at the plots shown in Figures 6.7 — 6.9, our theoretical solutions are a

good fit to the simulations for all these values of the parameters. There is a larger absolute

velocity as κ and dτ increase.

Simulations with dτ = 0.3.

When dτ = 0.3 we need to consider the limits imposed on κ and dτ as discussed earlier

(see Section 4.9.3). Looking at the plots shown in Figure 6.9, our theoretical solutions

are a good fit to the simulation results up to κ = 4, but when κ = 8 our theoretical

solution appears to overestimate the simulation results slightly. As discussed earlier when

calculating limits on the parameters (Section 4.9.3), this is only a very small relative error

when looking at just the average position but when calculating σ2y(t) this error can result

in nonsensical solutions.

Our theoretical solution for the mean y-position, Hy(t), with sinusoidal reorientation from

(4.93), appears to be a good fit to results from simulations, except for the extreme values

of the parameters κ and dτ . When one approaches the limiting values of these parameters

then our model starts to break down and produce an overestimating error that, although

relatively small, can produce nonsensical results when looking at σ2y(t) as discussed in

Section 4.9.3. This is due to the fact that we assumed dτ ≪ 1 and approximated higher

order moments in our derivation of the solution equations. As dτ and κ get larger then

these assumptions become less valid.

6.3.1.4 Results for the linear reorientation model

Plots comparing the mean position Hy(t) for our theoretical equation with linear reorien-

tation (5.93) to simulated random walks are shown in Figures 6.10 — 6.12. When dτ = 0

there is no bias and no overall drift so that Hy(t) = 0, and plots are omitted.

6.3.1.5 Comments on the linear reorientation model

The plots in Figures 6.10-6.12 show for various values of κ and dτ , the theoretical expected

value for the mean y-position calculated from our equation for Hy(t) with a linear reori-

entation model, and a 95% confidence interval calculated from the simulation mean and

variance.

We have similar results to the sinusoidal model although it should be noted that the linear

model has a larger absolute velocity than the sinusoidal model, as expected.


(a) κ = 0.1, dτ = 0.1. (b) κ = 0.5, dτ = 0.1.

(c) κ = 1, dτ = 0.1. (d) κ = 2, dτ = 0.1.

(e) κ = 4, dτ = 0.1. (f) κ = 8, dτ = 0.1.


ulated (· · ·), against time for linear reorientation with dτ = 0.1. (The scale used for each

plot is different.)


(a) κ = 0.1, dτ = 0.2. (b) κ = 0.5, dτ = 0.2.

(c) κ = 1, dτ = 0.2. (d) κ = 2, dτ = 0.2.

(e) κ = 4, dτ = 0.2. (f) κ = 8, dτ = 0.2.



plot is different.)


(a) κ = 0.1, dτ = 0.3. (b) κ = 0.5, dτ = 0.3.

(c) κ = 1, dτ = 0.3. (d) κ = 2, dτ = 0.3.

(e) κ = 4, dτ = 0.3. (f) κ = 8, dτ = 0.3.



plot is different.)


Simulations with 0 < dτ ≤ 0.3.

When dτ 6= 0 we have a bias towards the y-direction and there is an overall drift in this

direction. Looking at the plots shown in Figures 6.10 — 6.12, our theoretical solutions are

a good fit to the simulations for these values of the parameters. There is a larger absolute

velocity as κ increases.

It is not clear that our theoretical solution starts to break down and overestimate the simu-

lated results as in the sinusoidal model but as discussed previously, even very small relative

errors can produce nonsensical results when calculating theoretical values for σ2x1(t).

Our theoretical model for Hy(t), the mean y-position, for linear reorientation, appears

to be a good fit to results from simulations for all values of the parameters used. This

suggests that, even though we had to fit simple functions to the intractable integrals during

the derivation of the linear solution equations, the linear solution equations for Hy(t) are

possibly a better fit than the sinusoidal solution equations for the more extreme parameter

values.

6.3.2 Average velocity — V(t)

6.3.2.1 Absolute and instantaneous velocity

The equations derived in the previous chapters for V(t), give the instantaneous velocity

at a particular point in time. This is the average direction that the population is moving

in at that instant. Due to the random nature of our model and the simulations, this

instantaneous velocity will include an amount of random noise — even if the population is

moving steadily in the average direction there can still be large random fluctuations in the

instantaneous velocity at any time step. It is more useful to look at the absolute velocity,

H(t)/t. The absolute velocity will have less random fluctuations as t increases, and the

limiting values of both Vy(t) and Hy(t)/t are the same. We have compared theoretical

results to simulations for dτ = 0.1 and dτ = 0.3 only, to illustrate some of the points made

in the previous section.

6.3.2.2 Results for sinusoidal reorientation model

Plots comparing the absolute velocity Hy(t)/t for our theoretical model with sinusoidal

reorientation to simulated random walks are shown in Figure 6.13 and Figure 6.14.


The plots shown in Figures 6.13 and 6.14 confirm the results that we found for the average

position Hy(t). Allowing for the random noise in the simulations, the only plot that

seems to show a large error between theory and simulation is the plot with κ = 8 and


(a) κ = 0.1, dτ = 0.1. (b) κ = 0.5, dτ = 0.1.

(c) κ = 1, dτ = 0.1. (d) κ = 2, dτ = 0.1.

(e) κ = 4, dτ = 0.1. (f) κ = 8, dτ = 0.1.

Figure 6.13: Plots showing theoretical (—), and simulated (· · ·), absolute velocity

(Hy(t)/t) in the y-direction against time for sinusoidal reorientation with dτ = 0.1. (The

scale used for each plot is different.)


(a) κ = 0.1, dτ = 0.3. (b) κ = 0.5, dτ = 0.3.

(c) κ = 1, dτ = 0.3. (d) κ = 2, dτ = 0.1.

(e) κ = 4, dτ = 0.3. (f) κ = 8, dτ = 0.3.

Figure 6.14: Plots showing theoretical (—), and simulated (· · ·) absolute velocity (Hy(t)/t)

in the y-direction against time for sinusoidal reorientation with dτ = 0.3. (The scale used

for each plot is different.)


dτ = 0.3. In this case the theoretical solution is bigger than the solution obtained from

simulations. This fits with the conclusion made earlier that when κ and dτ are both large,

our theoretical solutions break down and overestimate the average position and hence the

absolute velocity as well.

In general, the simulation and theoretical solutions are very similar, although there is a lot

of random noise when t is close to zero — this is because we have a random spread of initial

directions. As time increases, the simulation solution tends to a constant limiting value

that is very close to that predicted by our theory (except for the extreme case discussed

above).

6.3.2.4 Results for linear reorientation model

Plots comparing the absolute velocity Hy(t)/t for our theoretical model with linear reori-

entation to simulated random walks are shown in Figure 6.15 and Figure 6.16.


The plots shown in Figures 6.15 and 6.16 confirm the results that we found for the average

position Hy(t). Allowing for the random noise in the simulations at smaller parameter

values and small t, all the theoretical and simulated plots show a good match.

6.3.3 Measure of spread about the origin — D2(t)

Our theory predicts spread about the origin in both the x and y-directions, D2(t) being the

sum of the spread in each of the directions. As the simulations do not seem to introduce

significant random noise in the average spread about the origin, D2(t), the comparison

between theory and simulation has been made by just comparing the theoretical values

with the simulation values.

As the average position becomes further and further away from the origin (as we expect in

a biased random walk), then D2(t) becomes less useful as a statistic. In the extreme cases

when one is almost moving linearly then the statistic D2(t) ∼ Hy(t)2. The spread about

the mean position σ2(t) is a more useful statistic to look at when dealing with a biased

random walk. When dτ = 0, there is no bias and the spread about the mean position is

the same as the spread about the origin, D2(t) = σ2(t) — this case has been included in

the section looking at σ2(t).


Plots comparing the mean squared displacement D2(t) for our theoretical equation with

sinusoidal reorientation to simulated random walks are shown in Figure 6.17 and Figure

6.18.


(a) κ = 0.1, dτ = 0.1. (b) κ = 0.5, dτ = 0.1.

(c) κ = 1, dτ = 0.1. (d) κ = 2, dτ = 0.1.

(e) κ = 4, dτ = 0.1. (f) κ = 8, dτ = 0.1.


(Hy(t)/t) in the y-direction against time for linear reorientation with dτ = 0.1. (The



(a) κ = 0.1, dτ = 0.3. (b) κ = 0.5, dτ = 0.3.

(c) κ = 1, dτ = 0.3. (d) κ = 2, dτ = 0.3.

(e) κ = 4, dτ = 0.3. (f) κ = 8, dτ = 0.3.


(Hy(t)/t) in the y-direction against time for linear reorientation with dτ = 0.3. (The



(a) κ = 0.1, dτ = 0.1. (b) κ = 0.5, dτ = 0.1.

(c) κ = 1, dτ = 0.1. (d) κ = 2, dτ = 0.1.

(e) κ = 4, dτ = 0.1. (f) κ = 8, dτ = 0.1.

Figure 6.17: Plots showing theoretical (—), and simulated (· · ·), D2(t) against time for

sinusoidal reorientation with dτ = 0.1. (The scale used for each plot is different.)


(a) κ = 0.1, dτ = 0.3. (b) κ = 0.5, dτ = 0.3.

(c) κ = 1, dτ = 0.3. (d) κ = 2, dτ = 0.3.

(e) κ = 4, dτ = 0.3. (f) κ = 8, dτ = 0.3.





The plots shown in Figures 6.17 and 6.18 show similar results to those found when looking

at the average position H(t) and absolute velocity H(t)/t — allowing for random noise

in the simulations, the plots show a good match between theory and simulation for all

values of the parameters except the extreme case when dτ = 0.3 and κ = 8. In this case

the theoretical solution seems to overestimate the simulation results as expected since the

theoretical equation for Hy(t) was also slightly larger than simulation results.

In general, the simulation and theoretical solutions are close, except for the extreme case

discussed above. For small values of the parameters the solutions appear to be linear (the

O(t) term in the theoretical solution is dominant), while for larger values of the parameters

there is a definite curve in the solution plots (the O(t2) term in the theoretical solution is

dominant).


Plots comparing the mean squared displacement D2(t) for our theoretical model with

linear reorientation to simulated random walks are shown in Figure 6.19 and Figure 6.20.


The plots shown in Figures 6.19 and 6.20 show a good match between theory and simu-

lation for all values of the parameters. For small values of the parameters the solutions

appear to be linear (the O(t) term in the theoretical solution is dominant), while for larger

values of the parameters there is a definite curve in the solution plots (the O(t2) term in

the theoretical solution is dominant).

6.3.3.5 Comparing spread in the x and y directions

Recall that in our biased random walk, the preferred direction of movement for our pop-

ulation of walkers is in the y-direction. As we introduce more bias into our random walk

we would expect the average position to move further along the y-direction. We would

also then expect a larger spread about the origin, but for the spread in the y-direction to

be larger than the spread in the x-direction.

The following plots in Figure 6.21 and Figure 6.22 show the spread about the origin in the

x and y directions, D2x(t) and D2

y(t), for dτ = 0.3 and various values of κ for the sinusoidal

and linear models respectively. For smaller values of dτ the difference between D2x(t) and

D2y(t) is less obvious but the general behaviour is the same. In each plot the dotted lines

(· · ·) show the spread about the origin from simulations and the solid lines (—) show the

theoretical values.

From Figures 6.21 and 6.22 one can see that there is a good match between theory and

simulation for both D2x(t) and D2

y(t) for the linear model, while the sinusoidal model is a


(a) κ = 0.1, dτ = 0.1. (b) κ = 0.5, dτ = 0.1.

(c) κ = 1, dτ = 0.1. (d) κ = 2, dτ = 0.1.

(e) κ = 4, dτ = 0.1. (f) κ = 8, dτ = 0.1.


linear reorientation with dτ = 0.1. (The scale used for each plot is different.)


(a) κ = 0.1, dτ = 0.3. (b) κ = 0.5, dτ = 0.3.

(c) κ = 1, dτ = 0.3. (d) κ = 2, dτ = 0.3.

(e) κ = 4, dτ = 0.3. (f) κ = 8, dτ = 0.3.




(a) D2

x(t) with κ = 0.1, dτ =

0.3.

(b) D2

y(t) with κ = 0.1, dτ =

0.3.

(c) D2

x(t) with κ = 2.0, dτ =

0.3.

(d) D2

y(t) with κ = 2.0, dτ =

0.3.

(e) D2

x(t) with κ = 8.0, dτ =

0.3.

(f) D2

y(t) with κ = 8.0, dτ =

0.3.

Figure 6.21: Plots showing D2x(t) and D2

y(t) against time for sinusoidal reorientation with

various values of the parameters. (The scale used for each plot is different.)


(a) D2

x(t) with κ = 0.1, dτ =

0.3.

(b) D2

y(t) with κ = 0.1, dτ =

0.3.

(c) D2

x(t) with κ = 2.0, dτ =

0.3.

(d) D2

y(t) with κ = 2.0, dτ =

0.3.

(e) D2

x(t) with κ = 8.0, dτ =

0.3.

(f) D2

y(t) with κ = 8.0, dτ =

0.3.

Figure 6.22: Plots showing D2x(t) and D2

y(t) against time for linear reorientation with

various values of the parameters. (The scale used for each plot is different.)


good match up to κ = 8 when we know our theoretical results are unreliable. It is clear

that as κ increases the difference between D2x(t) and D2

y(t) becomes greater. The plots

of D2x(t) are linear in t while those for D2

y(t) become more like t2 as κ increases. When

κ = 8, the sinusoidal model overestimates the true value of D2y(t), but this is expected

as our equation for Hy(t) also overestimates simulation results when the parameters are

this large. When κ = 8, the solution for the sinusoidal case for D2x(t) underestimates the

simulation results. In all cases, the linear model gives significantly larger values for D2y(t),

while D2x(t) is similar for both models.

6.3.4 Measure of spread about the mean position — σ2(t)

As was the case with D2(t), our theory predicts spread about the mean position in both

the x and y-directions, σ2(t) being the sum of the spread in each of the directions. As

the simulations do not seem to introduce significant random noise in the average spread

about the mean position, σ2(t), the comparison between theory and simulation has been

done by just comparing the theoretical values with the simulation values.


Plots comparing the spread about the mean position σ2(t) for our theoretical model with

sinusoidal reorientation to simulated random walks are shown in Figures 6.23 — 6.26.


The plots displayed in Figures 6.23 - 6.26 show, for various values of κ and dτ , the theo-

retical expected value for the average spread about the mean position calculated from our

equation for σ2(t) with sinusoidal reorientation. Looking at the solution plots it is clear

that, allowing for the random nature of the simulations, the theoretical solution equation

fits the simulations almost exactly for all smaller values of the parameters, and only breaks

down and underestimates the actual spread when dτ = 0.2 and κ = 8, or dτ = 0.3 and

κ ≥ 4. This is unsurprising as we know that our equations for the spread are likely to be

unreliable at these extreme values of the parameters.

In general, as we increase κ the spread σ2(t) increases until we reach the extreme values of

the parameters when the spread starts to decrease. This is because as we increase the size

of the parameters we are finding that the walkers are tending to all move in the preferred

direction. If we kept increasing the parameters, eventually we would expect motion in

a straight line directly in the y-direction, and the total spread about the mean position

would be close to zero.


(a) κ = 0.1, dτ = 0. (b) κ = 0.5, dτ = 0.

(c) κ = 1, dτ = 0. (d) κ = 2, dτ = 0.

(e) κ = 4, dτ = 0. (f) κ = 8, dτ = 0.

Figure 6.23: Plots showing theoretical (—), and simulated (· · ·), σ2(t) against time for

sinusoidal reorientation with dτ = 0. (The scale used for each plot is different.)


(a) κ = 0.1, dτ = 0.1. (b) κ = 0.5, dτ = 0.1.

(c) κ = 1, dτ = 0.1. (d) κ = 2, dτ = 0.1.

(e) κ = 4, dτ = 0.1. (f) κ = 8, dτ = 0.1.




(a) κ = 0.1, dτ = 0.2. (b) κ = 0.5, dτ = 0.2.

(c) κ = 1, dτ = 0.2. (d) κ = 2, dτ = 0.2.

(e) κ = 4, dτ = 0.2. (f) κ = 8, dτ = 0.2.




(a) κ = 0.1, dτ = 0.3. (b) κ = 0.5, dτ = 0.3.

(c) κ = 1, dτ = 0.3. (d) κ = 2, dτ = 0.3.

(e) κ = 4, dτ = 0.3. (f) κ = 8, dτ = 0.3.





Plots comparing the spread about the mean position σ2(t) for our theoretical model with

linear reorientation to simulated random walks are shown in Figures 6.27 — 6.29.

When dτ = 0, the motion is the same for both the sinusoidal and linear models — both

will have zero response to the bias. Thus plots for linear reorientation with dτ = 0 are

omitted as the behaviour is the same as that shown in Figure 6.23.


The plots displayed in Figures 6.23 — 6.26 show, for various values of κ and dτ , the

theoretical expected value for the average spread about the mean position calculated from

our equation for σ2(t) with linear reorientation. The same conclusions as found with the

sinusoidal model apply — up to the extreme values of the parameters there is an excellent

fit between theoretical and simulation results. At these more extreme parameter values

our theoretical solution underestimates the actual spread.

In general the spread about the mean σ2(t) is less for the linear model than the sinusoidal

model for the same values of the parameters.

6.3.4.5 Comparing spread in the x and y directions

Our theoretical models show a good match to the simulations for the statistic σ2(t) up to

the more extreme values of the parameters. However, as discussed when looking at D2(t),

this statistic is made up of components in the x and y-directions. If dτ 6= 0, these separate

components σ2x(t) and σ2

y(t) are not equal and it is useful to compare the spread about

the mean in each of these directions.

The following plots in Figures 6.30 — 6.33 show the spread about the mean position in the

x and y directions, σ2x(t) and σ2

y(t), for various values of the parameters for the sinusoidal

and linear models respectively. In each plot the dotted lines (· · ·) show the spread about

the mean position from simulations, and the solid lines (—) show the theoretical spread

about the mean position.

From Figures 6.30 — 6.33 one can see that for there is a very good match between theory

and simulation for both σ2x(t) and σ2

y(t) for both the sinusoidal and linear models, except

for the extreme case of dτ = 0.3 and κ = 8.

Sinusoidal model, dτ = 0.3, κ = 8

Looking at Figure 6.31, it is obvious that our model for σ2y(t) breaks down for these

values of the parameters — it is nonsensical to have ‘negative’ spread (see Section 4.9.3).

However, it is an important point to note that the simulated results do show a decrease

in the spread about the mean position. We might expect this spread to tend to zero as

the parameters kept increasing and the motion becomes more like straight line motion. In


(a) κ = 0.1, dτ = 0.1. (b) κ = 0.5, dτ = 0.1.

(c) κ = 1, dτ = 0.1. (d) κ = 2, dτ = 0.1.

(e) κ = 4, dτ = 0.1. (f) κ = 8, dτ = 0.1.




(a) κ = 0.1, dτ = 0.2. (b) κ = 0.5, dτ = 0.2.

(c) κ = 1, dτ = 0.2. (d) κ = 2, dτ = 0.2.

(e) κ = 4, dτ = 0.2. (f) κ = 8, dτ = 0.2.




(a) κ = 0.1, dτ = 0.3. (b) κ = 0.5, dτ = 0.3.

(c) κ = 1, dτ = 0.3. (d) κ = 2, dτ = 0.3.

(e) κ = 4, dτ = 0.3. (f) κ = 8, dτ = 0.3.




(a) κ = 0.1, dτ = 0.1. (b) κ = 0.1, dτ = 0.3.

(c) κ = 2.0, dτ = 0.1. (d) κ = 2.0, dτ = 0.3.

(e) κ = 8.0, dτ = 0.1. (f) κ = 8.0, dτ = 0.3.

Figure 6.30: Plots showing σ2x(t) against time for sinusoidal reorientation with various

values of the parameters. (The scale used for each plot is different.)


(a) κ = 0.1, dτ = 0.1. (b) κ = 0.1, dτ = 0.3.

(c) κ = 2.0, dτ = 0.1. (d) κ = 2.0, dτ = 0.3.

(e) κ = 8.0, dτ = 0.1. (f) κ = 8.0, dτ = 0.3.

Figure 6.31: Plots showing σ2y(t) against time for sinusoidal reorientation with various

values of the parameters. (The scale used for each plot is different.)


(a) κ = 0.1, dτ = 0.1. (b) κ = 0.1, dτ = 0.3.

(c) κ = 2.0, dτ = 0.1. (d) κ = 2.0, dτ = 0.3.

(e) κ = 8.0, dτ = 0.1. (f) κ = 8.0, dτ = 0.3.

Figure 6.32: Plots showing σ2x(t) against time for linear reorientation with various values

of the parameters. (The scale used for each plot is different.)


(a) κ = 0.1, dτ = 0.1. (b) κ = 0.1, dτ = 0.3.

(c) κ = 2.0, dτ = 0.1. (d) κ = 2.0, dτ = 0.3.

(e) κ = 8.0, dτ = 0.1. (f) κ = 8.0, dτ = 0.3.

Figure 6.33: Plots showing σ2y(t) against time for linear reorientation with various values

of the parameters. (The scale used for each plot is different.)


correlated but unbiased random walks the spread always increases as the motion becomes

more like a straight line (Bovet & Benhamou (1988)), and when dτ and κ are small this

is true for our results. However, for a fixed value of dτ , there appears to be a value of κ

after which the spread starts to decrease as the motion becomes more like a straight line.

So even though there is no bias in the x direction, the spread in the direction will start to

decrease. Looking at Figure 6.30(f) it seems that our model for σ2x(t) underestimates the

simulated results in a similar manner to D2x(t) (the statistics are equivalent as Hx(t) = 0).

From the simulation results in Figures 6.30 and 6.31 it is clear that the spread in the

non-preferred direction x is greater than the spread in the preferred direction y for the

extreme values of the parameters discussed above, although both show a decrease in the

spread compared to some smaller values of the parameters.

In all cases the simulated long-time limiting solutions for σ2x(t) and σ2

y(t) appear to be

linear in t, and the spread about the mean position is proportional to the square root of

time — this suggests that the dispersion about the mean position is diffusive.

Linear model, dτ = 0.3, κ = 8

The comments made above for the sinusoidal model apply to the linear model also, al-

though the solution for σ2x(t) is a much better match between theory and simulation. It is

nonsensical to have negative spread and the solution for σ2y(t) breaks down when dτ = 0.3

and κ = 8. In general the spread about the mean in both directions is less for the linear

model than the sinusoidal model for the same values of the parameters.

6.3.4.6 Comparison with an isotropic random walk

From Section 1.1.2.2, we have the expected spread for a completely random (isotropic)

walk

< r2 >=< x2 + y2 >= 4Dt. (6.3)

Recall that when taking the limit of the isotropic random walk we defined δ2/τ ≡ 2D,

thus in a normalized random walk where s = λ = 1, we have D = 12 . From (6.3) we

might expect for a completely random walk using our simulation model that σ2(t) = 2t

and σ2x(t) = σ2

y(t) = t. This is very close to the behaviour observed in the plots displayed

previously for κ = 0.1 (for all values of dτ ), and shows how our model is related to the

standard Brownian diffusion model.

6.4 The effect of the reorientation parameters on fixed time

solutions

In the previous section we have tested the validity of our theoretical solutions for the

sinusoidal and linear reorientation models by comparing to results of simulations. For


smaller values of the reorientation parameters κ and dτ there is a very good match between

simulations and theory for all the statistics of interest. In general, it is only the extreme

values of the parameters discussed previously where our theoretical solutions break down.

In this section we complete a more detailed study into the exact effect of the reorientation

parameters on the statistics of interest as we work through the parameter range.

6.4.1 Fixed time spatial distribution

To compare the effect of changing κ and dτ on the solution equations for the spatial

statistics we fix t = 100. Examples to illustrate the spatial distribution at t = 100 for

various parameter values are shown in Figure 6.34. The plots displayed are for simulation

runs with 500 walkers for clarity.

The dotted circle in the plots shows the maximum possible range of movement and hence

with unit speed s = 1, has radius r = 100. The solid circle show a measure of the spread

in each direction. This has been drawn as an ellipse of length rx and height ry with area

A, given by

A = πrxry = πσ2 = π(σ2x + σ2

y), (6.4)

where σx and σy are calculated from the simulation results. The length and width are

related byrxry

=σxσy. (6.5)

Solving (6.4) and (6.5) for rx and ry gives

rx =

√

σxσy +σ3x

σy, (6.6)

and

ry =

√

σxσy +σ3y

σx. (6.7)

Looking at the plots with κ = 8 and dτ = 0.3 in Figure 6.34, one can clearly see the spread

is greater in the x direction than the y direction. Looking at the other plots, one can see

that the spread does increase as κ increases, until one reaches the extreme values of the

parameters.

6.4.2 The effect of changing the reorientation parameters κ and dτ

In this section we carry out a more detailed study of the effect of the parameters dτ and

κ on the solution equations for the spatial statistics.

The following plots all show how the solutions at t = 100 change as the reorientation pa-

rameters vary. The plots of the results from simulations can be compared to our theoretical

solutions for sinusoidal and linear reorientation. The simulation results may contain some

random noise as we are only looking at one time position but the general behaviour should


(a) Sinusoidal κ = 0.1, dτ =

0.1.

(b) Linear κ = 0.1, dτ = 0.1.

(c) Sinusoidal κ = 2.0, dτ =

0.2.

(d) Linear κ = 2.0, dτ = 0.2.

(e) Sinusoidal κ = 8.0, dτ =

0.3.

(f) Linear κ = 8.0, dτ = 0.3.

Figure 6.34: Example plots of the population position and spread at t = 100.


be clear. All the plots show how the relevant statistic at t = 100 changes as κ increases

from 0 to 8, for 4 values of dτ .

6.4.2.1 Average position H(t)

The plots in Figure 6.35 show how the simulated and theoretical solutions for the average

position in the preferred direction at t = 100, Hy(100), change as the parameter κ increases

from 0 to 8, for dτ = 0, 0.1, 0.2 and 0.3.

(a) Sinusoidal simulation (b) Sinusoidal theory

(c) Linear simulation (d) Linear theory

Figure 6.35: Plots showing Hy(100) against κ for sinusoidal and linear reorientation with

dτ = 0 (—), dτ = 0.1 (· · ·), dτ = 0.2 (−−) and dτ = 0.3 (· − ·).

As observed in the previous section, there is a good match between the simulated and

theoretical solutions displayed in Figure 6.35. The plots all show similar behaviour — for

small values of κ the average position appears to increase linearly as κ increases, with the

greatest rate of increase for the larger values of dτ . Obviously this linear increase cannot


continue indefinitely, as in this non-dimensionalised system the maximum displacement at

t = 100 is given by Hy = 100. As κ increases further the rate of increase in Hy(t) slows

and tends asymptotically to the maximum value of 100 as κ→ ∞.

In general, the linear reorientation model gives a larger displacement in the preferred

direction, although at the extreme values of the parameters there is little difference between

the two models.

6.4.2.2 Spread about the origin D2(t)

The plots in Figure 6.36 show how the simulated and theoretical solutions for the spread

about the origin at t = 100, D2(100), change as the parameter κ increases from 0 to 8,

for dτ = 0, 0.1, 0.2 and 0.3. Figure 6.36 shows a good match between the theoretical and



Figure 6.36: Plots showing D2(100) against κ for sinusoidal and linear reorientation with

dτ = 0 (—), dτ = 0.1 (· · ·), dτ = 0.2 (−−) and dτ = 0.3 (· − ·).


simulated solutions. From the plots the general behaviour as κ increases seems to be a slow

increase in D2(100) at first, then a linear increase, before slowly tending to the asymptotic

limit, which for this system is D2(100) = 10, 000 (corresponding to Hy(100) = 100).


about the origin in the non-preferred direction at t = 100, D2x(100), change as the param-

eter κ increases from 0 to 8, for dτ = 0, 0.1, 0.2 and 0.3. Figure 6.37 shows a good match



Figure 6.37: Plots showing D2x(100) against κ for sinusoidal and linear reorientation with

dτ = 0 (—), dτ = 0.1 (· · ·), dτ = 0.2 (−−) and dτ = 0.3 (· − ·).

between the theoretical and simulated solutions. Recall that the average displacement in

the non-preferred direction is zero, Hx(t) = 0 ∀ t. Thus, the spread about the origin

D2x(t), is the same as the spread about the mean position σ2

x(t). If dτ 6= 0, from Figure

6.37 the general behaviour of D2x(100) as κ increases is a gradual increase before reaching a

maximum value and then decreasing towards zero. For a particular value of κ the spread is


greater for the smaller values of dτ . If dτ = 0, the spread increases linearly before tending

towards the asymptotic limit (which in our system is D2x(100) = 5000) as κ → ∞. Bovet

& Benhamou (1988) observed that, in an unbiased random walk the spread will increase

as the correlation (governed by the parameter κ in our model) increases. However, the

presence of bias in a random walk will decrease the spread in the non-preferred direction

in comparison to a completely unbiased random walk. In general, increasing the bias will

decrease the spread in the non-preferred direction for a fixed value of κ. As κ increases

the walkers will move more like a straight line in the preferred direction, leaving less of

the available movement to spread out in the non-preferred direction and this is exactly

the behaviour to be seen in Figure 6.37.


about the origin in the preferred direction at t = 100, D2y(100), change as the parameter κ

increases from 0 to 8, for dτ = 0, 0.1, 0.2 and 0.3. Figure 6.38 shows a good match between

the theoretical and simulated solutions. The behaviour shown in the plots in Figure 6.38

is almost identical to that shown in Figure 6.36 and the same comments apply. This is

unsurprising since for dτ 6= 0, the spread about the origin in the preferred direction D2y(t)

is always larger than the spread in the non-preferred direction D2x(t), and as κ increases

D2y(t) becomes dominant over D2

x(t), and D2y(t) ∼ D2(t).

6.4.2.3 Spread about the mean position σ2(t)

The plots in Figure 6.39 show how the simulated and theoretical spread about the mean

position at t = 100, σ2(100), change as the parameter κ increases from 0 to 8, for dτ =

0, 0.1, 0.2 and 0.3. In general, Figure 6.39 shows a good match between the theoretical and

simulated solutions. If bias is present then the spread about the mean position σ2(100)

initially increases as κ increases, before reaching a maximum and then starting to decrease

towards zero. For a particular value of κ the spread is less for the larger values of dτ (more

bias). If there is no bias and dτ = 0, then we have a simple correlated random walk and

σ2(100) has the same behaviour as D2(100) with no bias — the spread increases as κ

increases up to the limiting value as described by Bovet & Benhamou (1988). In general,

for the same parameter values the sinusoidal model has more spread about the mean

position than the linear model.

Since the average position in the x direction is zero, we have σ2x(t) = D2

x(t) and the

behaviour of σ2x(t) is given in Figure 6.37. The same comments and conclusions apply.

The plots in Figure 6.40 show how the solutions for the simulated and theoretical spread

about the origin in the preferred direction at t = 100, σ2y(100), change as the parameter

κ increases from 0 to 8, for dτ = 0, 0.1, 0.2 and 0.3. From Figure 6.40 we see that at large

values of κ our solutions break down and give the nonsensical ‘negative spread’ as described

earlier — this is due to the limitations introduced in our modelling process, see Section

4.9.3. However, the qualitative behaviour in the simulated and theoretical solutions is the




Figure 6.38: Plots showing D2y(100) against κ for sinusoidal and linear reorientation with

dτ = 0 (—), dτ = 0.1 (· · ·), dτ = 0.2 (−−) and dτ = 0.3 (· − ·).




Figure 6.39: Plots showing σ2(100) against κ for sinusoidal and linear reorientation with

dτ = 0 (—), dτ = 0.1 (· · ·), dτ = 0.2 (−−) and dτ = 0.3 (· − ·).




Figure 6.40: Plots showing σ2y(100) against κ for sinusoidal and linear reorientation with

dτ = 0 (—), dτ = 0.1 (· · ·), dτ = 0.2 (−−) and dτ = 0.3 (· − ·).


same — for dτ 6= 0 the solutions initially increase, before reaching a maximum and then

decreasing to zero. As κ increases the average motion becomes more like a straight line

in the preferred direction and the spread will decrease to zero corresponding to an exact

straight line when all the movement at each step is in the preferred direction only. As

discussed previously, for a particular value of κ the spread is greater for the sinusoidal

model and also for smaller values of dτ (less bias). For the same parameter values, the

spread about the mean in the preferred direction is less than the spread about the mean

in the non-preferred direction, σ2x(t) > σ2

y(t).

6.5 Simulations with parameters from experimental data

The motivation for including sinusoidal and linear reorientation in our velocity jump pro-

cess model comes from the experimental results of Hill & Hader (1997) as described in

the introductory chapter (Section 1.5). The main results of their experiments are given in

Section 1.5.3. From their data they were unable to determine a value for the angular vari-

ance per unit time σ20 (corresponding to our κ0), and were not certain as to which values

of the sampling time-step τs they should extrapolate back over to estimate the parameter

B−1. Two estimates for the parameter B−1 were given — one by fitting a straight line by

linear regression through the data points for all values of τs, and one by fitting a straight

line by linear regression through the data points for values of τs ≤ 0.4 s only.

The expected long-time angular distributions (1.100) and (1.102) are dependent onB−1/2σ20 ,

so using the observed long-time angular distribution they were able to use (1.100) and

(1.102) with the two given estimates for B−1 to obtain two corresponding estimates for

σ20 . Since both sets of parameter estimates give the same expected long-time angular dis-

tribution they were unable to determine which parameter set was the most realistic. In

this section we show that by observing and analysing the spatial data of experiments as

well as the angular data, it may be possible to overcome the problem described above,

since the two different parameter sets can produce different spatial statistics.

In Hill & Hader’s experiments the mean time step between recorded observations was

τ = 0.08 s. We will use this value as an estimate of the actual time between turns for

our velocity jump model, τ , although it should be noted that this value was obtained as

a consequence of experimental sampling limitations rather than the actual motion of the

algae.

6.5.1 Data set C1 (Sinusoidal model)

From Hill & Hader’s experiments (Section 1.5.3), the data set C1 is obtained from tracking

a population of C. nivalis in a vertical plane guided by negative gravitaxis, the swimming

behaviour being axisymmetric about the vertical axis — this corresponds to our sinusoidal

model. The average speed of movement for this population was found to be v = 55 µms−1.


6.5.1.1 Data set C1:a

The results obtained from the data after fitting a straight line by linear regression through

the data points for all values of τs are as follows

B−1 = 0.37,

σ20 = 1.7, which gives κ0 = 0.95.

With these values and τ = 0.08 s we get the following values for the reorientation param-

eters in our sinusoidal velocity jump model (see Section 4.2).

dτ = τB−1 ≈ 0.03,

σ2δ = τσ2

0 ≈ 0.136, which gives κ = 7.89.

Simulations using the sinusoidal reorientation model with these reorientation parameters

and λ = 1/τ = 12.5 s−1 and s = v = 55 µms−1 were completed and the results are

displayed in Figure 6.41.

Figure 6.41(a) shows the spatial distribution at t = 100 (see Section 6.4.1), while Figures

6.41(b) — 6.41(d) show the theoretical (—) and simulated (· · ·) values for the average

position Hy(t), and the spread in the x and y directions, σ2x(t) and σ2

y(t), respectively.

From Figures 6.41(b) — 6.41(d) it is clear that our theoretical model fits the simulated

results, and thus our theoretical equations for the spatial statistics should be able to

predict the experimental spatial statistics of a population similar to that observed by Hill

& Hader.

6.5.1.2 Data set C1:b

The alternative results obtained from the data after fitting a straight line by linear regres-

sion through the data points for values of τs ≤ 0.4 s only are as follows

B−1 = 0.8,

σ20 = 4, which gives κ0 = 0.27.


eters in our sinusoidal velocity jump model (see Section 4.2).

dτ = τB−1 = 0.064,

σ2δ = τσ2

0 = 0.32, which gives κ = 3.73.

Simulations using the sinusoidal reorientation model with these reorientation parameters

and λ = 1/τ = 12.5 s−1 and s = v = 55 µms−1 were completed and the results are

displayed in Figure 6.42.

From Figures 6.42(b) — 6.42(d) it is clear that our theoretical model fits the simulated

results, and thus our theoretical equations for the spatial statistics should be able to


(a) C1:a at t = 100 (b) C1:a Hy(t)

(c) C1:a σ2

x(t) (d) C1:a σ2

y(t)

Figure 6.41: Plots showing (a) final position and spread (t = 100), (b) Hy(t), (c) σ2x(t)

and (d) σ2y(t) for reorientation parameters from data set C1:a.


(a) C1:b at t = 100 (b) C1:b Hy(t)

(c) C1:b σ2

x(t) (d) C1:b σ2

y(t)


and (d) σ2y(t) for reorientation parameters from data set C1:b.


predict the experimental spatial statistics of a population similar to that observed by Hill

& Hader.

Comparing the results for the two data sets C1:a and C1:b, we see that although Hill

& Hader observed that both data sets have the same long-time angular distribution, the

spatial statistics of a population moving with these parameters are different. The mean

position Hy(t) is slightly larger for set data C1:a but this may be hard to distinguish in

experimental data. However, the spread is significantly larger for data set C1:a (which has

the larger reorientation parameter κ). This corresponds to the simulated results observed

previously that if dτ is small, the spread will increase as κ increases.

In their experiments Hill & Hader could not distinguish which of the data sets C1:a or C1:b

was most appropriate for the observed population just by analysing the angular statistics.

From this example we can see that if the two data sets have a large difference in the

reorientation parameter κ then the observed spatial spread will be significantly different,

and thus it should be possible to distinguish between the two data sets in experiments by

considering the spatial statistics as well as the angular statistics.

6.5.2 Data set C3 (Linear model)

From Hill & Hader’s experiments, the data set C3 is obtained from tracking a population of

C. nivalis in a horizontal plane subject to illumination of 80 klux from the side, resulting

in positive phototaxis — this corresponds to our linear model. The average speed of

movement for this population was found to be v = 60 µms−1.




B−1 = 0.44,

σ20 = 3.2, which gives κ0 = 0.41.


eters in our linear velocity jump model (see Section 5.2).

dτ = τB−1 ≈ 0.035

σ2δ = τσ2

0 = 0.256, which gives κ = 4.48.

Simulations using the linear reorientation model with these reorientation parameters and

λ = 1/τ = 12.5 s−1 and s = v = 60 µms−1 were completed and the results are displayed

in Figure 6.43.

Figures 6.43(b) — 6.43(d) show a good match between theoretical and simulated results.


(a) C3:a at t = 100 (b) C3:a Hy(t)

(c) C3:a σ2

x(t) (d) C3:a σ2

y(t)







B−1 = 0.62,

σ20 = 4.4, which gives κ0 = 0.22.



dτ = τB−1 ≈ 0.05,

σ2δ = τσ2

0 = 0.352, which gives κ = 3.45.



in Figure 6.44.


Comparing the results for the two data sets C3:a and C3:b, we see that similarly to data

set C1, the spatial statistics are different, albeit not as significantly different as data

set C1. The mean position Hy(t) is similar for both C3:a and C3:b, but the spread is

slightly larger for data set C3:a (which has the larger reorientation parameter κ). This

corresponds to the simulated results observed previously that if dτ is small, the spread will

increase as κ increases. The difference in spread between data sets C3:a and C3:b is not as

significant as the difference observed in data set C1, because the difference in the values

of the reorientation parameter κ is not as large. The difference in spread seems significant

though, so as with dataset C1, by looking at both the spatial and angular statistics of

experimental data it should be possible to distinguish between populations moving with

the two different sets of reorientation parameters.

6.5.3 Data set C4 (Linear model)

From Hill & Hader’s experiments, the data set C4 is obtained from tracking a population of

C. nivalis in a horizontal plane subject to illumination of 200 klux from the side, resulting

in positive phototaxis — this corresponds to our linear model. The average speed of

movement for this population was found to be v = 59 µms−1.




B−1 = 0.19,


(a) C3:b at t = 100 (b) C3:b Hy(t)

(c) C3:b σ2

x(t) (d) C3:b σ2

y(t)




σ20 = 0.8, which gives κ0 = 1.84



dτ = τB−1 ≈ 0.015

σ2δ = τσ2

0 = 0.064, which gives κ = 16.14.



in Figure 6.45.

(a) C4:a at t = 100 (b) C4:a Hy(t)

(c) C4:a σ2

x(t) (d) C4:a σ2

y(t)








B−1 = 0.61,

σ20 = 2.8, which gives κ0 = 0.51.



dτ = τB−1 ≈ 0.05,

σ2δ = τσ2

0 = 0.224, which gives κ = 5.03.



in Figure 6.46.


Comparing the results for the two data sets C4:a and C4:b, we see that similarly to data

sets C1 and C3, the spatial statistics are different, with the difference in the spread being

even more significant with this data set. As in the previous cases, the mean position

Hy(t) is similar for both C4:a and C4:b, but the spread is significantly larger for data set

C4:a (which has the larger reorientation parameter κ). This corresponds to the simulated

results observed previously that if dτ is small, the spread will increase as κ increases.

The difference in spread between data sets C4:a and C4:b is more significant than the

differences observed in data sets C1 and C3, because the difference in the values of the

reorientation parameter κ is larger. As with data sets C1 and C3, by looking at both

the spatial and angular statistics of experimental data it should be possible to distinguish

between populations moving with the two different sets of reorientation parameters.

Assuming our velocity jump process model is a reasonable approximation of how a pop-

ulation of the algae C. nivalis moves, then our theory and simulations can give us useful

results to look for in any future experiments. Irrespective of which of the two parameter

estimates used, the algae with gyrotactic motion (sinusoidal model) appear to move less

far than the algae with phototactic motion (linear model), while the phototactic motion

produces the largest displacement with the more intense light (200 klux).

6.6 Conclusions

In this chapter we have presented a computer algorithm to simulate the movement of a

population of random walkers moving with either the linear or sinusoidal velocity jump


(a) C4:b at t = 100 (b) C4:b Hy(t)

(c) C4:b σ2

x(t) (d) C4:b σ2

y(t)




process models introduced in Chapters 4 and 5. Using these simulations, it is possible to

test the validity of the theoretical equations for the spatial statistics derived in Chapters

4 and 5. In general there is good agreement between simulation and theoretical results,

although as expected the theoretical model does start to break down at extreme values of

the allowed reorientation parameter range for dτ and κ.

In general, the mean position in the preferred direction, Hy(t) always increases as the

reorientation parameters increase up to a maximum limit where the motion is a straight

line in the preferred direction. For the same parameter values, the linear model produces

a greater displacement in the preferred direction than the sinusoidal model.

For a particular value of the parameter κ, the spread about the mean position in either

direction is always larger for the smaller values of dτ (less bias). For a fixed non-zero

value of dτ , the spread increases as κ increases up to a maximum value and then starts to

decrease towards zero, the limiting value that corresponds to motion in a straight line in

the preferred direction. If dτ = 0 then we have an unbiased but correlated random walk

similar to Bovet & Benhamou (1988), and the spread about the mean position is the same

as the spread about the origin and will increase up to a limiting value (given by s2t2) as

κ increases.

A useful application of the simulation model has been to complete simulations with pa-

rameter values from the experiments of Hill & Hader. Although populations of walkers

moving with two different sets of reorientation parameters can have the same long-time

angular distribution and similar values for the mean position Hy(t), the spatial spread

σ2(t) can be significantly different between the two populations if there is a large differ-

ence between the values of the two concentration parameters (κ) that correspond to the

spread in the reorientation distributions. This may prove to be a useful result to use if

analysing experimental data and (as in Hill & Hader’s results) it is hard to distinguish

which set of parameters are appropriate for a particular population of random walkers.


• A computer algorithm has been designed to simulate and analyse the movement of

a population of random walkers all moving with the velocity jump process models

with sinusoidal or linear reorientation as introduced in Chapters 4 and 5.

• In general, there is a good match between the theoretical and simulated results for

all the spatial statistics of interest for most of the parameter values in the range

investigated.

• For more extreme parameter values in the range, the models start to break down

— in particular the asymptotic solution for the sinusoidal model for Hy(t) and

the asymptotic solutions for both reorientation models for the spread in the non-

preferred direction D2y(t), seem to produce unrealistic results for extreme values of

the parameters.


• The mean position Hy(t) will increase as the reorientation parameters both increase,

up to a maximum value corresponding to straight line motion (st).

• The average spread about the origin in the non-preferred direction is the same as

the spread about the mean position, D2x(t) = σ2

x(t), since Hx = 0. If dτ = 0 the

spread always increases as κ increases up to a limiting value (s2t2/2). If dτ 6= 0 then

as κ increases, the spread initially increases before reaching a maximum value and

then decreases to zero. For a particular value of κ the spread is greater for smaller

values of dτ .

• The average spread about the origin in the preferred direction D2y(t) increases as

both the reorientation parameters increase, up to a maximum value corresponding

to straight line motion (s2t2).

• The average spread about the mean position in the preferred direction σ2y(t), be-

haves in a similar manner to σ2x(t). If dτ 6= 0 then as κ increases, the spread initially

increases before reaching a maximum value and then decreases to zero. For a partic-

ular value of κ the spread is greater for smaller values of dτ . For the same parameter

values, σ2x(t) > σ2

y(t).

• By analysing the spatial statistics of experimental data it should be possible to

distinguish between random walks with two different sets of reorientation parameters

that produce the same long-time angular statistics as seen in Hill & Hader (1997).

Chapter 7

Angular statistics and the effect of

sampling length

7.1 Introduction

In the previous chapter we used simulations to test the validity of our asymptotic equations

for the spatial statistics of a population of random walkers as well as to investigate the

effect of the reorientation parameters κ and d0 on the spatial statistics. The fact that

we have a good match between theoretical and simulation results, not only means that

we can be confident our asymptotic equations are valid but also that our simulations

work. In this chapter we revisit the results of Hill & Hader (1997) and investigate how

accurate their method is when using simulation data. One of the problems that they found

when analysing experimental data was noise due to heat convection currents and other

environmental factors, and also the fact that for some experiments they only had a small

number of data points. Both these problems can be avoided by running our simulations,

collecting the angular statistics and then completing a similar analysis to Hill & Hader. We

run simulations using known values for the reorientation parameters, compare simulation

results to theory for the expected long-time absolute angular distribution, and see what

effect changing the sampling length has on the turning angle statistics.

7.2 The long-time absolute angular distribution

Section 1.5 gives details of how Hill & Hader (1997) used a simple model of a random walk

on a circle to derive a Fokker–Planck equation (1.93) for the absolute angular distribution

f(θ). Using the assumption that the following results hold as τ → 0,

µδ(θ, τ) = µ0(θ)τ, (7.1)

σ2δ (θ, τ) = σ2

0(θ)τ, (7.2)

208

CHAPTER 7: Angular statistics and the effect of sampling length 209

so that the mean and variance have a linear dependence on τ and both tend to zero as

τ → 0, the following results were obtained.

Sinusoidal reorientation

Assuming that σ20(θ) = σ2

0 is a constant and that

µ0(θ) = −B−1 sin(θ − θ0) (−π ≤ θ, θ0 < π), (7.3)

where θ0 = 0 is the preferred direction and B is the average reorientation time, the long-

time steady state solution to (1.93 ), the Fokker–Planck equation plus boundary conditions,

is given by

f(θ) = M(θ; θ0, 2/Bσ20) =

1

2πI0(2/Bσ20)

exp

(

2

Bσ20

cos(θ − θ0)

)

, (7.4)

where I0(2

Bσ2

0

) is the modified Bessel function of the first kind and zero order.

Linear reorientation

Assuming that σ20(θ) = σ2

0 is a constant and that

µ0(θ) = −B−1θ (−π ≤ θ < π), (7.5)

where θ0 = 0 is the preferred direction and B is the average reorientation time, the long-

time steady state solution to (1.93 ), the Fokker–Planck equation plus boundary conditions,

is given by


, λ = B−1/σ20 , (7.6)

where B(λ) is the normalization function defined by

B(λ) =

(∫ π

−πe−λθ

2

dθ

)

−1

=√λ(√

πerf(π√λ))

−1. (7.7)

See Section 1.5 for further details.

The experimental results of Hill & Hader (1997) were a reasonable fit to the expected

theoretical distributions but they had a lot of noisy data. Using simulations we can verify

that given a population of walkers moving with certain reorientation parameters, the

expected long-time angular distributions in (7.4) and (7.6) are a reasonable fit to data.

When deriving the asymptotic solution equations for the statistics of interest in Chapters

4 and 5 we assumed that we could approximate the higher order angular moments of

our unknown underlying spatial distribution by higher moments of the long-time angular

distributions in (7.4) and (7.6), see Sections 4.6.1 and 5.3.4.1. Using simulations we can

also verify that these higher moments are small in the parameter range that our asymptotic

solution equations are valid, and the approximation using the moments of (7.4) and (7.6)

is reasonable.


7.2.1 Validating the approximation for M0(t)

When closing the systems of differential equations for the linear reorientation model in

Chapter 5, we approximated the higher order moments F3(t), K3(t), M3(t) and also the

moment M0(t) as time-independent constants using the steady state solution (7.6). Using

simulations we can demonstrate that the approximation for M0(t) as a time-independent

constant is reasonable. Figure 7.1 compares values of M0(t) for simulated data and ap-

proximations using the steady state solution (7.6). Note that assuming there is a uniform

spread of initial directions, M0(0) = π2/3 ≈ 3.29. The simulations have been completed

with three values of κ and all have dτ = 0.2 and λ = 1. It is clear from Figure 7.1 that

Figure 7.1: Plots of M0(t) against t. Legend: (- -) simulation κ = 1, (· · ·) simulation

κ = 2, (− · −) simulation κ = 4, (+) approximation κ = 1, (*) approximation κ = 2, (♦)

approximation κ = 4.

the simulated values reach a steady state very quickly. Thus, although the approxima-

tions using Hill & Hader’s steady state solution, (7.6), seem to slightly underestimate the

simulated values, it seems reasonable to approximate M0(t) as a time-independent con-

stant as we have done in Chapter 5. The reason for this slight underestimation of the

simulation results is discussed later in this chapter and seems likely to be because Hill &

Hader’s steady state approximation, (7.6), is valid for continuous random walks. If we

use a smaller time step between turns (larger value of λ) then the approximations become

closer to the simulated results, although the simulations now take longer to reach the

steady state solutions.

7.2.2 Comparing theoretical distributions to simulation results

For the following results we have run simulations of 2000 walkers moving for 500 time

steps, where each time step τ = 0.1 s, and the reorientation parameters are taken from


Hill & Hader’s experimental results so that we can make direct comparisons with their

data.


Plots of the long-time angular distribution using parameters from Hill & Hader’s experi-

mental results with data set C1 are shown in Figure 7.2. Data set C1 is the set of data

collected on algae moving with sinusoidal reorientation due to gyrotaxis, set C1:a being

the reorientation parameter values estimated using all sampling time steps, and set C1:b

being the reorientation parameter values estimated using only sampling time steps for

τs ≤ 0.4 s (see Section 1.5.3).

Simulation p.d.f.Theoretical p.d.f.

Legend

0

0.05

0.1

0.15

0.2

0.25

0.3

–3 –2 –1 1 2 3θ

(a) C1:a : κ0 = 0.95, B−1 = 0.37.


Legend

0

0.05

0.1

0.15

0.2

0.25

0.3

–3 –2 –1 1 2 3θ

(b) C1:b : κ0 = 0.27, B−1 = 0.8.

Figure 7.2: Plots showing theoretical and simulated long-time p.d.f., f(θ), with parameter

values taken from Hill and Hader’s experiments with data set C1.

Both sets of parameter values produce the same long-time theoretical probability distri-

bution for f(θ). This is expected since Hill & Hader used the observed long-time angular

distribution from experiments to estimate σ20 given two different estimates of B−1 (Section

1.5.3). Even with 2000 data points our simulation data is fairly noisy, but the simulated

results seem a reasonable fit to the theoretical distribution. It would be possible to do

various tests to see how good the fit between theory and simulation is (see for example,

Fisher (1993), or Mardia & Jupp (1999)), but for our purposes, it is easier to compare by

looking at the moments of the distributions as we do in the next section.



Plots of the long-time angular distribution using parameters from Hill & Hader’s exper-

imental results with data set C3 and C4 are shown in Figures 7.3 and 7.4 respectively.

Data set C3 is the set of data collected on algae moving with linear reorientation due to

phototaxis with an 80 klux light source, while C4 is the set of data collected on algae

moving with linear reorientation due to phototaxis with a 200 klux light source. Sets

C3:a and C4:a are the reorientation parameter values estimated using all sampling time

steps, and set C3:b and C4:b are the reorientation parameter values estimated using only

sampling time steps for τs ≤ 0.4 s (see Section 1.5.3).


Legend

0

0.05

0.1

0.15

0.2

0.25

0.3

–3 –2 –1 1 2 3θ

(a) C3:a : κ0 = 0.41, B−1 = 0.44.


Legend

0

0.05

0.1

0.15

0.2

0.25

0.3

–3 –2 –1 1 2 3θ

(b) C3:b : κ0 = 0.22, B−1 = 0.62.



As with the sinusoidal model, sets C3:a and C3:b, and C4:a and C4:b produce the same

long-time theoretical probability distribution for f(θ) respectively.

As with the sinusoidal model, there is a reasonable fit between simulation and theoretical

results. We will comment more on the fit by looking at the moments of the distributions

in the next section.

7.2.2.3 Limitations of the theoretical model

The previous simulations were completed with a time step of τ = 0.1 s and produced a

reasonable fit between theoretical and simulation results. Since τ = 0.1 s, the variance

of the turning angle was small, σ2δ = 0.1σ2

0 . The solutions for the long-time angular

distributions in (7.4) and (7.6) are independent of τ since dτ/σ2δ = B−1/σ2

0 , and predict



Legend

0

0.05

0.1

0.15

0.2

0.25

0.3

–3 –2 –1 1 2 3θ

(a) C4:a : κ0 = 1.84, B−1 = 0.19.


Legend

0

0.05

0.1

0.15

0.2

0.25

0.3

–3 –2 –1 1 2 3θ

(b) C4:b : κ0 = 0.51, B−1 = 0.61.



the same theoretical distribution for all values of τ . However when deriving the Fokker–

Planck equation, Hill & Hader made the assumption that (7.3) and (7.5) hold only as

τ → 0.

The following simulations have been run with the same values for σ20 and B−1 but with

τ = 1 s. Thus, there is now a large variance in the turning angle σ2δ when compared to

the previous section.

Figure 7.5 show plots of the long-time angular distribution for data set C1, while Figure

7.6 show the distribution for data set C4, both completed with simulations with τ = 1 s.

From Figures 7.5 and 7.6, it is clear that the simulated data is now not a good fit to

the theoretical expected long-time angular distribution. The fit seems to be worse for

the smaller κ values corresponding to a large value of σ2δ . This makes sense — if one

increased the time step used by each walker so that σ2δ became very large, then at some

point the variance of the turning angle at each step would be greater than the variance of

the expected long-time angular distribution.

This is obviously nonsensical — consider the unlikely situation of a population all facing

the preferred direction after a long-time period. After one further time step all walkers will

now be distributed according to the turning angle distribution and have a larger variance

than the expected theoretical absolute angle distribution. If the walkers start with a

distribution that is not all facing the preferred direction then the subsequent variance

after one step will be even greater. This result is confirmed when looking at the moments

of the long-time angular distribution in the next section.



Legend

0

0.05

0.1

0.15

0.2

0.25

0.3

–3 –2 –1 1 2 3θ

(a) C1:a : κ0 = 0.95, B−1 = 0.37.


Legend

0

0.05

0.1

0.15

0.2

0.25

0.3

–3 –2 –1 1 2 3θ

(b) C1:b : κ0 = 0.27, B−1 = 0.8.

Figure 7.5: Plots showing theoretical and simulated long-time p.d.f., f(θ), for data set C1

with τ = 1.


Legend

0

0.05

0.1

0.15

0.2

0.25

0.3

–3 –2 –1 1 2 3θ

(a) C4:a : κ0 = 1.84, B−1 = 0.19.


Legend

0

0.05

0.1

0.15

0.2

0.25

0.3

–3 –2 –1 1 2 3θ

(b) C4:b : κ0 = 0.51, B−1 = 0.61.

Figure 7.6: Plots showing theoretical and simulated long-time p.d.f., f(θ), for data set C4

with τ = 1.


7.2.3 Moments of the long-time absolute angular distribution

From Section 1.2 the angular moments of a circular distribution f(θ) are given by

an =

∫ π

−πcos(nθ)f(θ) dθ, bn =

∫ π

−πsin(nθ)f(θ) dθ, (7.8)

where bn = 0 ∀n if the distribution is symmetric about θ = 0. We are interested in the

moments for n ≤ 4, corresponding to the moments we defined for our unknown underlying

spatial distribution Fn(t) in (4.15). The moments F3 and F4 were assumed to be time

independent and approximated by a3 and a4, the moments of the expected long-time

angular distribution f(θ) from (7.4) and (7.6) in order to close our system of differential

equations, see Section 4.6.1.

From Section 4.6.1, the angular moments for the expected long-time angular distribution

with sinusoidal reorientation are simply

an =In(z)

I0(z), (7.9)

where z = 2/Bσ20 . The angular moments for the expected long-time angular distribution

with linear reorientation are not as simple and are given in Section 5.3.4.1.

In the following simulations we have calculated the simulated angular moments using

an =1

N

N∑

i=1

cosnθF,i, (7.10)

where each simulation is run with N = 2000 walkers, and θF,i is the final angle of facing

of the i-th walker.

7.2.3.1 First angular moment, n = 1

The plots in Figures 7.7 and 7.8, show how the first angular moment a1 changes as the

concentration parameter for unit time κ0 increases, for sinusoidal and linear reorientation

respectively. Two different plots show results for B−1 = 0.1 and B−1 = 0.5, while each

plot shows the theoretical result, the simulated result using a time step of τ = 0.1 s, and

the simulated result using a time step of τ = 1 s. The latter will thus have a much larger

variance in the turning angle σ2δ (and a larger value of B−1).

Figures 7.7 and 7.8 confirm the main result observed in the previous section — the simu-

lated results for τ = 0.1 s are a reasonable fit to the theoretical expected angular moments,

but the simulated results for τ = 1 s are consistently smaller than the theoretical expected

moments, with the greatest difference being when κ0 is small corresponding to a large

variance in the turning angle distribution. In general, a1 is larger for the linear model

for the same reorientation parameters — this corresponds to the linear model giving a

larger absolute displacement as observed in previous chapters (the moment a1 is the same

moment that results in the equation for the absolute velocity Vy(t) in previous chapters).


(a) a1 , B−1 = 0.1 (b) a1, B−1 = 0.5

Figure 7.7: Plots showing the first angular moment a1 against k0 for the sinusoidal re-

orientation model, with (a) B−1 = 0.1, (b) B−1 = 0.5. Legend: theoretical results (—),

simulation results with τ = 0.1 s (- -), simulation results with τ = 1 s (· · ·).

(a) a1 , B−1 = 0.1 (b) a1, B−1 = 0.5

Figure 7.8: Plots showing the first angular moment a1 against k0 for the linear reorientation

model, with (a) B−1 = 0.1, (b) B−1 = 0.5. Legend: theoretical results (—), simulation

results with τ = 0.1 s (- -), simulation results with τ = 1 s (· · ·).


Similar results are obtained for the angular moment a2; the results are not presented as

we have made no previous assumptions about this moment in our analysis in Chapters 4

and 5.

7.2.3.2 Third angular moment, n = 3

The plots in Figure 7.9 show how the third angular moment a3 changes as the concentration

parameter for unit time κ0 increases and B−1 = 0.5, for sinusoidal and linear reorientation

respectively. Each plot shows the theoretical result, the simulated result using a time step

of τ = 0.1 s, and the simulated result using a time step of τ = 1 s.

(a) Sinusoidal a3 , B−1 = 0.5 (b) Linear a3, B−1 = 0.5

Figure 7.9: Plots showing the third angular moment a3 against k0 with B−1 = 0.5, for (a)

sinusoidal reorientation model (b) linear reorientation model. Legend: theoretical results

(—), simulation results with τ = 0.1 s (- -), simulation results with τ = 1 s (· · ·).

When B−1 = 0.1 the simulated and theoretical values of a3 ≈ 0, and there is a large

amount of relative noise in the simulation data. For this reason results are not shown.

The plots shown in Figure 7.9 show similar results to that seen with the moment a1 —

there is a reasonable fit between the theoretical results and simulations with τ = 0.1

s, but the theoretical solutions overestimate the simulation results when τ = 1 s. In

general, the higher moment a3 is relatively small even with large values of the reorientation

parameters — it seems that our approximation using Hill & Hader’s long-time expected

angular distribution for the higher moments in our system derived in the previous chapters

is reasonable. For the same reorientation parameter values, the linear model has a higher

value of a3.


7.2.3.3 Fourth angular moment, n = 4

The plots in Figure 7.10 show how the fourth angular moment a4 changes as the con-

centration parameter for unit time κ0 increases and B−1 = 0.5, for sinusoidal and linear

reorientation respectively. Each plot shows the theoretical result, the simulated result

using a time step of τ = 0.1 s, and the simulated result using a time step of τ = 1 s.

(a) Sinusoidal a4 , B−1 = 0.5 (b) Linear a4, B−1 = 0.5

Figure 7.10: Plots showing the fourth angular moment a4 against k0 with B−1 = 0.5,

for (a) sinusoidal reorientation model (b) linear reorientation model. Legend: theoretical

results (—), simulation results with τ = 0.1 s (- -), simulation results with τ = 1 s (· · ·).

The plots in Figure 7.10 show similar results to those in Figure 7.9 and the same comments

apply. For the same reorientation parameter values and allowing for simulation noise,

a3 > a4 for both reorientation models.

The long-time expected angular distributions of Hill & Hader, (7.4) and (7.6), are only

likely to be a reasonable fit to experimental or simulation data if the time step used by

the walkers τ is small and the subsequent variance of the turning angle distribution is

also small. If we have a large value for dτ and σ2δ then the moments of the theoretical

distribution will overestimate the true angular moments.

There is a reasonable fit between simulation and theory for the higher order moments that

we approximated in Section 4.6.1. However, in Section 4.6.1 we also made the assumption

that the approximations were time independent and so for small time they may not be as

valid. In general these higher moments are approximately zero except for large values of the

reorientation parameters and this is unlikely to affect our asymptotic solution equations

(as the results in Chapter 6 suggest).


7.3 The effect of sampling length on the angular statistics

of a velocity jump process

In the following analysis of the angular statistics of random walks we refer to the sampling

length, τs, as the fixed time between observations of the spatial position of each trajectory.

By changing the sampling length we change the time between observations — this results

in a linear interpolation of the original trajectory data for each sampling length.

Recall the assumption made by Hill & Hader that, as τ → 0,

µδ(θ, τ) = µ0(θ)τ,

σ2δ (θ, τ) = σ2

0(θ)τ.

Using this assumption Hill & Hader applied different sampling lengths, τs, to their data

and calculated the parameter µ0 for each value of τs. Assuming the linear dependence

on τs they were able to extrapolate back and estimate the value of B−1. This resulted in

two different estimates for B−1, depending on whether they used all values of τs or just

those values such that τs ≤ 5τ (where τ is the average of the observed time step between

turns subject to the limitations of the experimental set-up), see Section 1.5. They knew

that the assumption that the observed value of µδ(θ, τ) = µ0(θ)τs was only valid for small

τs, but their data was quite noisy hence the two possible estimates for B−1. Using these

estimates and the observed long-time angular distribution, they were able to use (7.4) and

(7.6) to estimate the parameter σ20 .

It is possible to repeat the method of Hill & Hader using simulations with a lot more data

and with reorientation parameters that we fix for the whole population of walkers. Thus,

we can compare the known reorientation parameters that were used in the simulations to

the reorientation parameters that are calculated using Hill & Hader’s method.

It should be made clear that from now on when we refer to τ we mean the time step

between turns used by the population of walkers in the original random walk. When we

refer to τs we mean the new time step between points on the trajectories imposed by

changing the sampling length.

When changing the sampling length we use a linear interpolation of the spatial position

at each time step. The algorithm is given in Section 7 of Hill & Hader (1997). Using

this simple linear interpolation the observed spatial statistics will be unchanged, but the

angular statistics will be different. In a biased random walk, Hill & Hader predicted that

as τs → ∞, the p.d.f. of the direction of movement, θ, will become more peaked about

the preferred direction, with a consequent reduction in the mean turning angle µδ(τs)

(averaged over all swimming directions) and thus in σ2δ (τs). This is clearer when looking

at the example plots in the next section.


7.3.1 Examples of changing the sampling length

Figure 7.11(a) shows an individual random walk using the sinusoidal simulation model

(where the turning rate is given by a Poisson process with parameter λ) for 500 time

steps, with s = λ = 1 and reorientation parameters κ = 1 and dτ = 0.3. Figures 7.11(b)—

(f), show the same random walk rediscretized with different fixed sampling lengths.

It is clear from Figure 7.11 that as τs increases the observed speed decreases and the total

distance travelled appears less, although the overall absolute displacement is the same.

Increasing the sampling length, τs, has the effect of smoothing out the trajectory and a

lot of information about the original random walk is lost.

7.3.2 Angular statistics of a velocity jump process with sinusoidal re-

orientation

For the following study we ran the velocity jump process simulation with sinusoidal reori-

entation for 200 time steps with a population of 2500 walkers. Thus, for a unit sampling

length we would expect approximately 500,000 turning angles to collect statistics from

(Hill & Hader had only approximately 5000 turning angles to work with and encountered

problems with small numbers of data points with large sampling lengths). The simulation

has been run with ‘typical’ parameters from Hill & Hader’s experimental results. We as-

sume that τ = 0.08 s, B−1 = 0.6, and σ20 = 2, so that in the simulations we use dτ = 0.05,

σ2δ ≈ 0.2 and correspondingly κ = 6. These parameter values are comparable to the values

in data sets C1 — C4 from Hill & Hader, see Section 1.5.3. As we are only interested in the

angular statistics, we assume that we can rescale with respect to time so that s = λ = 1

for convenience (Section 4.9.2).

Using the data from this simulation run we then apply different sampling lengths τs and

calculate a new set of absolute angles (θ) and corresponding turning angles (δ) for each

sampling length. Each new set of absolute angles is split into 18 bins of size π9 rads, and for

each bin we calculate the mean turning angle µδ(θ) and angular variance σ2δ (θ). Sampling

lengths of less than τ/2 are not used to avoid spurious correlations.

Recall from (7.3), that we expect µδ(θ) = −dτs sin θ and σ2δ (θ) = σ2

δ = constant. Using

the assumption that µδ(θ, τ) decreases linearly as τ decreases we can then calculate an

estimate for B−1, which should be the gradient of the graph of dτs plotted against τs (and

we might expect to find B−1 = 0.05). This is exactly the method used by Hill & Hader

so we are able to compare results.

7.3.2.1 Mean turning angle, µδ(θ)

Plots showing µδ(θ) against θ for various sampling lengths τs are shown in Figure 7.12.

Other sampling lengths were also used but plots are omitted — see Table 7.1.

Hill & Hader encountered a lot of noise in their experimental results and also had only


(a) Poisson time step (λ = 1). (b) τs = 1.

(c) τs = 2. (d) τs= 5.

(e) τs = 10. (f) τs = 50.

Figure 7.11: Plots showing the effect of changing the sampling length τs of an individual

random walk.


(a) τs = 0.6τ (b) τs = τ (c) τs = 2τ

(d) τs = 5τ (e) τs = 10τ (f) τs = 20τ

Figure 7.12: Plots showing how µδ(θ) changes with θ for the sinusoidal model with various

sampling lengths τs. Simulation results for angular bins of π9 rads (—), and functions fitted

by inspection to the data (- -).


a small number of data points for larger sampling lengths. Consequently, they used a

method of least squares to fit a sinusoidal function to their data. In the plots displayed in

Figure 7.12, we have simply fitted a sinusoidal function by inspection — this is reasonable

as there seems to be little noise in the data (except for τs = 20).

7.3.2.2 Variance of the turning angle, σ2δ (θ)

Plots showing σ2δ (θ) against θ for various sampling lengths τs are shown in Figure 7.13.


(a) τs = 0.6τ (b) τs = τ (c) τs = 2τ

(d) τs = 5τ (e) τs = 10τ (f) τs = 20τ

Figure 7.13: Plots showing how σ2δ (θ) changes with θ for the sinusoidal model with various

sampling lengths τs. Simulation results for angular bins of π9 rads (—), and the mean from

the data averaging over all θ (- -).

In the above plots the straight line plotted is the mean of the variance, σ2δ (θ), averaged

over all angles, which Hill & Hader assumed to be a constant. It is clear however, that

for larger sampling lengths the variance is not constant over θ, but small for θ ≈ 0

and significantly larger for θ ≈ ±π. Hill & Hader also encountered this effect in their

experimental results and suggested that there was possibly a weak sinusoidal dependence

of the form σδ(θ) = a− b sin θ. However, from the above (and subsequent) results, if σ2δ (θ)


is a constant for the original random walk but the sampling length is such that τs ≫ τ

then the variance of the turning angle will become small close to θ = 0 and large close to

θ = ±π. It seems likely that this is the effect observed by Hill & Hader and this suggests

that their sampling length was too large (although it was not possible for them to sample

at shorter time scales due to experimental constraints).

A possible explanation for this effect is as follows: walkers that are moving close to the

preferred direction (θ ≈ 0) will become increasingly likely to still be orientated towards

the preferred direction as the sampling length increases, and thus there will be a small

relative variance in σ2δ (θ) for θ ≈ 0. However, a walker moving opposite to the preferred

direction (θ ≈ ±π) will become increasingly likely to turn towards the preferred direction

as the sampling length increases, and this seems to introduce a large relative variance in

σ2δ (θ) for θ ≈ ±π. As suggested by Hill & Hader, if the sampling length of a correlated

and biased random walk keeps on increasing then eventually we would expect to observe a

decrease in the variance of the turning angle as almost every turn will be back towards the

preferred direction — this is observed in later results. Bovet & Benhamou (1988) showed

that if the sampling length of a correlated but unbiased random walk keeps on increasing

then the variance of the turning angle tends to infinity and the walk appears completely

random (Brownian) and all correlation effects are lost.

7.3.2.3 Estimating the reorientation parameters

Table 7.1 shows the amplitude of the functions fitted by inspection for the plots of µδ(θ)

v θ, and the values of the mean calculated over all θ for σ2δ for the sinusoidal model.

τs/τ 0.6 1 2 3 4 5 6

dτs 0.02 0.03 0.06 0.09 0.12 0.15 0.18

σ2δ 0.09 0.15 0.29 0.49 0.59 0.72 0.89

τs/τ 8 10 12 14 16 18 20

dτs 0.24 0.28 0.37 0.42 0.49 0.55 0.60

σ2δ 1.17 1.41 1.73 1.98 2.21 2.41 2.51

Table 7.1: Estimated value for the amplitude of the mean turning angle µδ(θ), and calcu-

lated mean value of σ2δ , for the sinusoidal model with rescaled sampling length τs/τ .

The data from Table 7.1 is plotted in Figure 7.14. In each plot the observed values

from Table 7.1 are represented by a dashed line (- -), and the expected value by a solid

line (—). The expected value is simply calculated from the reorientation parameters

used in the original simulation (including τ = 0.1) so that E(dτs) = B−1τs = 0.5τs and

E(σ2δ (τs)) = σ2

0τs = 2τs.

From Figure 7.14(a) it appears that for the sampling lengths we have used the observed


(a) dτsv τs/τ (b) σ2

δ v τs/τ

Figure 7.14: Plots showing (a) the amplitude of the mean turning angle dτs , (b) variance

of the turning angle σ2δ , against rescaled sampling length τs/τ for the sinusoidal model.

reorientation parameter dτs does increase linearly as the sampling length increases. The

gradient of the plot appears linear for all values of τs unlike the results of Hill & Hader

that appeared to level off if the sampling length was too large. The value of B−1 estimated

from the gradient of the observed values (and recalling that τ = 0.1) is given by

B−1 = dτs/τs = 0.3. (7.11)

So the assumption made by Hill & Hader that the observed reorientation parameter dτs

increases linearly with τs appears to hold for these sampling lengths but the gradient

calculated from these observed values, B−1 = 0.3, underestimates the expected value,

B−1 = 0.5.

From Figure 7.14(b) it can be seen that the mean variance in the turning angle σ2δ increases

linearly with τs only for small values of τs, and for larger values the increase is non-linear

and appears to level off. This fits with the previous argument and Hill & Hader’s comment

that the variance in the turning angle will start to decrease if the sampling length is large

enough. As with the estimate of dτs , the value of σ20 estimated from the gradient of the

plot for small τs, given by

σ20 = σ2

δ/τs ≈ 1.5, (7.12)

is also an underestimate of the known variance used in the original simulation, σ20 = 2.

Even if we use a sampling length of τs = τ so that we are sampling with the same average

time between turns as the original trajectory, we still underestimate the known value of dτ

and σ2δ that we know the walkers are using. Recall that our velocity jump process model

uses a Poisson process to model the time between turns, where τ is only the average time

between turns. It seems likely that by applying a fixed sampling length to the Poisson time


steps in the original random walk some smoothing of the trajectory will have taken place

and/or some ‘turns’ will be wrongly recorded as δ = 0 if a particular part of a trajectory

is in the same direction for a larger time than the sampling length. This could explain

why using a unit sampling length (τs = τ) underestimates the reorientation parameters.

However, as we shall see later in Section 7.4.1, this does not completely explain why at

larger sampling lengths the reorientation parameters are underestimated.

7.3.3 Angular statistics of a velocity jump process with linear reorien-

tation

We use the same simulation parameters and method as described in Section 7.3.2 to look

at the effect of the sampling length on the angular statistics of a velocity jump process with

linear reorientation. Recall from (7.3), that we expect µδ(θ) = −dτsθ and σ2δ (θ) = σ2

δ =

constant. Using the assumption that dτs and σ2δ decrease linearly with τs we can estimate

the reorientation parameters from the observed data exactly as Hill & Hader did.

7.3.3.1 Mean turning angle, µδ(θ)

Plots showing µδ(θ) against θ for various sampling lengths τs are shown in Figure 7.15.


As for the sinusoidal model, in the plots displayed in Figure 7.12 we have simply fitted a

linear function by inspection — this is reasonable as there seems to be little noise in the

data (except for θ ≈ ±π). It would be possible to use a method of least squares as Hill &

Hader did with their data but this seems unnecessary for the level of accuracy we require.

7.3.3.2 Variance of the turning angle, σ2δ (θ)

Plots showing σ2δ (θ) against θ for various sampling lengths τs are shown in Figure 7.16.


The plots in Figure 7.16 show similar behaviour to the sinusoidal model — the variance

in the turning angle is independent of θ for small sampling lengths, but at large sampling

lengths the variance is much greater when θ ≈ ±π compared to the variance when θ ≈ 0.

Possible reasons for this have been discussed previously.


Table 7.2 shows the amplitude of the functions fitted by inspection for the plots of µδ(θ)

v θ, and the values of the mean calculated over all θ for σ2δ for the linear reorientation

model.

The data from Table 7.2 is plotted in Figure 7.17. In each plot the observed values

from Table 7.2 are represented by a dashed line (- -), and the expected value by a solid

line (—). The expected value is simply calculated from the reorientation parameters


(a) τs = 0.6τ (b) τs = τ (c) τs = 2τ

(d) τs = 5τ (e) τs = 10τ (f) τs = 20τ

Figure 7.15: Plots showing how µδ(θ) changes with θ for the linear model with various

sampling lengths τs. Simulation results for angular bins of π9 rads (—), and functions

fitted by inspection to the data (- -).

τs/τ 0.6 1 2 3 4 5 6

dτ 0.019 0.032 0.065 0.09 0.12 0.15 0.18

σ2δ 0.091 0.150 0.296 0.446 0.589 0.731 0.879

τs/τ 8 10 12 14 16 18 20

dτ 0.25 0.30 0.36 0.42 0.5 0.55 0.6

σ2δ 1.154 1.416 1.683 1.804 1.858 1.827 1.766


lated mean value of σ2δ , for the linear model with rescaled sampling length τs/τ .


(a) τs = 0.6τ (b) τs = τ (c) τs = 2τ

(d) τs = 5τ (e) τs = 10τ (f) τs = 20τ

Figure 7.16: Plots showing how σ2δ (θ) changes with θ for the linear model with various

sampling lengths τs. Simulation results for angular bins of π9 rads (—), and the mean from

the data averaging over all θ (- -).


used in the original simulation (including τ = 0.1) so that E(dτs) = B−1τs = 0.5τs and

E(σ2δ (τs)) = σ2

0τs = 2τs.


δ v τs/τ


of the turning angle σ2δ , against rescaled sampling length τs/τ for the linear model.

From Figure 7.17 we have very similar behaviour to the results observed from the sinusoidal

model. From Figure 7.17(a), the estimated value of the reorientation parameter is B−1 =

0.3 which underestimates the known value of B−1 = 0.5, and is the same value as observed

in the sinusoidal model.

From Figure 7.17(b), the estimated value of the mean variance of the turning angle for

small sampling lengths, τs < 10τ , is σ20 = 1.5 which underestimates the known value of

σ20 = 2, and is the same value as observed in the sinusoidal model. For large sampling

lengths the mean variance in the turning angle starts to decrease as observed and discussed

previously.

As with the sinusoidal model, the observed values of the reorientation parameters always

underestimate the known reorientation parameters even with a unit sampling length, τs =

τ .

7.4 Limitations of using the angular statistics to estimate

the reorientation parameters of a velocity jump process

In the previous section we have used the method of Hill & Hader to calculate the reori-

entation parameters of a velocity jump process from the observed data from simulations

using different sampling lengths. For small sampling lengths, the assumption made by

Hill & Hader that the amplitude of the mean turning angle, dτs , and the mean variance

of the turning angle, σ2δ , decrease linearly as τs decreases holds. However, even using a


unit sampling length, τs = τ , the observed values of the reorientation parameters seem to

underestimate the known values used in the simulations.

One possible reason for this is that by applying a unit sampling length, τs = τ , to the

trajectory of a random walk where the time between turns is given by a Poisson process

with average τ , we smooth out the data and some information about the random walk is

lost. Another possible reason is that the method of Hill & Hader is only valid when the

original random walk is continuous and by using a velocity jump process as an approxima-

tion in the simulations we cannot accurately estimate the reorientation parameters from

observed data with different sampling lengths.

7.4.1 The effect of sampling length on the angular statistics of a velocity

jump process with a fixed time between turns

To test if the variable Poisson step length in the original random walk causes the under-

estimation of the reorientation parameters, we carry out simulations in exactly the same

manner and using the same parameters as in Section 7.3.2, except we now use a fixed time

step between turns, τ = 0.1.

7.4.1.1 The mean and variance of the turning angle

Example plots showing µδ(θ) and σ2δ (θ) against θ for fixed τ and τs = τ and τs = 10τ , are

shown in Figures 7.18 and 7.19, for the sinusoidal and linear models respectively. Other

sampling lengths were also used but plots are omitted — see Tables 7.3 and 7.4 and Figures

7.20 and 7.21.

From Figures 7.18 and 7.19, it is clear that when we have unit sampling length τs = τ ,

then dτ = 0.05 and σ2δ ≈ 0.2 as we expect. However, for τs = 10τ it appears that the

observed parameter values underestimate the known values used in the simulations, and

the plots are very similar to those observed in the previous section for the same sampling

length.


Tables 7.3 and 7.4 show the amplitude of the functions fitted by inspection for the plots

of µδ(θ) v θ, and the values of the mean calculated over all θ for σ2δ , for the sinusoidal and

linear reorientation models with fixed time between turns respectively.

The data from Tables 7.3 and 7.4 is plotted in Figures 7.20 and 7.21. In each plot the

observed values from Tables 7.3 and 7.4 are represented by a dashed line (- -), and the

expected values by a solid line (—). The expected values are simply calculated from

the reorientation parameters used in the original simulation (where τ = 0.1) so that

E(dτs) = B−1τs = 0.5τs and E(σ2δ (τs)) = σ2

0τs = 2τs.


(a) τs = τ (b) τs = 10τ

(c) τs = τ (d) τs = 10τ

Figure 7.18: Plots showing how µδ(θ) and σ2δ change with θ for the sinusoidal model with

fixed time between turns.

τs/τ 0.6 1 2 5 10 20

dτ 0.024 0.05 0.08 0.16 0.3 0.65

σ2δ 0.095 0.226 0.339 0.773 1.506 2.755


lated mean value of σ2δ , for the sinusoidal model with fixed time between turns and with

rescaled sampling length τs/τ .


(a) τs = τ (b) τs = 10τ

(c) τs = τ (d) τs = 10τ

Figure 7.19: Plots showing how µδ(θ) and σ2δ change with θ for the linear model with fixed

time between turns.

τs/τ 0.6 1 2 5 10 20

dτ 0.02 0.05 0.08 0.17 0.35 0.66

σ2δ 0.010 0.226 0.340 0.773 1.493 1.653

Table 7.4: Estimated value for the amplitude of the mean turning angle µδ(θ), and cal-

culated mean value of σ2δ , for the linear model with fixed time between turns and with

rescaled sampling length τs/τ .



δ v τs/τ


of the turning angle σ2δ , against rescaled sampling length τs/τ for the sinusoidal model

with fixed time between turns.


δ v τs/τ


of the turning angle σ2δ , against rescaled sampling length τs/τ for the linear model with

fixed time between turns.


Although estimates of the reorientation parameters from the observed data for a velocity

jump process with fixed τ match the known values of the parameters for a unit sampling

length τs = τ , the subsequent behaviour as the sampling length increases is very similar

to that observed in a velocity jump process with variable τ , see Figures 7.14 and 7.17.

From the observed data, the estimates of the reorientation parameters using sampling

lengths larger than the unit length, τs > τ , always underestimate the known values used

in the simulations. It appears that if observing experimental/simulation data from a

velocity jump process with a fixed time between turns, the estimates for the values of the

reorientation parameters underestimate the true values, unless the sampling length used

is the same as the time between turns of the original random walk.

7.4.2 Estimating the reorientation parameters for large and small values

of τ and τs

A possible explanation for the fact that our estimates of the reorientation parameters

using observed data from simulated velocity jump processes, is that the method of Hill &

Hader only holds for a continuous random walk. Thus, our velocity jump processes that

approximate a continuous random walk are likely to produce only approximate estimates

for the reorientation parameters.

It is also worth considering at this point the validity of Hill & Hader’s method in more

extreme velocity jump process models.

7.4.2.1 The sampling resolution and effect of large sampling lengths

We have discussed earlier the comment from Hill & Hader that with a biased and correlated

random walk, the variance in the turning angle will start to decrease if the sampling length

is large enough (unlike an unbiased and correlated random walk where the variance will

keep increasing as the sampling length increases and the walk appears more random). The

simulation results shown earlier confirm this result.

It is also worth considering the effect that increasing the sampling length has on the

amplitude of the mean turning angle, dτs . Consider the linear reorientation velocity jump

model — as the sampling length, τs, increases the turns in the observed trajectory will

become increasingly likely to be back towards the preferred direction whatever the previous

direction of movement. Thus we might expect the mean turning angle to take the form

µδ(τs, θ) = −θ, (7.13)

when τs is sufficiently large. Thus the estimate for dτs is likely to be 1. Increasing the

sampling length further will not affect the form of (7.13) and the estimate of dτs is still

likely to be 1. Consequently we would not expect the increase in dτs to be linear for

large sampling lengths but to gradually tend to the limiting value of 1. The value of


B−1 using these estimates of dτs and the large sampling lengths, τs, is likely to make a

large underestimate of the true value of B−1. A similar argument could be made for the

sinusoidal model although it is not quite as obvious what the form of µδ(τs, θ) will be.

This problem was encountered by Hill & Hader with their experimental data and was the

reason that they made two different estimates for the reorientation parameter B−1. It

seems likely that their estimate using only smaller sampling lengths τs ≤ 0.4 s will be

closer to the true value, although even these smaller sampling lengths are likely to be too

large. However, they also encountered a lot of experimental noise so by only using a few

data points corresponding to τs ≤ 0.4 s they could introduce more errors into the estimate.

7.4.2.2 How the values of the reorientation parameters used in the original

random walk affect observed estimates

Consider our linear velocity jump process model with parameters τ = 1 time unit (t.u.),

B−1 = 1 t.u.−1, so that the non-dimensional parameter dτ = 1 (possibly an unrealistic

model). From (7.5) the mean turning angle will on average move a walker back towards the

preferred direction at each turn. Thus using a similar argument to the previous section,

we can see that by using sampling lengths τs > τ we are unlikely to observe a situation

where dτs > 1, as dτs = 1 corresponds to the case where the mean turning angle is such

that it takes all the walkers back to the preferred direction. Thus, as we increase τs we will

not observe an increase in dτs past 1, and any estimates of the reorientation parameters

using Hill & Hader’s method will give a large underestimate of the true value.

Applying this argument to velocity jump processes with τ < 1 t.u., it seems likely that

the larger our value of τ and the less like a continuous random walk our model is, the

less useful Hill & Hader’s method will be to estimate the reorientation parameters. We

illustrate this point in the next section.

7.4.2.3 Estimating the parameter B−1 using the linear velocity jump model

with different values of τ

Using a similar method as in the previous sections we have studied simulated results for

the linear velocity jump process model with parameters B−1 = 0.5, σ20 = 2 and various

values of τ , so that the parameters dτ and κ are different for each walk but the expected

observed values should be the same.

We define the parameter ρ to be the ratio between the expected observed reorientation

parameter B−1 and the estimate from the observed data B−1,

ρ =B−1

B−1, (7.14)

so that ρ = 1 corresponds to a perfect estimate of B−1 from the data, ρ < 1 corresponds to

an underestimate, and ρ > 1 an overestimate. Table 7.5 shows the values of ρ calculated


from velocity jump processes using different values for τ , the average time between turns

in the original random walk (not to be confused with τs which is the various sampling

lengths applied later).

τ 0.2 0.15 0.1 0.05 0.01 0.001

ρ 0.58 0.59 0.61 0.70 0.8 0.98

Table 7.5: Values of ρ, the ratio between the expected and observed values of B−1 with

the corresponding average time step between turns in the original random walk, τ .

In general, when estimating the observed reorientation parameters for the above random

walks in Table 7.5 we found that the smaller the value of τ , the larger the relative value of

τs could be before the linear relationship between dτs and τs started to break down. This

fits with earlier comments that the linear relation breaks down when dτ is large and dτs

becomes close to 1 for small sampling lengths.

The data in Table 7.5 has been plotted as a log-plot (with base 10) in Figure 7.22.

Figure 7.22: Log-plot of − log10(τ) against ρ.

The results that are plotted in Figure 7.22 confirm the suggestion that as our velocity jump

process becomes closer to a continuous random walk, as the average time between turns

τ decreases, the method of Hill & Hader produces estimated values of the reorientation

parameters that are much closer to the true values used in the simulations. For the values

of τ used, the slope in Figure 7.22 appears almost linear — it seems unlikely this will be

the case if we decrease τ even further, as in such cases we would expect ρ→ 1.

A similar simple was study was completed looking at the variance of the turning angle,

σ2δ , and the main conclusion was the same as above — as τ decreases the estimate of σ2

0

becomes closer to the known value of σ20 used in the simulations. However, for the same

value of τ the value of ρ for the parameter B−1 was greater than ρ for σ20 — there was a

greater relative error with the estimate of σ20 . This suggests that the estimate of σ2

0 is less

reliable than the estimate of B−1 for a particular value of τ .


7.5 Conclusions

In this chapter we have revisited the experimental results and method of analysis of the

angular statistics of Hill & Hader (1997). Their equation for the long-time absolute angular

distribution is shown to be a good fit to simulation results if the time step between turns, τ ,

used in the simulation is small. If τ is large and we use the same value for the unit variance

σ20 , then the subsequent variance of the mean turning angle σ2

δ is large and the absolute

angular distribution will not have an angular variance that is less than the variance of

the turning angle. Thus the moments of the expected distribution will overestimate the

simulated values in this case.

In general, the higher order moments, F3 and F4, approximated in the derivation of the

asymptotic solution equations in Chapters 4 and 5 and assumed to be time independent,

are either close to zero or a good fit to the expected values using Hill & Hader’s expected

long-time angular distribution. However, for large reorientation parameter values the

approximation breaks down, as observed in simulations in Chapter 6.

It is possible to use a similar method to Hill & Hader to estimate the reorientation parame-

ters used in simulated velocity jump processes. For small sampling lengths the assumption

made by Hill & Hader that the observed amplitude of the mean turning angle and the ob-

served mean variance of the turning angle will decrease linearly with the sampling length

appears to hold. However, the estimates from the observed data underestimate the known

values used in the simulations. Decreasing the average time step between turns in the

original random walk, so that the velocity jump process becomes more like a continuous

random walk, results in estimates for the reorientation parameters that are much closer to

the known values used in simulations. In general, if the sampling length used to estimate

the parameters is too large then the assumption that there is a linear relation between

the sampling length τs and the observed parameters dτs and σ2δ (τs) breaks down, so that

estimates of B−1 and σ20 are likely to be too small. As the sampling length increases,

each observed turn is more likely to result in a move back towards the preferred direction

and consequently we would expect the variance to decrease and this is observed in simu-

lations. We also expect the observed amplitude of the mean turning angle dτs to tend to

the limiting value of 1 (for the linear model) and not keep on increasing indefinitely.

Returning to Hill & Hader’s experimental results, it seems likely that their estimate using

only the small sampling lengths, τs ≤ 0.4 s will give a better estimate of the true value

of B−1 than an estimate using all the sampling lengths. However, comparing their results

to the results of simulations it appears that even their smaller sampling lengths were too

large and may underestimate the parameter values. It should be noted however that by

using only the smallest sampling lengths to estimate the parameters there are fewer data

points and there is more chance of errors being introduced into the estimate.



• The theoretical long-time angular distributions derived from the Fokker–Planck

equation by Hill & Hader for the sinusoidal and linear reorientation models are

only valid when the variance of turning angle distribution at each step, σ2δ , is small.

For random walks that approximate a continuous random walk such that σ2δ = σ2

0τ ,

this implies that for the theoretical p.d.f. to be valid, the time step between turns,

τ , must be small.

• The higher order moments F3(t) and F4(t) approximated in Chapter 4 using the long-

time distribution of Hill & Hader, are approximately zero except for large values of

the reorientation parameters when there is a good fit between theory and simulation

if σ2δ is small. The assumption in Chapter 4 assumed time independence of these

higher order moments — this is unlikely to have an effect on the asymptotic solution

equations unless the reorientation parameters are large.

• Using our simulations of the velocity jump models described in Chapters 4 and

5, it is possible to use the method of Hill & Hader to estimate the reorientation

parameters B−1 and σ20 from the observed data using different sampling lengths,

τs. The assumption made by Hill & Hader that the parameters decrease linearly as

the sampling length decreases appears to hold. However, estimates of the parameter

values calculated from the observed results underestimate the known values used in

the simulations.

• Estimates of the reorientation parameters from observed data of simulated velocity

jump processes with fixed time between turns, τ , give similar results — except for a

unit sampling length, τs = τ , the estimates from the data always underestimate the

known values used in the simulations.

• If the sampling length, τs, becomes too large then the variance in the turning angle

will start to decrease, and the amplitude of the mean turning angle dτs will tend to 1

(in the linear model) and there is no longer a linear relationship with τs. Thus, esti-

mates of the values of the reorientation parameters B−1 and σ20 with large sampling

lengths are likely to underestimate the true values. It seems likely that the sampling

lengths used by Hill & Hader were too large, although their smallest sampling length

used was governed by experimental constraints.

• If the average time between turns, τ , in the original velocity jump model is made

smaller the subsequent estimates of the reorientation parameters B−1 and σ20 are

likely to be closer to the true values than estimates made from simulations with

larger values of τ .

Chapter 8

Further modelling with computer

simulations

8.1 Introduction

The asymptotic equations for the statistics of interest we derived in Chapters 4 and 5 are

only valid for certain parameter ranges (namely ‘small’ κ and dτ ). However, when running

computer simulations as described in Chapter 6, there is no need to be restricted to these

‘small’ values of the parameters and using simulations it is also possible to investigate

other scenarios such as spatially dependent parameters or a changing preferred direction.

In particular, we shall study

i) the effect that extreme values of the reorientation parameters κ and dτ have on the

spatial statistics (i.e. parameter values where the theoretical equations are no longer

valid);

ii) allowing the reorientation parameters κ and dτ to be spatially dependent and the

subsequent effect on the spatial statistics;

iii) allowing a variable preferred direction or a preferred direction that is dependent on

the spatial position, a simple example being if the preferred direction is to a point

source.

8.2 Simulations with parameter values outside the limits of

the theoretical models

The plots in the following sections show results of simulations run using extreme values

of the parameters, using the same computer program as described and used in Chapter

6. We fix the parameters for the speed and turning frequency, λ = s = 1, so that we

effectively have a non-dimensionalised system dependent on only the parameters κ and dτ

239

CHAPTER 8: Further modelling with computer simulations 240

(see Section 4.9.2). Using the same notation as Chapter 6, we assume that the preferred

direction is always the y direction.

8.2.1 The effect of the parameter κ

The concentration parameter κ can be thought of as the ‘swimming ability’ (see Section

4.2.4) and gives a measure of the spread of the reorientation angle distribution. For large

values of κ the reorientation distribution is sharply peaked and the average motion becomes

less random and more like a straight line.

8.2.1.1 Average position — Hy(t)

The plots in Figure 8.1 show how the simulated mean position at t = 100, Hy(100), for

sinusoidal and linear reorientation changes as the parameter κ increases over a large range,

for dτ = 0.1 and dτ = 0.3 (compare to Figure 6.35).

(a) Sinusoidal (b) Linear

Figure 8.1: Plots showing Hy(100) against κ for sinusoidal and linear reorientation for

dτ = 0.1 (—), and dτ = 0.3 (· · ·).

The plots in Figure 8.1 show the same behaviour observed previously in Figure 6.35. For

both reorientation models, there is initially a linear increase in Hy(100) as κ increases,

but as κ increases further the rate of increase of Hy(100) slows down and the solution

tends asymptotically to the maximum possible value given by Hy(100) = 100 for this

non-dimensionalised system. For the same parameter values, the linear model produces

a larger displacement in the preferred direction. For a particular value of κ, the larger

displacement in the preferred direction is for the larger value of dτ .

If we assume that the ‘optimal’ motion is that which produces the largest displacement in

the preferred direction (i.e. a straight line) then clearly the larger the value of κ the more


‘optimal’ the motion will be. However, as the rate of increase of Hy(100) decreases as κ

increases, the greatest relative increase in Hy(100) is when κ is small.

In a biological sense it could be argued that if a population has a poor average orientat-

ing/swimming ability (small κ) then a small increase in this ability will produce a large

difference in the average displacement. However, if the population has a good average

orientating/swimming ability (large κ), then a similar increase in the ability would not

produce much difference to the average displacement.

8.2.1.2 Spread about the mean position — σ2(t)

The plots in Figures 8.2 and 8.3 show the simulated spread in the x-direction and y-

direction about the mean position at t = 100, σ2x(100) and σ2

y(100) respectively. Plots

show how the spread for sinusoidal and linear reorientation changes as the parameter κ

increases over a large range, for dτ = 0.1 and dτ = 0.3 (compare to Figures 6.37 and 6.40).


Figure 8.2: Plots showing σ2x(100) against κ for sinusoidal and linear reorientation for

dτ = 0.1 (—), and dτ = 0.3 (· · ·).

Allowing for the random nature of the simulations, the plots in Figures 8.2 and 8.3 all

show the same qualitative behaviour as seen previously in Figures 6.37 and 6.40. Initially

there is an increase in the spread as κ increases up to a maximum value, and then the

spread decreases asymptotically towards zero as κ increases further. In general, the spread

is greater for the smaller value of dτ , and smaller for the linear model — as the motion

becomes more like a straight line in the preferred direction (more ‘optimal’), the spread

will decrease. For the same parameter values, the spread in the non-preferred direction

(x) is greater than the spread in the preferred direction (y).



Figure 8.3: Plots showing σ2y(100) against κ for sinusoidal and linear reorientation for

dτ = 0.1 (—), and dτ = 0.3 (· · ·).

8.2.1.3 Examples of the population spread at t = 100

The plots in Figure 8.4 show examples of the spread at t = 100 for a population of walkers

moving with dτ = 0.1 and various values of κ. As in Section 6.4.1, the maximum possible

displacement is marked as a dotted circle, while an estimation of the simulation average

spread is marked as a solid circle.

The plots in Figure 8.4 illustrate the points made previously — for a fixed value of dτ , as

κ increases the average motion becomes more like a straight line directly in the preferred

direction and consequently the average displacement increases and the average spread de-

creases. For the same parameter values the linear model has a larger average displacement

and a smaller spread.

8.2.2 The effect of the parameter dτ

The parameter dτ can be thought of as ‘sensing ability’ (see Section 4.2.4) and it controls

by how much an individual walker will reorientate itself back to the preferred direction at

each step. It is not necessarily the case that we will get ‘optimal’ motion by increasing dτ

to be as large as possible (as was the case with changing κ).



sinusoidal and linear reorientation changes as the parameter dτ increases from 0 to 3 for

the sinusoidal model, and from 0 to 2 for the linear model, for κ = 1 and κ = 4.

It is clear from the plots in Figure 8.5 that there is an optimal value of dτ that produces

the largest displacement in the preferred direction. Unlike the parameter κ, if we increase


(a) Sinusoidal κ = 0.1 (b) Linear κ = 0.1

(c) Sinusoidal κ = 10 (d) Linear κ = 10

(e) Sinusoidal κ = 50 (f) Linear κ = 50

Figure 8.4: Plots showing distribution at t = 100 for sinusoidal and linear reorientation

for dτ = 0.1 and κ = 0.1, κ = 10 and κ = 50.



Figure 8.5: Plots showing Hy(100) against dτ for sinusoidal and linear reorientation for

κ = 1 (—), and κ = 4 (· · ·).

dτ further then there is a decrease in the average displacement.

The results for the sinusoidal model in Figure 8.5(a) show that there is a different optimal

value of dτ for each value of κ. When κ = 1 the maximum average displacement is when

dτ ≈ 1.5, but when κ = 4 the maximum average displacement is when dτ is slightly larger

than 1. The results for the linear model in Figure 8.5(b) show that for both values of κ

the maximum average displacement appears to be when dτ = 1.





show how the spread for sinusoidal and linear reorientation changes as the parameter dτ

increases from 0 to 3 for the sinusoidal model, and from 0 to 2 for the linear model, for

κ = 1 and κ = 4 (compare to Figures 6.37 and 6.40).

From Figure 8.6 the behaviour of σ2x(100) as dτ increases is similar for both the linear

and sinusoidal reorientation models. For small dτ the spread is much larger for the larger

value of κ, but as dτ initially increases the spread decreases at a faster rate for the larger

value of κ, and for larger values of dτ the spread is larger for the smaller value of κ. For

the larger parameter values the motion becomes more like a straight line and the spread

is smaller. When dτ increases past 1 it seems that there is a slight increase in the spread

for the larger value of κ although this may be due to random noise in the simulations.

From Figure 8.7(a) the behaviour of σ2y(100) as dτ increases for the sinusoidal model is

similar to the behaviour of σ2x(100) and the same comments apply.

From Figure 8.7(b) the behaviour of the linear model as dτ increases is clear. For smaller



Figure 8.6: Plots showing σ2x(100) against dτ for sinusoidal and linear reorientation for

κ = 1 (—), and κ = 4 (· · ·).


Figure 8.7: Plots showing σ2y(100) against dτ for sinusoidal and linear reorientation for

κ = 1 (—), and κ = 4 (· · ·).


values of dτ the spread is larger for the larger value of κ, and the spread decreases to a

minimum when dτ at which point the spread is smaller for the larger value of κ (since the

motion is more like a straight line). As dτ increases past 1, the spread increases and the

plot actually appears to be symmetric about dτ = 1 so that the spread when dτ = 2 is

approximately the same as when dτ = 0. We shall explain this result in the next section.



moving with κ = 4 and various values of dτ . As in Section 6.4.1, the maximum possible

displacement is marked as a dotted circle, while an estimation of the simulation average

spread is marked as a solid circle.

The plots in Figure 8.8 illustrate the points made previously — for a fixed value of κ, as

dτ increases up to 1 the motion becomes more like a straight line in the preferred direction

and the average displacement increases and the spread decreases for both reorientation

models. Increasing dτ further past 1 has little effect on the sinusoidal model, but the linear

model shows completely different behaviour — there is no average displacement and the

spread is much larger in the preferred direction than in the non-preferred direction. This

behaviour is explained in the next section.

8.2.3 Theoretical optimal value of dτ

We have seen in the plots in Figures 8.5(a) and 8.5(b) that if we increase the parameter

dτ too much then we produce motion that has a smaller average displacement in the

preferred direction. However the sinusoidal and linear models appear to have different

values of ‘optimal’ dτ that produce the maximum displacement in the preferred direction.

Recall that at each turn in an individual random walk there is a mean correction term, µδ,

that is dependent on the absolute direction of movement θ. This mean correction term is

different for our two reorientation models (see Section 1.5).

8.2.3.1 Sinusoidal model

In this model we have

µδ = −dτ sin θ.

At any particular turning point in the random walk we assume the walker will be facing

an absolute direction θ and is about to turn through an angle δ. To turn back to the

preferred direction (θ = 0) in one turn would require the walker to turn through δ = −θ.If the walker turns through a smaller angle (δ < θ) then it will not have corrected enough

to be moving in the preferred direction, while if it turns through a larger angle (δ > θ)

then it will have corrected too much and will have turned past the preferred direction.


(a) Sinusoidal dτ = 0.1 (b) Linear dτ = 0.1

(c) Sinusoidal dτ = 1 (d) Linear dτ = 1

(e) Sinusoidal dτ = 2 (f) Linear dτ = 2


for κ = 4 and dτ = 0.1, dτ = 1 and dτ = 2.


Equal angular spread

The optimal value of dτ will depend on what angle θ the walker is facing and will be

different for each walker. However, in our simple model the parameter dτ is a fixed

constant so it is necessary to find the value of dτ that minimizes the difference between θ

and dτ sin θ for all the possible values of θ.

Let Td represent the total difference between θ and dτ sin θ over all values of θ for the

population for a fixed value of dτ . The function θ− dτ sin θ can be positive or negative so

to find the total difference we must look at the absolute value of the function, |θ−dτ sin θ|.Thus, we have

Td = 2

∫ π

0|θ − dτ sin θ| dθ = 2

∫ a

0(dτ sin θ − θ) dθ + 2

∫ π

a(θ − dτ sin θ) dθ,

where a is the value of θ such that θ = dτ sin θ. Evaluating the integrals gives

Td =π2

2− 2dτ cos a− a2. (8.1)

By definition θ = a when θ = dτ sin θ, so a = dτ sin a, or dτ = a/ sin a. Substituting this

into (8.1) gives

Td =π2

2− 2a cos a

sin a− a2. (8.2)

To find the optimal value of a and hence the optimal value of dτ , we must first calculate

the minimum value of Td.dTdda

=−2 cos a

sin a+

2a cos2 a

sin2 a.

The minimum of Td occurs when dTda = 0, so a = π/2. If a = π/2 then dτ = π/2. So

averaging over all θ in the sinusoidal model the optimal value is dτ = π/2.

Non-equal angular spread

The above analysis is perfectly valid if we assume that all angles θ are equally likely to

occur throughout the whole population and then the optimal value of dτ can be found by

minimizing the function Td for these θ. However, from Hill & Hader’s analysis we know

that for a biased random walk with sinusoidal reorientation, the long-time limit of the

angular distribution is not uniform and from (1.100) and the results of Chapter 7, a good

approximation is given by

f(θ) =1

2πI0(z1)e(z1 cos(θ)), (8.3)

where z1 = 2B−1/σ20 . From Section 4.2, we have B−1 = dτ/τ and σ2

0 = σ2/τ =

−2 ln(A1(κ))/τ , where A1(κ) = I1(κ)I0(κ) , and substituting this into the above equation gives

f(θ) =1

2πI0(z2)e(z2 cos(θ)), (8.4)


where z2 = − dτ

lnA1(κ) Thus, a better estimate for the optimal value of dτ assuming a

non-equal angular spread would be found by minimizing the function

Td = 2

∫ π

0f(θ)| θ − dτ sin θ| dθ, (8.5)

where f(θ) is given by (8.4). From (8.4), Td will be dependent on both κ and dτ . If we

fix κ we can then find the value of dτ that minimizes Td. This has been done numerically

using Maple 8. Figure 8.9 shows how the optimal value of dτ , which we write as dopt,

changes as κ increases. It is clear from Figure 8.9 that when κ ≈ 0, and the angular

Figure 8.9: Plot of dopt against κ for sinusoidal reorientation.

distribution is almost uniform, then dopt ≈ π2 as calculated previously. As κ increases,

and the angular distribution becomes more peaked about θ = 0, dopt initially decreases

linearly but then decreases asymptotically tending to the value 1, which agrees with the

behaviour suggested by the simulation results in Figure 8.5(a).

Because of the nature of the sinusoidal reorientation model and the fact that dopt > 1,

it seems the optimal reorientation behaviour is that which results in some walkers (those

close to θ = 0) overcorrecting and moving past the preferred direction, while others (those

close to θ = π) do not reorient far enough back to the preferred direction. When κ is large

and the population angular distribution is more peaked about θ = 0, it makes sense that

the dopt parameter decreases as there are now more walkers close to θ = 0 that are liable

to overcorrect if dτ > 1.

8.2.3.2 Linear model

The analysis for this case is more straightforward. Looking at the simulation results in

Figure 8.5(b), it is clear that the largest displacement seems to occur when dτ = 1 for

both values of κ. We can show that dτ = 1 is the optimal value in a similar way to the


sinusoidal model by minimizing the total difference between θ and dτθ. Let Td be this

difference, defined by

Td = 2

∫ π

0g(θ) |θ − dτθ| dθ, (8.6)

where g(θ) is the long term angular distribution defined in (1.102). This is clearly min-

imized when Td = 0, corresponding to dτ = 1. Thus, dopt = 1 is the optimal value for

dτ in the linear reorientation model. If dτ < 1 then the population will all under-correct

on average, while if dτ > 1 the population will all overcorrect on average, and this agrees

well with the results displayed in Figure 8.5(b).

We can now explain the results observed in Figures 8.7(b) and 8.8(f). We observed that

if dτ > 1 the spread in the preferred direction increases as dτ increases up to a maximum

at dτ = 2. If for example, dτ = 2 then the mean turning angle is given by µδ = −2θ

and all walkers will on average always overcorrect past the preferred direction. With this

mean turning angle, a walker moving parallel to the preferred y-direction with θ = 0

or θ = π (where π and −π are equivalent), will keep moving in the same direction on

average, while a walker moving parallel to the non-preferred direction with θ = ±π2 , will

reorientate to move completely the opposite direction as previously. Hence, the spread in

the non-preferred direction is likely to be small and the spread in the preferred direction is

likely to be large, while the average displacement in either of the directions will be small.

This is exactly the behaviour observed in Figures 8.5(b), 8.6(b), 8.7(b) and 8.8(f).

Although we have found the optimal values of dτ for the sinusoidal and linear models,

it does not mean that a population with dτ = dopt will always turn to the preferred

direction. The parameter dopt is the amplitude of the mean turning angle and although

the mean turning angle will result in an average reorientation that is close to the preferred

direction (sinusoidal) or exactly the preferred direction (linear), the reorientation is still

highly dependent on the parameter κ which governs how much randomness there is in the

reorientation distribution. In a biological sense, it is no good for a swimmer to have a

perfect sensing ability if it does not have the swimming or orientating ability to act upon

it.

The above optimal values of dτ for the sinusoidal and linear models are based on the

assumption that the optimal motion is that which produces the largest average displace-

ment in the preferred direction. This may not always be the ‘optimal’ motion in a natural

environment, for example searching a particular area in detail.

Our asymptotic solution equations for the spatial statistics derived in Chapters 4 and 5,

could not use values of dτ this large as we made the assumption that dτ ≪ 1 in order to

close and solve the systems of differential equations.


8.2.4 Biological relevance of larger reorientation parameter values

Our simulation models can use any values for the reorientation parameters that we choose.

Assuming that the largest displacement in the preferred direction is optimal, the optimal

values of the reorientation parameters is κ→ ∞ and correspondingly dτ = 1 for both the

sinusoidal and linear models. With these reorientation parameters we have motion where

a walker will always turn back to the preferred direction at each step. If κ is smaller then

there will be some randomness in the choice of direction but the average behaviour will

be to turn back to the preferred direction at every step (or close to the preferred direction

for the sinusoidal model). There is no problem doing this with our abstract simulations

but this is not necessarily biologically realistic.

Consider the continuous random walk model of Hill & Hader from Section 1.5. They

assumed that the underlying motion of the algae was continuous, and only when fixed

sampling lengths are imposed, is it possible to determine the reorientation parameters.

If the time step between turns τ → 0 corresponding to a continuous random walk, then

our parameter κ→ ∞ and we have dτ = B−1τ where B is the average reorientation time

(time taken to turn back to the preferred direction). Thus, if τ → 0 then it is nonsensical

to assume that in this continuous random walk the underlying movement could have a

value of dτ ≈ 1 unless B → 0 also. However, we know that B has been measured and is

O(1) so this is unrealistic. When fitting a fixed sampling time step length to a continuous

walk it is certainly possible to observe a value of dτ ≈ 1 if the sampling time step is large

enough, but this will not be the same time step that the actual underlying motion will

have, see Chapter 7.

For biological motion that is modelled as a velocity jump process that is not a continuous

random walk then there is no reason not to allow values of dτ ≈ 1. This may be appro-

priate to model for example butterflies moving between ovipositing sites as in Kareiva &

Shigesada (1983), or if we just wish to approximate a continuous random walk using a

certain fixed time step τ 6= 0.

8.3 Simulations with non-constant parameters

The theoretical equations for the spatial statistics derived earlier were dependent on fixed

constant values for the parameters κ and dτ . Using our simulations we can investigate the

effect of making these parameters spatially dependent.

8.3.1 Spatial dependence of κ

We have seen in Section 8.2.1 that there is no limit on the value of κ for the simulations

to work. As κ → ∞ we observed that the average motion becomes more like a straight

line in the preferred direction. When κ → ∞ our original probability distribution for the


turning angle will become more and more peaked about the mean turning angle (which is

itself dependent on the absolute angle a walker is facing).

8.3.1.1 The model for κ(x)

We assume that the parameter κ increases as a walker moves further in the y-direction

only. Thus κ is dependent on the y-position only and is independent of the x-position.

In a sense, we are saying that the ‘orientating/swimming ability’ will increase as a walker

moves further along the preferred direction. If the walker is moving away from the preferred

direction we assume κ decreases down to a minimum value of zero. We also specify the

initial value of κ at y = 0, namely κI . The simplest model that has the behaviour described

above is an exponential growth function:

κ(y) = κIepy, (8.7)

where p is a parameter that controls the rate of increase of κ with movement in the y-

direction. From (8.7), κ(y) → 0 as y → −∞, κ(0) = κI , and κ(y) → ∞ as y → ∞. The

function in (8.7) has been chosen for simplicity and is fairly arbitrary — there is no reason

why other functions could not be used. Figure 8.10 shows examples of κ(y) with various

parameter values: p = 0.01, 0.05 and 0.1.

Figure 8.10: Plot of κ(y) against y with κI = 1, for p = 0.01 (—), p = 0.05 (· · ·), and

p = 0.1 (- -).

Our original theoretical model is a special case of this general model, where p ≡ 0.

8.3.1.2 Examples of individual random walks

The plots in Figure 8.11 show examples of simulations of individual random walks using

the model for spatially dependent κ(y). We fix κI = 1, and fix the parameters dτ and p


as constants in all simulations. Each simulation is run for t = 0 up to t = 1000.

(a) p = 0.01, dτ = 0. (b) p = 0.01, dτ = 0.1.

(c) p = 0.05, dτ = 0. (d) p = 0.05, dτ = 0.1.

Figure 8.11: Plots showing individual random walks for sinusoidal reorientation with κ(y)

for various parameter values. (The scale of each plot is different)

The plots in Figure 8.11 show plots using sinusoidal reorientation only. Similar simulations

have been completed using linear reorientation and as the qualitative results are similar,

the plots are omitted.

Looking at the plots in Figure 8.11 one can see that for small dτ and p there is not an

obvious preferred direction and the motion is quite random. When the parameter p is

increased slightly, the walker is now much more sensitive to the increase in κ along the

preferred direction. It should be noted that because κ increases as a walker moves along

the preferred direction we can introduce a bias to the motion even though the parameter

dτ = 0. In this case, the walker is likely to keep moving in the same direction (since κ

increases), and the further in the preferred direction it moves the less likely it is to turn,

even if the movement is not directly in the preferred direction. If dτ 6= 0 then the walker

is directed back to the preferred direction and once moving in this direction, is unlikely


to turn away from it as κ will keep increasing as long as the walker keeps moving in the

same direction.

The above comments apply only to a particular individual random walk. In a whole

population it is unlikely that all walkers will move in exactly the same way. Simulations

for a whole population of walkers moving with this model for κ(y) can be completed and

the average motion analysed.



the sinusoidal and linear reorientation models with spatially dependent κ, changes as the

parameter p increases from 0 to 1, for dτ = 0.1 and dτ = 0.3.


Figure 8.12: Plots showing Hy(100) against p for sinusoidal and linear reorientation for

dτ = 0.1 (—), and dτ = 0.3 (· · ·).

Figure 8.12 shows similar behaviour to Figure 8.1 in that the average displacement in the

preferred direction increases asymptotically towards Hy(100) = 100 as the parameter p

increases, with the greatest rate of increase for the smaller values of p. However, unlike

Figure 8.1, if p = 0 then Hy 6= 0 — this is simply because p measures how the parameter

κ changes with the y-position. If p = 0 then we simply have our previous spatially

independent model where κ is fixed for all (x, y). As observed in previous simulation, the

greatest displacement is for the larger values of dτ , and the linear model produces a larger

displacement than the sinusoidal model for the same parameter values.






show how the spread for sinusoidal and linear reorientation with spatially dependent κ

changes as the parameter p increases from 0 to 1, for dτ = 0.1 and dτ = 0.3.


Figure 8.13: Plots showing σ2x(100) against p for sinusoidal and linear reorientation for

dτ = 0.1 (—), and dτ = 0.3 (· · ·).


Figure 8.14: Plots showing σ2y(100) against p for sinusoidal and linear reorientation for

dτ = 0.1 (—), and dτ = 0.3 (· · ·).

Figure 8.13 shows similar behaviour to previous plots, for example Figure 8.2. There is

initially an increase in the spread as p increases (p = 0 corresponding to a fixed value of


κ), and then a decrease towards zero as p increases further. The larger spread is for the

smaller value of dτ .

Figure 8.14 does not show the same behaviour as previously. There is a large initial

increase in the spread as p increases, but then only a slight decrease as p increases further

(with dτ = 0.3 and the linear model there is actually an increase in the spread). In

general, the spread is greater for the smaller value of dτ . We have a large spread in

the y-direction because of the y dependence of κ — a walker starting to move in the

positive y-direction will become increasingly unlikely to turn away to a different direction

as it moves further. Conversely, a walker moving in the negative y-direction will become

more random in its movement as it moves further in the direction. Hence, there will be

some walkers still moving around near the origin, while other walkers are far along in the

positive y-direction, and the result will be a large spread about the mean position in the

y-direction.



moving with dτ = 0.1 and various values of p with the spatially dependent κ reorientation

model.

The plots in 8.15 illustrate the points made previously very well. When p is small there

are only a few walkers that have a large displacement in the preferred direction, but when

p is large the majority of the walkers have a large displacement in the preferred direction.

However, even when p is large, there are a still a large number of walkers that are close to

the origin and this results in a large spread. Walkers that start off moving in the wrong

direction and don’t quickly reorientate back to the preferred direction seem likely to be

left behind.

8.3.2 Spatial dependence of dτ

We have seen in Section 8.2.2 that assuming straight line motion is optimal, then the

optimal value of dτ is dopt = π/2 for the sinusoidal reorientation model and dopt = 1 for

the linear reorientation model.

Recalling the definition of dτ from (4.2), we have

dτ = B−1τ. (8.8)

Since τ , the average time-step between turns, is assumed to be fixed, an increasing value

of dτ corresponds to a decreasing value of B, the average time taken to reorientate back

to the preferred direction. If dτ increases to dopt, then the smallest value that B can take

is

B =τ

dopt. (8.9)


(a) Sinusoidal p = 0.05 (b) Linear p = 0.05

(c) Sinusoidal p = 0.5 (d) Linear p = 0.5


for dτ = 0.1 and p = 0.05 and p = 0.5.


Since dopt = π/2 for the sinusoidal model and dopt = 1 for the linear model, we will have

either B = 2τ/π for the sinusoidal model, or B = τ for the linear model.

8.3.2.1 The model for dτ (x)

We assume that the parameter dτ increases as a walker moves further in the y-direction

only, up to a maximum value of dopt. Thus dτ is dependent on the y-position only and

is independent of the x-position — the ‘sensing ability’ will increase as a walker moves

further along the preferred direction only. If the walker is moving away from the preferred

direction we assume that dτ decreases down to a minimum value of zero. We also specify

the initial value of dτ at y = 0, namely dint, where dint 6= 0. A suitable model for dτ (y) is

dτ (y) = dopt dint

(

1 + tanh(qy)

dopt(1 − tanh(qy)) + dint2 tanh(qy)

)

, (8.10)

where q is a parameter that controls the rate of increase of dτ with movement in the

y-direction. From (8.10), we have dτ (y) → 0 as y → −∞, dτ (0) = dint, and dτ (y) → dopt

as y → ∞. As with (8.7), the function in (8.10) is arbitrary and we have chosen it as it

clearly illustrates the desired behaviour that we have assumed.

Figure 8.16: Plot of dτ (y) against y with dint = 0.1 and dopt = 1, for q = 0.01 (—),

q = 0.05 (· · ·), and q = 0.1 (- -).

Figure 8.16 shows examples of dτ (y) with various parameter values: q = 0.01, 0.05 and

0.1.

Our original theoretical model is a special case of this general model, where q ≡ 0.


8.3.2.2 Examples of individual random walks

In the following plots we show examples of simulations of individual random walks using

the model for spatially dependent dτ (y). We fix dint = 0.1, and fix the parameters κ and

q as constants in all simulations. Each simulation is run for t = 0 up to t = 1000.

(a) q = 0.01, κ = 1. (b) q = 0.01, κ = 4.

(c) q = 0.05, κ = 1. (d) q = 0.05, κ = 4.

Figure 8.17: Plots showing individual random walks for sinusoidal reorientation with dτ (y)

for various parameter values. (The scale of each plot is different)

The plots in Figure 8.17 show plots using sinusoidal reorientation only. Similar simulations

have been completed using linear reorientation and as the qualitative results are similar,

plots are omitted.

Looking at the plots in Figure 8.17, one can see that when κ = 1 there is some randomness

in the motion but for all values of q there is a definite movement in the preferred direction.

As q increases, the motion becomes slightly less random and the movement in the average

direction increases. When κ = 4, the motion tends to a straight line very quickly and the

motion is similar for all values of q. Note that even when dτ = dopt there is still some

randomness in the choice of direction for both values of κ — this is perhaps a more realistic


model for animal and micro-organism movement than our model for κ(y). It makes more

sense for an animal to increase its ability to sense the preferred direction (dτ value) as

it moves further along it, than to be able to improve its swimming/orientating ability (κ

value) if the environment is homogenous.

The above comments apply only to a particular individual random walk. In a whole

population it is unlikely that all walkers will move in exactly the same way. Simulations

for a whole population of walkers moving with this model for dτ (y) can be completed and

the average motion analysed.



the sinusoidal and linear reorientation models with spatially dependent dτ , changes as the

parameter q increases from 0 to 1, for κ = 1 and κ = 4.


Figure 8.18: Plots showing Hy(100) against q for sinusoidal and linear reorientation for

κ = 1 (—), and κ = 4 (· · ·).

The plots in Figure 8.18 show that as the parameter q increases there is a slow gradual

increase in the average displacement in the preferred direction. In general, the linear

model produces the larger displacement, and there is also larger displacement for the

larger values of κ. When q = 0 there is a fixed (non-zero) value of dτ and hence Hy(100)

is also non-zero. For larger values of q the plots seem to tend to an asymptotic limit

that is not the maximum allowed displacement. The parameter q controls how quickly

the walkers reach the optimal value of dτ = dopt as they move in the positive y-direction.

However, even if dτ = dopt for all (x, y) the average displacement is still dependent on the

value of κ, see Figure 8.5, and this κ value will control what this asymptotic limit will


be. A population of walkers with perfect ‘sensing ability’ (dτ = dopt) will not have a very

large average displacement if there is too much randomness in the movement (small κ).





show how the spread for sinusoidal and linear reorientation with spatially dependent dτ

changes as the parameter q increases from 0 to 1, for κ = 1 and κ = 4.


Figure 8.19: Plots showing σ2x(100) against q for sinusoidal and linear reorientation for

κ = 1 (—), and κ = 4 (· · ·).


Figure 8.20: Plots showing σ2y(100) against q for sinusoidal and linear reorientation for

κ = 1 (—), and κ = 4 (· · ·).


From Figure 8.19, the spread in the non-preferred direction decreases as q increases, al-

though the rate of decrease for κ = 1 is very small. The spread is larger for the larger

value of κ.

Figure 8.14 shows similar behaviour to Figure 8.20 — as q increases the spread in the

preferred direction increases to a large value. The spread is larger for the larger value of

κ. The reason for this increase in the spread is for the same reason as discussed in the

spatially dependent κ model — the walkers that move a distance in the preferred direction

will get better at sensing and become less likely to turn away, while those that have not

moved very far in the preferred direction will have a much more random motion. Hence,

there will be a large spread about mean position in the preferred direction.



moving with κ = 4 and various values of q with the spatially dependent dτ reorientation

model.

The plots in 8.21 illustrate the points made previously. When q is small there are only a

few walkers that have a large displacement in the preferred direction and there is a large

spread. When q is larger, the majority of the walkers have a large displacement in the

preferred direction but there are still a number of walkers close to the origin, which results

in a large spread in the preferred direction.

8.3.3 Biological relevance of spatially dependent reorientation parame-

ters

We have seen that our simulations can easily be adapted to include spatial dependence

of the reorientation parameters, but as in Section 8.2.4 we should question whether these

models are realistic. The parameter κ is a measure of the variance of the reorienta-

tion probability distribution, and as discussed previously can be considered as the ‘swim-

ming/orientating ability’ of the walker. The parameter dτ is a measure of how quickly a

walker will reorientate back to the preferred direction and can be considered as the ‘sensing

ability’ of the walker.

Our model that includes spatial dependence of the parameter κ on the y-position is not

necessarily a realistic model for biological motion. It seems unlikely that an animal could

increase its ability to overcome the inherent randomness in the environment (due to exter-

nal turbulence or internal mechanisms) simply by moving further in the preferred direction.

It could be argued that if the environment was non-homogenous then this model could be

valid — a fish swimming from the open sea to a lagoon for example. However, if we assume

a homogeneous environment then it seems sensible to have a fixed constant ‘orientating

ability’ and hence a fixed value of κ.


(a) Sinusoidal q = 0.01 (b) Linear q = 0.01

(c) Sinusoidal q = 0.1 (d) Linear q = 0.1


for κ = 4 and q = 0.01 and q = 0.1.


The model that includes spatial dependence of the parameter dτ on the y-position is

perhaps more realistic. An animal that moves closer to the source of the signal that is

producing the bias response is likely to be able to sense that source better — for example

a fish larvae responding to the noise of a reef will arguably be able to hear better the closer

it is, or phototactic algae that are closer to a light source may be able to sense it better

(although some algae lose their phototactic ability if the light source is too strong).

If we include spatially dependent parameters in the model then we must also be aware of

the comments made earlier in Section 8.2.4. It may not be sensible to allow the reorien-

tation parameters to increase indefinitely or to an extreme limiting value as the spatial

position changes.

8.4 Simulations with a changing preferred direction

In all the simulations and theoretical results so far we have assumed that the preferred

direction is fixed and walkers with bias will on average keep moving in this direction for

all time. This is a reasonable model if the distance a walker is moving at each step is

far smaller than the distance to the source of the bias, for example algae moving under

gyrotaxis. In reality this may not always be the case, and the source of the bias may

not always be in the same direction relative to a walker, for example fish larvae swimming

towards a small reef. In such cases the source of the bias is better modelled as a point source

and at each step of the random walk the preferred direction will be towards this point

source and hence the preferred direction will change depending on the spatial position.

8.4.1 Reorientation models for a changing preferred direction

If the preferred direction is always towards a fixed point source then the preferred absolute

angle of movement at each step θ0, will be different depending on the spatial position of

the walker (previous models have all assumed that θ0 = 0 ∀ (x, y) for simplicity).

Assume we have a fixed point source at position (xp, yp) and a population of walkers that

move with similar reorientation models as described in Section 4.2 or Section 5.2, except

that now at each step each walker will try to reorientate to be facing the point source

rather than a fixed direction. Assume that the walkers move around the x, y plane, and

have a direction θ, where θ = 0 corresponds to the positive y-direction and θ is measured

clockwise from here. Under these assumptions a walker at position (x, y) will have a

preferred direction of movement given by

θ0 =

tan−1(

xp−xyp−y

)

if y < yp

tan−1(

xp−xyp−y

)

+ π if y > yp

±π2 if y = yp.

(8.11)


Without loss of generality we can choose our y-axis such that the point source and the

origin both lie upon it and hence xp = 0.

In the previous models we found that by increasing the reorientation parameters to their

optimal values we produced a larger displacement in the fixed preferred direction and a

smaller spread. However, if we have movement towards a point source we may expect that

the walkers will reach the preferred point and stay in the vicinity of it rather than carrying

on moving in the same direction indefinitely. However, it may be the case that we are only

interested in modelling the movement of the walker before it reaches the point source and

are not concerned with what happens afterwards, for example fish larvae recruiting to a

reef (see examples in Chapter 10).

The following simulations have been run using the simulation model presented in Chapter

6, with the addition of a point source for the source of the preferred direction — the

reorientation parameters are fixed constants and not spatially dependent.

8.4.2 Examples of individual random walks

The plots in Figure 8.22 show examples of simulations of individual random walks using

the model for a point source as the preferred direction. We have dτ = 0.3 and κ = 4 as

constants in all simulations, and the position of the point source is given by (xp, yp) =

(0, 100). Each simulation is run for t = 0 up to t = 1000.


Figure 8.22: Plots showing individual random walks for sinusoidal and linear reorientation

where the preferred direction is to a point.

Figure 8.22 illustrates the behaviour that we may have expected — the walkers move

towards the point source and then stay in the vicinity of it rather than moving in the


same direction indefinitely. There appears to be less randomness in the linear model, both

approaching the point source and subsequent movement in its vicinity.

8.4.3 Average position — Hy(t)

The plots in Figure 8.23 show how the simulated mean position Hy(t), for the sinusoidal

and linear reorientation models with a point source as the preferred direction, changes as

t increases from 0 to 100 for various reorientation parameter values. In all the simulations

we have (xp, yp) = (0, 40).

(a) Sinusoidal κ = 1, dτ = 0.1 (b) Linear κ = 1, dτ = 0.1

(c) Sinusoidal κ = 4, dτ = 0.3 (d) Linear κ = 4, dτ = 0.3

Figure 8.23: Plots showing the average position in the y-direction, Hy(t), against t for

sinusoidal and linear reorientation.

When the reorientation parameters are small as in Figures 8.23(a) and 8.23(b), there is

a linear increase in the average displacement with time. This is exactly the behaviour

observed in the simple simulation model in Chapter 6 with the same parameter values,


see Figures 6.7(c) and 6.10(c). If the walkers do not move with a large enough absolute

velocity to get near to the point source, then they move in a very similar manner to that

observed when the preferred direction is fixed.

When we have larger parameter values as in Figures 8.23(c) and 8.23(d), the initial be-

haviour for small t is the same as described above, but then when the walkers reach the

point source the average displacement stops increasing and stays at y = 40, as might be

expected. The average displacement of the linear model reaches the point source quicker

than the average displacement of the sinusoidal model.

8.4.4 Spread about the mean position — σ2(t)

The plots in Figures 8.24 and 8.25 show how the simulated spread about the mean posi-

tion in the x and y directions σ2x(t) and σ2

y(t) respectively, for the sinusoidal and linear

reorientation models with a point source as the preferred direction, change as t increases

from 0 to 100 for various reorientation parameter values. In all the simulations we have

(xp, yp) = (0, 40).

Figures 8.24 and 8.25 both show similar results. When the reorientation parameters are

small, there is a linear increase in the spread in each direction with time. This is exactly the

behaviour observed in the simple simulation model in Chapter 6 with the same parameter

values, see Figures 6.30, 6.30, 6.32 and 6.33. If the walkers do not move with a large

enough absolute velocity to get near to the point source, then they move in a very similar

manner to that observed when the preferred direction is fixed.

When the reorientation parameters are large then the spread initially increases linearly,

but then quickly decreases and tends to a constant value corresponding to the population

all moving to be close to the point source. The final limiting spread is larger for the

sinusoidal model than the linear model.



moving with various reorientation parameter values and a point source as the preferred

direction.

The plots in Figure 8.26 illustrate the points made earlier — for small values of the

reorientation parameters there is little difference between moving to a point source and

moving towards a fixed preferred direction (simple simulation model) as the absolute

velocity of the walkers does not allow them to get near to the point source in the given

time. For larger values of the reorientation parameters the population all move to be close

to the point source and stay nearby. There is still a random spread about the point source

as the walkers are continuing to move — this limiting spread is larger for the sinusoidal

model than for the linear model.




Figure 8.24: Plots showing the spread in the x-direction, σ2x(t), against t for sinusoidal

and linear reorientation.




Figure 8.25: Plots showing the spread in the y-direction, σ2y(t), against t for sinusoidal






where the preferred direction is to a point.


8.5 Conclusions

In this chapter we have extended the simple computer simulation described in Chapter 6

to investigate models where our theoretical results are no longer valid.

The theoretical equations for the spatial statistics from Chapters 4 and 5 are valid for

only a limited parameter range due to assumptions made when deriving the model (see

Section 4.9.3). Using simulations we can investigate the effect of extreme values of the

reorientation parameters on the subsequent motion. In general, the parameter κ can take

any positive value, and (assuming dτ 6= 0) as κ → ∞ the subsequent motion becomes

more like a straight line directly in the preferred direction. Thus the average displacement

increases and the spread decreases as κ→ ∞. The parameter dτ has an optimal value given

by dopt = 1 for the linear model, and 1 ≤ dopt ≤ π/2 depending on the value of κ for the

sinusoidal model. For the sinusoidal model the optimal motion is thus when on average part

of the population ‘overcorrects’ and reorientates past the preferred direction, so that the

rest of the population do not ‘under-correct’ too much. These extreme parameter values

may not be biologically realistic — dτ ≈ 1 is certainly not appropriate for a continuous

random walk model unless the average reorientation time tends to zero which is unrealistic.

Using simulations it is also possible to make the movement model more complex by includ-

ing spatially dependent reorientation parameters κ(y) and dτ (y). We now have two new

parameters p and q which control the rate at how quickly the parameters increase towards

their optimal values as the y-position increases. Similarly to previous results, the aver-

age displacement increases and the spread decreases as the parameters p and q increase.

In the spatially dependent model for dτ , even if we have a population that very quickly

reaches the optimal value of dτ , the average displacement and spread is still highly depen-

dent on κ and the choice of κ will limit the maximum average displacement at t = 100.

Conversely, as long as dτ is non-zero, the spatially dependent κ model will always tend

asymptotically towards the maximum theoretical displacement at t = 100 (y = 100 in this

non-dimensionalised system). In a biological sense this means that it is no good having a

perfect sensing ability if a walker cannot overcome the randomness inherent in the envi-

ronment to be able to reorientate to the preferred direction. The spatially dependent dτ

model seems more realistic then the spatially dependent κ model — it makes more sense

for an animal to increase its ability to sense the preferred direction (dτ value) as it moves

further along it, than to be able to improve its swimming/orientating ability (κ value).

We can also model a changing preferred direction where the preferred movement is towards

a point source and not a fixed direction. In this model the results for small reorientation

parameters are indistinguishable from the results in Chapter 6 for the simple simulation

model. For larger parameter values the population of walkers will tend to cluster around

the point source although they will still be moving and have a certain amount of spread.

Possible further extensions could also be modelled using our simulation approach. Ran-


dom walks with a choice of two preferred directions have recently been used by Plank &

Sleeman (2003), when modelling angiogenesis using reinforced random walks. This might

also be applicable to for example, fish larvae recruiting where there are a choice of two

possible reefs to move towards, or other similar biological situations. It would be simple

to introduce time dependency into the reorientation parameters, for example a decaying

dτ parameter might be used to model a signal that is getting weaker with time.


• Simulations with extreme κ values have been completed. The largest average dis-

placement in the preferred direction is found as κ → ∞ for both the sinusoidal and

linear models.

• Simulations with extreme dτ values have been completed. The largest average dis-

placement in the preferred direction is found when dτ = dopt, where dopt is different

for the sinusoidal and linear models. In the sinusoidal model dopt = π/2 when there

is a uniform angular spread (corresponding to κ ≈ 0), and dopt → 1 as κ → ∞. In

the linear model dopt = 1 for all values of κ.

• Simulations with a spatial dependence for κ have been completed. In general (as-

suming dτ 6= 0), as the parameter that controls the change in κ with the y-position

(p) increases, the average displacement increases asymptotically to the theoretical

maximum and the spread decreases to zero.

• Simulations with a spatial dependence for dτ have been completed. In general (as-

suming dτ 6= 0), as the parameter that controls the change in dτ with the y-position

(q) increases, the average displacement increases and the spread decreases. The lim-

iting values for the average displacement and spread are determined by the value of

κ used even if dτ very quickly reaches dopt.

• The above models may not be biologically realistic if for example we are trying

to approximate a continuous random walk, or the walkers are moving around a

homogenous environment.

• Simulations with movement towards a point source rather than a fixed preferred

direction have been completed. In general for t = 100, for small reorientation pa-

rameter values the results are the same as the simple simulation model with a fixed

preferred direction. For larger parameter values the population tends to cluster

around the point source with a certain spread that is larger for smaller κ values and

also larger for the sinusoidal model compared to the linear model.

Chapter 9

Mean dispersal distance of

correlated random walks

9.1 Introduction

In previous chapters we have looked at the mean squared displacement about the origin,

D2(t), and the mean squared displacement about the average position, σ2(t), as well

as considering the spread in each direction separately. The square root of the mean

squared displacement is only an approximate measure of the actual average dispersal of

the population. A better measure is the mean dispersal distance (MDD) which is the

beeline distance of each walker from the mean position averaged over all the population.

However, equations for the mean squared displacement are far simpler to derive whether

considering chains of polymers as in Tchen (1952) and Flory (1969), or correlated random

walks of animals or cells as in Skellam (1973), Lovely & Dahlquist (1975), Hall (1977),

Okubo (1980), Dunn (1983), Kareiva & Shigesada (1983) etc. An equation for the mean

dispersal distance seems more difficult as stated by McCulloch & Cain (1989), who derived

an approximate formula which was very complex to compute for a limited number of moves.

Bovet & Benhamou (1988) suggest an equation for the MDD which is a correction of the

root of the mean squared displacement, but their equation is only valid after a long-time

period as it assumes a Normal spatial distribution which occurs only when all the initial

correlation effects are lost. Byers (2000, 2001) uses simulation results of a correlated and

unbiased random walk with a fixed step length to calculate a complex multivariate least

squares cubic polynomial for the correction factor, Z, that is dependent on the number of

steps in the walk and the amount of correlation at each turn. In this chapter we extend the

results of Byers to include correlated random walks with variable step lengths and suggest

an alternative and simpler model for the correction factor, Z. We also show how some

results can be extended to biased and correlated random walks, but that the correction

factor is more complex and dependent on both the reorientation parameters, κ and dτ ,

273

CHAPTER 9: Mean dispersal distance of correlated random walks 274

and time t. Using simulations to find the correction factor for biased random walks with

different reorientation models and reorientation parameters, it is possible to gain more

information about the long-time spatial distribution of the population.

9.2 The mean squared displacement

The standard measure of dispersal is the mean squared displacement. As we have seen

previously, the dispersal (or spread) about the mean position is defined as

σ2(t) =

∫

R2

∫ π

−π‖x − H(t)‖2 p(x, t) dθdx. (9.1)

In an unbiased random walk H(t) = (0, 0) and the spread about the mean position is the

same as the spread about the origin. In Chapter 3 we showed how to derive an equation

for σ2(t) for an unbiased random walk and this is given in (3.43).

9.2.1 Comparing the mean squared displacement for unbiased discrete

random walks and velocity jump processes

Kareiva & Shigesada (1983) derived an equation for the mean squared displacement for a

correlated random walk with a discrete number of time steps n and a symmetric turning

angle distribution g(θ), see (1.75). We can relate their discrete random walk to our

continuous velocity jump process by assuming that n = λt, so that the number of discrete

turns is just the product of the turning frequency and the total time. We now assume

that g(θ) in their model is the same as our von Mises reorientation distribution with no

bias (see (4.9) and (5.2)). The turning frequency in our velocity jump process is governed

by a Poisson process with parameter λ. The time between events in a Poisson process is

exponentially distributed with mean 1/λ and variance 1/λ2. Thus, for our velocity jump

process we have E(l) = sλ , Var(l) = s2

λ2 and E(l2) = 2s2

λ2 . With these results (1.75) becomes

E(R2t ) =

2s2

λ0

(

t− 1

λ0(c− cλt+1)

)

, (9.2)

where c = I1(κ)/I0(κ), and λ0 = λ(1 − c).

For an unbiased random walk, the velocity jump process model gives a solution for the

mean squared displacement, see (3.43). In the absence of bias we have

σ2 =2s2

λ0

(

t− 1

λ0(1 − eλ0t)

)

, (9.3)

where λ0 = λ(1 − c) and c = I1(κ)/I0(κ).

Since 0 < c < 1, the long-time limits of (9.2) and (9.3) are given by

E(R2t )∞ =

2s2

λ0

(

t− c

λ0

)

, (9.4)

and σ2(t)∞ =2s2

λ0

(

t− 1

λ0

)

. (9.5)


Thus the equations for spread from the velocity jump process model and the model of

Kareiva and Shigesada are very similar with only a slight difference in the constant term,

and the two solutions have the same long-time limiting behaviour.

9.2.2 Mean squared displacement for variable and fixed step lengths

It is worth mentioning that (1.75) highlights the difference between the mean squared

displacement of a random walk with a fixed step length l, so that E(l)2 = E(l2) = l2, and

a random walk with a variable step length with a given mean. An example is the Poisson

process time step in our velocity jump model where our mean step length is l = s/λ and

E(l)2 = (s/λ)2 and E(l2) = 2s2/λ2.

For example, in a random walk with a fixed step length l, E(l)2 = E(l2) = l2, and if we

assume that n = λt and l = s/λ, (1.75) reduces to

E(R2t ) =

s2t(1 − c2) − 2s2c(1 − cλt)

λ20

, (9.6)

where λ0 = λ(1 − c).

From (9.3), a random walk with a step length that is exponentially distributed due to a

Poisson process with parameter λ, has E(l)2 = l2 and E(l2) = 2l2, and if we assume that

n = λt and l = s/λ, (1.75) reduces to

E(R2t ) =

2s2t(1 − c) − 2s2c(1 − cλt)

λ20

. (9.7)

It is immediately clear that for c < 1 the mean squared displacement for the random walk

with the variable step length (9.7), is always larger than the mean squared displacement

for the random walk with the fixed step length (9.6), even though both have the same

mean step length. As c → 0 the former becomes almost twice as large. The variability

of the step length has an important effect on the expected mean square displacement.

This suggests that it is not sensible to model the movement of animals or micro-organisms

with fixed step lengths, if there is likely to be variability in the actual step length used in

the original random walk and one wants to predict information about the mean squared

displacement.

9.3 The mean dispersal distance of unbiased random walks

The mean squared displacement is not always a useful statistic as it is much larger than

the mean dispersal distance (MDD), around which the population will be distributed

(Byers (2001)). Using the notation of the previous chapters, the mean dispersal distance,

MDD(t), is defined as

MDD(t) =

∫

R2

∫ π

−π‖x − H(t)‖ p(x, t) dθdx. (9.8)


It is not possible to derive a differential equation for MDD(t) for a velocity jump process

as we did for σ2(t) in Chapters 3, 4 and 5. However, equations have been found for

MDD(n), the mean dispersal distance after n steps in a discrete random walk.

9.3.1 Calculating the mean dispersal distance from the mean squared

displacement in a discrete random walk

Using a similar equation for the mean squared displacement given in (1.68), Bovet &

Benhamou (1988) derived an equation for the mean dispersal distance, MDD(n), after a

large number of steps, n. Similar equations to (1.68) have also been used by Flory (1969),

Skellam (1973), Lovely & Dahlquist (1975), Hall (1977), Okubo (1980), Dunn (1983) etc.

when modelling correlated and unbiased random walks.

Assuming a fixed step length l, and a von Mises distribution for the turning angle so that

c = I1(κ)/I0(κ), (1.68) gives

E(R2n) = l2

1 + c

1 − cn− l2

2c(1 − cn)

(1 − c)2. (9.9)

(9.9) has the same limiting behaviour as the equation of Kareiva & Shigesada (1.75) with

a fixed time step for which E(l)2 = E(l2) = l2, see also (9.6). As n becomes large, cn

becomes small since c < 1, and the second term on the right hand side of the above

becomes negligible. The long-time solution is then

E(R2n) ∼ l2

1 + c

1 − cn. (9.10)

Bovet & Benhamou used the following argument to calculate an equation for the mean

dispersal distance MDD(n). Splitting R2n into components gives

R2n = X2

n + Y 2n .

Tchen (1952) showed that when n is large, the two components Xn and Yn are normally

distributed and statistically independent. In a correlated but unbiased random walk there

is no preferential orientation in space and

E(Xn) = E(Yn) = 0, s.d.(Xn) = s.d.(Yn) = δ,

where s.d. represents standard deviation. From this it follows that

E(X2n) = E(Y 2

n ) = δ2,

and therefore

E(R2n) = E(X2

n) + E(Y 2n ) = 2δ2,

or

δ =

√

E(R2n)

2.


Now let D be the beeline distance from the origin , and let u = Xn/δ and v = Yn/δ, so that

u and v are two independent random variables distributed according to the normal law

N(0, 1). The variable d = D/δ =√u2 + v2 is thus a random variable distributed according

to the χ law with two degrees of freedom. From Evans et al. (2000), the expected value

of a χ law is given by

E(z) =

√2Γ(1

2(µ+ 1))

Γ(µ/2), (9.11)

where µ is number of degrees of freedom. For two dimensions, µ = 2, and we have

E(d) =

√

π

2

and

E(D) = δE(d) =

√

E(R2n)π

4. (9.12)

Thus if E(D) = MDD(n), we can write

MDD(n) =

√

π

4

√

E(R2n), (9.13)

and substituting for (9.10) gives the equation already seen in (1.78):

MDD(n) = l

√

0.79(1 + c)

(1 − c)n. (9.14)

The equation in (9.14) is only valid for large n, as we have only used the long-time limiting

solution for R2n given in (9.10), and because Tchen’s result that the distribution of Xn and

Yn are normally distributed only holds for n large enough so that all correlation effects

have been lost and the spread appears diffusive.

9.3.2 A better model for MDD(n)

The equation derived in the previous section for MDD(n), (9.14), is only valid for large n.

For smaller n the assumption that the spatial distribution is Normal breaks down. Recall

(1.43), the equation for the mean squared displacement of the 1-d telegraph equation:

< x2(t) >=v2

λ

(

t− 1

2λ(1 − e−2λt)

)

. (9.15)

λ is the probability of reversing direction, and thus governs the correlation — if λ is

small the walk is highly correlated. For small λ and small t, < x2(t) >∼ v2t2, which is

characteristic of a wave propagation process, and for large t, < x2(t) >∼ v2t/λ, which is

characteristic of a diffusion process with diffusion coefficient D = s2/2λ.

9.3.2.1 The spread of a two-dimensional correlated random walk

We might expect a similar result to (9.15) in two dimensions — if the random walk is

highly correlated, the initial behaviour will be to spread out linearly from the origin with


few turns, similar to a wave propagating outwards. As t increases the walk will start to

lose the initial correlation effects and the population spread will become diffusive. If the

walk is not highly correlated we would expect the behaviour to become diffusive more

quickly. This predicted behaviour can be clearly seen in Figure 9.1, for correlated random

walks with κ = 1 (low correlation) and κ = 50 (very high correlation) after a short time.

(a) κ = 1. (b) κ = 50.

Figure 9.1: Plots of the spread of a population of 500 walkers after t = 100, moving as

an unbiased and correlated velocity jump process with (a) κ = 1, (b) κ = 50. The dotted

circle shows the maximum possible displacement at t = 100.

At t = 100 the highly correlated random walk is still spreading out linearly with time

and the spatial distribution is definitely not Normal. After the same time period, the

walk with low correlation appears to be spreading diffusively and the spatial distribution

appears Normal.

This illustrates that the equation of Bovet & Benhamou for the mean dispersal distance,

(9.14), is likely to be valid for smaller n if there is low correlation, but if the walk is highly

correlated, (9.14) will only be valid for very large n.

9.3.2.2 Correction factor method

Byers (2001) compares the equation of Bovet & Benhamou for MDD(n), (9.14), to a

corrected form of the equation of Kareiva & Shigesada (1983):

MDD(n) = Z√

E(R2n), (9.16)

where Z is a correction factor dependent on the number of steps, n, and the degree of

correlation, c, and E(R2n) is calculated from (1.75). The correction factor is needed as

the mean dispersal distance can differ from the root of the mean squared displacement

by as much as 12%. Byers simulated simple random walks with a fixed step length for

various degrees of correlation (giving a range of values of c) and various numbers of steps


n. Byers fitted a complicated multivariate least squares polynomial to the simulation data

to find the correction factor Z. This was a reasonable fit over only a limited range so five

constraints were also needed. The main result was that 0.89 ≤ Z ≤ 1, with a higher value

of Z for smaller numbers of steps n and higher correlation (c ≈ 1). Byers then shows how

the correction factor equation (9.16) is a better fit to simulation data then the equation of

Bovet & Benhamou, (9.14), for small numbers of steps and high correlation. This seems

obvious as Bovet & Benhamou made it clear that their equation was only a long-time

limiting solution and one would not expect it to be valid if the spatial distribution is not

Normal (which corresponds to small numbers of steps n, and/or high correlation c).

The result obtained by Byers (2001), that the correction factor lies in the range 0.89 ≤Z ≤ 1, is not surprising when one considers the equation of Bovet & Benhamou, (9.14)

and the results in 9.3.2.1. In a correlated random walk, the long-time spatial distribution

will become Normal as n increases (Tchen, 1952), where highly correlated walks will take

a larger number of steps to have a Normal distribution when compared to walks with low

correlation. Bovet & Benhamou show that if the spatial distribution is Normal, then the

ratio between MDD(n) and√

E(R2n) is given by

√

π/4 ≈ 0.89. If the spatial distribution

is similar to a linear spread in time (as in Figure 9.1(b)), then we would expect the ratio

between MDD(n) and√

E(R2n) to be close to 1. If the number of steps is very small then

we also expect the ratio between MDD(n) and√

E(R2n) to be close to 1 (and exactly

equal to 1 for n ≤ 1).

9.3.3 The mean dispersal distance of an unbiased velocity jump process

with a variable time step

We have shown that our equation for σ2(t), (9.3), has the same behaviour as the equation

derived for the mean squared displacement by Kareiva and Shigesada (9.2), when we

assume that n = λt. From (9.13) and the results of Byers, we suggest the following

equation for the mean dispersal distance of an unbiased velocity jump process with a

variable time step:

MDD(t) = Z(c, t)√

σ2(t), (9.17)

where the correction factor Z(c, t) will be dependent on the amount of correlation, c, and

the time, t. In a highly correlated unbiased random walk we would expect Z(c, t) →√

π/4

slowly as t increases, while in a walk with low correlation Z(c, t) →√

π/4 more quickly as

t increases. As long as the walk is unbiased and c < 1, the long-time spatial distribution

will always be Normal and we will always have Z(c,∞) =√

π/4.

The multivariate least squares polynomial fitted by Byers is complicated, has a limited

range, and is calculated from simulation data from a random walk with a fixed step length.

Taking into account the comments on how we expect Z(c, t) to behave, we suggest the


following simple model for the correction factor

Z(c, t) =

√

π

4+

(

1 −√

π

4

)

exp (−a(1 − c)t) , (9.18)

where a is a constant to be determined and c = I1(κ)/I0(κ). The equation in (9.18) is a

simple decaying function that behaves in the manner described previously, i.e. Z(c, 0) = 1

and Z(c, t) →√

π/4 as t→ ∞, with a slower decrease in Z(c, t) for larger values of c < 1.

9.3.3.1 Estimating Z(c, t)

The equation in (9.18) was fitted against simulation data for a range of values of c and t

to see if the model is reasonable and to find an estimate for the constant a. The simulated

decrease in Z(c, t) does appear to be an exponential decay and our simple model seems

reasonable. The best fit to simulated data was given when a = 16 , although we cannot

justify why it should be this value exactly. Plots comparing the simulated results for Z(c, t)

to the expected values from (9.18) with a = 16 are shown in Figure 9.2 for 0 ≤ t ≤ 500 and

four values of κ.

Allowing for simulation noise, there is a reasonable match between the expected and

simulated values of Z(c, t) as shown in Figure 9.2. The decrease in the simulated Z(c, t)

appears to be exponential and in all the plots Z(c, t) → 0.89 as t increases.

9.3.3.2 Comparing the equation for MDD(t) to simulated random walks

Now that we have a reasonable model for Z(c, t) we can compare the equation for the

mean dispersal distance, MDD(t), given in (9.17) to simulated random walks and also to

other equations in the literature.

In Figure 9.3 we compare our equation for MDD(t) from (9.17) and using the correction

factor in (9.18) with a = 16 , to two other equations for the mean dispersal distance in

the literature — a correction of (9.2) from Kareiva & Shigesada using (9.18) a = 16 , and

the equation of Bovet & Benhamou, (9.14). In Figure 9.4 we demonstrate how our time

dependent correction factor (9.18) with a = 16 , results in a much better estimate for the

mean dispersal distance for highly correlated random walks (large κ) than simply using

the constant long-time limiting value of (9.18).

Figure 9.3 shows two plots of MDD(t) v t for 0 ≤ t ≤ 100. In each plot, we have compared

simulation data with i) the equation for MDD(t) from (9.17) using σ2(t) from (9.3) and

Z(c, t) from (9.18), ii) the equation for MDD(t) from (9.17) using Kareiva & Shigesada’s

equation for R2t , (9.2), in place of σ2(t) and Z(c, t) from (9.18), iii) Bovet & Benahmou’s

equation for MDD(n) from (9.14), using n = λt.

From Figure 9.3 it is hard to distinguish between our equation and the equation of Kareiva

& Shigesada, and both are a very good fit to simulation results. From Figure 9.3(a), for

κ = 1 the equation of Bovet & Benhamou underestimates both the simulated results and


(a) κ = 1, c = 0.446 (b) κ = 4, c = 0.864

(c) κ = 10, c = 0.949 (d) κ = 20, c = 0.975

Figure 9.2: Plots comparing expected values of Z(c, t) (—) to simulated results (+) for

(a) κ = 1, (b) κ = 4, (c) κ = 10, (d) κ = 20.


(a) MDD(t) with κ = 1. (b) MDD(t) with κ = 20.

Figure 9.3: Plots of MDD(t) v t for the velocity jump process model (—), Kareiva &

Shigesada’s model (· · ·), Bovet & Benhamou’s model (- -), and simulation results (+).

the results expected from the other two equations. This is because of the point made

in Section 9.2.2 — the equation of Bovet & Benhamou does not take into account the

variable step length that is present. From Figure 9.3(b), for κ = 20 the equation of Bovet

& Benhamou overestimates both the simulated results and the results expected from the

other two equations. This is because the equation of Bovet & Benhamou is only valid

as the long-time limiting solution when the spatial distribution has settled down and is

Normal. A highly correlated random walk (κ = 20) will take a longer time before Bovet &

Benhamou’s equation is valid. If we look at a longer time period (not shown) the equation

of Bovet & Benhamou starts to underestimate the simulated results because of the point

made above about the variable step length.

Figure 9.4 shows plots comparing the results of simulated random walks with i) the equa-

tion forMDD(t) from (9.17) using σ2(t) from (9.3) and Z(c, t) from (9.18), ii) the equation

for MDD(t) from (9.17) using σ2(t) from (9.3) and Z(c, t) = 0.89 ∀ c, t, for 0 ≤ t ≤ 200

and four values of κ.

From Figure 9.4 it is clear that, for all values of κ, there is a good match between simulated

results and expected results for the equation for MDD(t) from (9.17) using σ2(t) from

(9.3) and Z(c, t) from (9.18). For small κ there is little difference between using Z(c, t)

from (9.18) and Z = 0.89 — this is because for a walk with low correlation the spatial

distribution very quickly becomes Normal. When κ is large the expected values using

Z = 0.89 underestimate the simulated values and the expected values using Z(c, t) from


(a) κ = 1 (b) κ = 4

(c) κ = 10 (d) κ = 20

Figure 9.4: Plots of MDD(t) v t for velocity jump process model with Z(c, t) (—), Z =

0.89 (- -), and simulation results (+).


(9.18) — this is because the walk is highly correlated and the spatial distribution is not

Normal for these values of t.

9.3.4 The mean dispersal distance in each direction for an unbiased

velocity jump process

In Chapters 4 and 5 we considered not only the total spread about the mean position,

σ2(t), but also the spread about the mean position in each direction, σ2x(t) and σ2

y(t). In a

similar manner we can define MDDx(t) and MDDy(t) to be the mean dispersal distance

in the x and y directions respectively. In an unbiased random walk we would expect

MDDx(t) ≈ MDDy(t) as from previous Chapters we know that if the walk is unbiased,

then σ2x(t) = σ2

y(t).

From Section 9.3.1,we might expect a similar relation between the mean dispersal in each

direction and the spread about the mean in each direction as existed for the total spread

and total mean dispersal. In fact from Evans et al. (2000), the expected value of a χ law

is given in (9.11), and if we consider only one direction, µ = 1 and

E(d) =

√

2

π

and

MDDx(t) =

√

2

πσx(t) ≈ 0.798σx(t), (9.19)

MDDy(t) =

√

2

πσy(t) ≈ 0.798σy(t). (9.20)

Simulations confirm these equations as the long-time limiting relations. The behaviour of

Zx(c, t) and Zy(c, t) is similar to that of Z(c, t) — the ratio between the mean dispersal

distance and the root of the mean squared displacement is equal to 1 at t = 0 and tends

to the limiting value of 0.798 as t increases, with a slower decrease in Z(c, t) for a more

correlated random walk (c ≈ 1). The decrease appears exponential as we found with

Z(c, t), but we have not attempted to fit a function to simulation data.

In a biased random walk we known from Chapters 4, 5 and 6 that the spread is greater

in the non-preferred direction (x) so we would expect the mean dispersal distance to be

greater in the x direction also.

9.4 The mean dispersal distance of biased random walks

In a biased random walk it is not as easy to calculate the mean dispersal distance for

two reasons: i) the spread is larger in the non-preferred direction, and ii) the spread is

not normally distributed. Figure 9.5 shows the spread at t = 100 of a population of 500

walkers moving as a velocity jump process with with parameter dτ = 0.1 and (a) κ = 1

and sinusoidal reorientation, (b) κ = 50 and sinusoidal reorientation, (c) κ = 1 and linear

reorientation, (d) κ = 50 and linear reorientation. Similar results are shown in Figure 8.4.


(a) Sinusoidal, κ = 1. (b) Sinusoidal, κ = 50.

(c) Linear, κ = 1. (d) Linear, κ = 50.

Figure 9.5: Simulated plots of the spread of a population of 500 walkers after t = 100,

moving as a biased and correlated velocity jump process with dτ = 0.1 and (a) sinusoidal

reorientation, κ = 1, (b) sinusoidal reorientation, κ = 50, (c) linear reorientation, κ = 1,

(d) linear reorientation, κ = 50.


By inspection of the plot in Figure 9.5(a) and (c), the random walks with small correlation

and bias appear to be approximately normally distributed with similar spread about the

mean in each direction. However, the plots in Figures 9.5(b) and (d) clearly show that the

random walks with bias and high correlation have greater average spread in the x direction,

and the distributions in the y direction do not look Normal and appear skewed towards

the preferred direction (with a larger ‘tail’ for the sinusoidal model). The distributions in

the x direction do appear to be approximately Normal.

9.4.1 The limiting value of the correction factor

From the theoretical and simulated results in Chapters 4, 5, 6 and 8, we know that for

a fixed value of dτ , σ2x(t) → 0 and σ2

y(t) → 0 as κ → ∞, but for a particular value of κ,

σ2x(t) > σ2

y(t).

Now, suppose that the mean dispersal distance in each direction is given by

MDDx(t) = Zxσx(t), (9.21)

MDDy(t) = Zyσy(t), (9.22)

where Zx and Zy are correction factors. In an unbiased random walk we expect Zx, Zy →0.798 as t increases. By definition, we have σ2(t) = σ2

x(t) + σ2y(t) and MDD(t) = Zσ(t),

and combining these and (9.21) and (9.22) gives

Z =MDD(t)

√

MDD2x(t)

Z2x

+MDD2

y(t)

Z2y

. (9.23)

If the walk is weakly biased and correlated then as t → ∞ we will have σ2x(t) ≈ σ2

y(t)

and we would expect Z → 0.89 as in the unbiased random walk. However, if the walk is

highly correlated and biased and σ2x(t) ≫ σ2

y(t) and σ2y(t) ≈ 0, then MDD ≈MDDx and

MDDy ≈ 0. If this is the case then from (9.23), we would expect Z ≈ Zx as t→ ∞.

9.4.2 Simulated behaviour of the limiting value of the correction factor

We have completed simulations of 1000 random walkers, looking at the value of the correc-

tion factor Z(κ, dτ , t) for various values of κ and dτ . In general, the correction factor is 1

when t = 0 and then tends to some limiting value Z(κ, dτ ,∞), but we have not attempted

to fit a decaying exponential function as in the previous section. Simulations were run

up to t = 1000, which gives a good estimate of the long-time limiting behaviour even for

large κ. Figure 9.6 show how the correction factors Z(κ, dτ , 1000), Zx(κ, dτ , 1000) and

Zy(κ, dτ , 1000) change as κ increase from 0 to 50, for dτ = 0.1, 0.5 and 1, and sinusoidal


Allowing for simulation noise, Figures 9.6(a) and (b) show the behaviour that we pre-

dicted. For small κ, Z(κ, dτ , 1000) ≈ 0.89 which suggests that the spatial distribution


(a) Sinusoidal Z(κ, dτ , 1000) (b) Linear Z(κ, dτ , 1000)

(c) Sinusoidal Zx(κ, dτ , 1000) (d) Linear Zx(κ, dτ , 1000)

(e) Sinusoidal Zy(κ, dτ , 1000) (f) Linear Zy(κ, dτ , 1000)

Figure 9.6: Plots of values of Z(κ, dτ , t), Zx(κ, dτ , t), and Zy(κ, dτ , t) as a function of κ at

t = 1000 from numerical simulations of sinusoidal and linear reorientation with dτ = 0.1

(- -), dτ = 0.5 (· · ·), and dτ = 1 (- · -). The solid lines (—) correspond to Z = 0.798 or

Z = 0.89 respectively, the expected values if the distribution is Normal.


could be Normal. As κ increases, we start to get MDDx ≫MDDy and Z(κ, dτ , 1000) →Zx(κ, dτ , 1000) ≈ 0.798 as suggested in (9.23). As κ increases, the correction factor de-

creases faster for the larger values of dτ — this is because as κ increases, the spread in the

y direction decreases faster when dτ is larger, as observed in Chapters 4, 5, 6, and 8.

The plots in Figures 9.6(c) and (d), show similar behaviour allowing for simulation noise.

The correction factor for the mean dispersal distance in the non-preferred direction,

Zx(κ, dτ , 1000) appears to be approximately equal to 0.798 for all values of κ in the

simulation. Recall that if the spatial distribution in the x direction is normal then

Zx(κ, dτ ,∞) = 0.798. From the simulation results, we cannot say that the spatial distri-

bution in the x direction is Normal, but we do not have evidence to reject the statement

either. Statistical tests could be used to confirm this result or calculate confidence limits,

but we have not attempted to do this.

The plots in Figure 9.6(f) shows similar behaviour to Figures 9.6(c) and (d) allowing for

simulation noise. We cannot reject the possibility that the spread in the y direction for the

linear model is Normal. However, with the sinusoidal model, the plots in Figure 9.6(e) show

that the correction factor, Zy(κ, dτ , 1000) decreases as κ increases, with the largest decrease

for the larger values of dτ . It appears that the spatial distribution in the y direction for

the sinusoidal reorientation model is not Normal, since Zy(κ, dτ , 1000) 6= 0.798. Recall

the plot of the spatial distribution in Figure 9.5 (b) which suggested a spatial distribution

in the y direction, skewed in the positive y direction with a long tail. In such a spatial

distribution, the mean squared displacement is likely to be a larger relative size than the

mean dispersal distance when compared to a Normal distribution, and hence the correction

factor will be smaller.

Recall that as κ→ ∞, the long-time solutions for the mean dispersal distance will behave

like MDDy(t) → 0 and MDDx(t) ≫ MDDy(t), so that MDDx(t) ≈ MDD(t). This

explains why the plots of Z(κ, dτ , 1000) in Figure 9.6(b) decrease as κ increases, even

though the plots of Zx(κ, dτ , 1000) and Zy(κ, dτ , 1000) both appear to be ≈ 0.798 for all

values of κ.

9.5 Conclusions

In this chapter we have extended the correction factor method of Bovet & Benhamou

(1988) and Byers (2000, 2001), to calculate an equation for the mean dispersal distance

of a correlated and unbiased velocity jump process that is valid for all time. In general, it

is not possible to easily extend the model to include biased random walks as the spatial

distribution does not seem to be Normal. However, by looking at the correction factors

Z(κ, dτ ,∞), Zx(κ, dτ ,∞) and Zy(κ, dτ ,∞) it is possible to gain some information about

the actual spatial distribution in each direction of simulated random walks.



• The mean squared displacement (MSD) of an unbiased and correlated velocity jump

process is equivalent to the MSD of an unbiased and correlated discrete random

walk, if we assume that the number of steps is given by n = λt.

• An unbiased and correlated random walk with a variable step length can have an

MSD as much as twice as large as an unbiased and correlated random walk with a

fixed step length, given that both walks have the same mean step length, E(l) = l.

• The mean dispersal distance (MDD) of a correlated and unbiased random walk can

be calculated from the square root of the MSD and a correction factor. If the number

of steps, n is large enough so that all initial correlation effects have been lost, then

the population spread is Normally distributed and the correction factor is given by

Z ≈ 0.89. If n is small or the walk is highly correlated, then the population spread

is not Normally distributed and the correction factor lies in the range 0.89 ≤ Z ≤ 1.

• The multivariate least squares polynomial used by Byers (2001) to approximate

Z(c, t) is complicated and limited by constraints. A simple exponential decay model

for Z(c, t) fits simulated data very well for an unbiased and correlated velocity jump

process with a variable step length.

• In a biased and correlated random walk the MSD and MDD are not equal in each

direction as they are in an unbiased walk. If bias is present then MDDx(t) >

MDDy(t), where y is the preferred direction. The spread in the preferred direction

is not normally distributed in general, but the spread in the non-preferred direction

could be normally distributed.

• In a biased and correlated random walk the correction factor lies in the range

0.798 < Z(κ, dτ , t) < 1, while the long-time correction factor lies in the range

0.798 ≥ Z(κ, dτ ,∞) ≥ 0.89. Z(κ, dτ ,∞) is smallest for a random walk with the least

spread in the preferred direction (high correlation and bias), and Z(κ, dτ ,∞) ≈ 0.89

for a walk with small bias and correlation as the walk is similar to an unbiased

walk and the spatial distribution is approximately Normal. The correction factor

Z(κ, dτ , t) is dependent on the correlation and the bias parameters, as well as time

— a simple function for Z(κ, dτ , t) has not been found.

• Simulation results suggest that the spatial distribution in the preferred direction

is not Normal for a biased random walk with sinusoidal reorientation and large

values of the reorientation parameters, as the long-time limiting correction factor

Zy(κ, dτ ,∞) 6= 0.798. For small values of reorientation parameters (κ < 5) in the

sinusoidal model, and all values of κ in the linear model, the spatial distribution in the

y direction could be Normal as Zy(κ, dτ ,∞) ≈ 0.798. The spatial distribution in the

x direction for both reorientation models could be Normal as Zx(κ, dτ ,∞) ≈ 0.798.

Chapter 10

Random walks to a barrier and

the recruitment of fish larvae

10.1 Introduction

In this chapter the random walk models developed in previous chapters are used to inves-

tigate the probability of surviving to reach an absorbing barrier, given that we introduce

a simple mortality model. This model is appropriate to the movement and subsequent

recruitment of fish larvae that typically have a very small chance of survival (recruitment

meaning that the immature larvae are considered to become part of the adult population).

Fish larvae foraging in the open sea have been modelled by Pitchford & Brindley (2001)

and Pitchford et al. (2003), and the variability in an individual’s foraging rate is found to

be critical to its survival probability. Deterministic models of reef fish larvae returning and

recruiting to the reef have been studied by Armsworth (2000, 2001), who demonstrates

that the ability of the larvae to orientate to the reef and direct their motion dramatically

increases the chance of survival when compared to passively advected fish larvae.

We consider three basic reef models: a simple ‘infinite’ linear reef, a finite circular reef,

and a finite circular reef with a constant cross-current.

There are two main conclusions to be taken from our results:

i) If there is a very low survival probability then the variability in the system has an

important effect. Deterministic models are likely to underestimate the true survival

probability.

ii) The survival probability for a fish larva attempting to recruit to a reef is highly

sensitive to its sensing and orientating/swimming abilities. Small changes in the

environment can have a critical impact on the survival probability of a typical reef

fish.

Deterministic models are unlikely to be valid if there is a very low survival probability,

290

CHAPTER 10: Random walks to a barrier and the recruitment of fish larvae 291

and in such cases the variability in the environment or in an individual’s motion cannot

be ignored.

10.2 Background to fish larval movement and recruitment

10.2.1 Recruitment of fish larvae in the open sea

Cushing & Horwood (1994) used a deterministic model to predict the survival probability

of fish larvae in the open sea. Alvarez (2000) showed that if stochastic fluctuations are

included in a system where the growth rate is convex (a positive increasing function with

positive second derivative) then the expected population density will be larger. This is

related to Jensen’s inequality: if f(x) is convex and X is a random variable with mean

X then E(f(X)) > f(X), i.e. the average of the function is greater than the function of

the average (Houston & McNamara, 1999). Using this result, Pitchford & Brindley (2001)

showed that the deterministic model of Cushing & Horwood (1994) will underestimate the

survival probability — the variability in the system is important. More recently, Pitchford

et al. (2003) show that it is not necessarily the fact that the growth rate is convex that is

important, but that it is stochastic and includes an amount of variability. They compare

a deterministic model with a fixed constant growth rate to a stochastic model with the

same fixed constant growth rate plus some white noise. By modelling the weight of the

larvae as a random walk with an absorbing barrier they show that the survival probability

for the stochastic model is always larger than the survival probability for the deterministic

model. The greatest difference between the two models occurs with a large death rate,

large variance and small growth rate.

10.2.2 Recruitment of reef fish larvae

Most reef fishes have a dispersing larval stage which is thought to provide the larvae a

greater chance of avoiding reef predators (Bonhomme & Planes, 2000). This dispersing

stage ends when the larvae leave the pelagic environment and recruit into adult reef

populations. The supply rate of larvae to reefs for recruitment is believed to be a critical

determinant of the structure of reef fish populations (Doherty & Williams, 1988), and the

importance of considering supply rates is acknowledged by those managing reef fisheries

and other industries exploiting these populations, see for example Done et al. (1997).

The dispersal paths of pelagic larvae are determined by local advection in the hydrody-

namic regime around reefs, and initial models assumed that fish larvae were passive during

the dispersal and recruitment process (Dight et al., 1990a,b). Such models predicted a

very small recruitment probability and it was thought that recirculatory features in the lee

of reefs could entrap dispersing larvae, see for example Dight & Black (1991). This idea

led to considerable investment in fine-scale numerical simulations of the hydrodynamic


features of flows around individual reefs (Wolanski & Sarenski, 1997).

However, more recent evidence shows that fish larvae can exhibit directed motion and

are not simply passive (Leis et al., 1996). If swimming of the larvae is included in the

models then the probability of recruitment is increased, see Wolanski et al. (1997). The

most recent theoretical models in the literature that include swimming effects are mainly

deterministic using the assumption that the introduced component of velocity due to

swimming of the larvae aims directly at the target destination, see Wolanski et al. (1997),

Armsworth (2000) and (2001).

We shall show that such deterministic models may be too simple and that the random

variability in the movement of the fish larvae can have a significant effect on the probability

of recruitment.

10.2.3 Theoretical models of fish larvae returning to a reef

Armsworth & Bode (1998) examine the effect of including directed motion in the ensuing

population dynamics and spatial structure. Armsworth (2000) looked at the effects of

directed motion on recruitment by considering 4 cases — strong or weak swimmers, and

current-dependent or current-independent orientation cues. The extent of sensory capa-

bilities of the larvae is critical and the rate of recruitment depends sensitively upon it

for both current-dependent and current-independent orientation cues. Armsworth (2001)

uses a deterministic model to find the most efficient swimming strategy for various reef

environments. To swim efficiently, fish larvae should exploit favourable currents and avoid

unfavourable currents by either directed movement or altering depth of swimming. Such

models do not take into account the variable nature of both the environment and the

swimming of the fish larvae.

10.2.4 Experimental data for fish larvae returning to a reef

There is a large amount of experimental data in the literature mainly from observations

made around the Great Barrier Reef (G.B.R.) in Australia for various coral reef fish species.

Data on current speeds near to reefs

Frith et al. (1986) collected data on current speeds around Lizard Island on the G.B.R.

They found typical current speeds to be 10 − 17 cms−1, but could range from < 5 cms−1

to as much as 60 cms−1.

Data on swimming ability of fish larvae

Leis & Carson-Ewart (1999) made in situ observations of the coral trout (Plectropomus

leopardus) and found that they swam directionally and/or changed depth and were not

simply passive. In general, the swimming speed appeared to be greater moving away from


reefs. In a further study, Leis & Carson-Ewart (2000) observed the movement of four coral

reef fish species and again found that they swam directionally. The larvae swam deeper

in the open ocean and could detect predators 3–6 m away and could change depth and/or

direction to avoid them. They also make the point that in situ observations of reef fish

larvae are extremely difficult due to the depth and nature of the movement. In addition,

the development of swimming ability in larvae of three reef fish species was studied by

Fisher et al. (2000), and development was considerably faster than expected.

Data on swimming speed of fish larvae

Leis & Carson-Ewart (1997) made in situ observations of swimming speeds of fish larvae

near Lizard Island on the G.B.R. and found an average speed of 20.6 cms−1 which is

considered strong. Stobutzki & Bellwood (1997) measured swimming speed duration of

various fish larvae — some larvae can maintain speeds of 13.5 cms−1 for several days with-

out rest. Leis & Carson-Ewart (1999) measured swimming speeds of 7.2 cms−1 towards

the reef and 17.9 cms−1 away from the reef. Sustained swimming times were measured

by Fisher & Bellwood (2002). Speeds between 4 cms−1 and 16 cms−1 were measured and

the average swimming time for each speed recorded. The smallest speed gave the longest

swimming time and also the largest total distance travelled. Bellwood & Fisher (2001)

measured swimming speed for different developmental stages of larvae. The greatest rel-

ative speed (body lengths per second) was for the smaller larvae, but the critical speed

(when leaving pelagic stage and returning to the reef) was determined by developmental

stage and size.

Data on orientation cues

Leis et al. (1996) observed directed motion relative to the reef from over 1 km away.

The orientation cue was unknown, but the average movement was away from the reef,

although measurements were made in daylight. Plumes of warmer water up to 2 km south

of One Tree Reef on the G.B.R. were observed by Doherty et al. (1996), which may pro-

vide a signal or temperature gradient for fish larvae to orientate to. McCauley & Cato

(1998) documented nocturnal peaks in noise levels on reefs from 15 km away, which was

attributed to adult reef fish calling. Similarly, McCauley & Cato (2000) collected statis-

tics of reef noise and found fish calling either en masse or individually ad nauseum, with

the highest activity at night. One call type was found to have a lunar trend. Nocturnal

orientation in response to sound originating from the reef was also measured by Stobutzki

& Bellwood (1998). A review of possible orientation cues available to fish larvae is pre-

sented by Montgomery et al. (2001). Possible cues that were discussed include an innate

sensing ability that may be some sort of magnetic compass sense, ambient sound from the

reef, chemo-sensory signals at small scales, ocean swell and/or wave direction, and visual


location at small distances. Logistical constraints limit experiments designed to establish

which cues are used at small scales of the order of metres, as discussed by for example,

Sweatman (1988) and Stobutzki & Bellwood (1998).

We use the above data from the literature to estimate realistic values for the swimming

speed, current speed, and reef distance in the subsequent random walk models. The exact

values used are given with each model.

Data on reorientation parameters

There is little data available on the turning and orientating behaviour of fish larvae on

both individual and population levels. In the absence of data, a biased random walk

model is plausible because we have evidence of directed swimming plus random changes of

direction — either intrinsic or due to local fluid velocity and turbulence. We do not have

any data to use to estimate the values of the reorientation parameters, κ and dτ , or the

turning frequency, λ, so in the following models we use values that are ‘typical’ of those

observed in swimming micro-organisms by Hill & Hader (1997).

10.3 The effect of variability on fish larvae recruitment

10.3.1 Model 1: simple reef environment

We initially assume a very simple model for the reef environment so that we can use the

asymptotic results derived in Chapters 4 and 5. We assume our fish larvae move around

the (x, y) plane (where the preferred direction is y = x1, and x = x2 using the notation of

Chapters 4 and 5), with no current or flow effects and no interactions between individuals.

We assume that there exists an infinite linear reef at a distance R from the origin in the

y-direction. Thus, any fish larva reaching the position (x, yR) where yR ≥ R and x can

be any value, is assumed to have recruited successfully. Figure 10.1 illustrates this simple

environment.

This ‘infinite’ reef model is reasonable if we assume that the size of the reef is much greater

than the distance the larvae need to move to recruit, R.

As in the velocity jump process models described in Sections 4.2 and 5.2, we assume

that all the fish larvae start at the origin (0, 0) and the population is initially orientated

uniformly around the unit circle, so that for t = 0 we have E(cos(θ)) = 0. We assume

that each fish moves with a fixed speed s, a fixed turning frequency λ, and turns at each

step using either the sinusoidal or linear models with reorientation parameters κ and dτ .


Figure 10.1: Simple ‘infinite’ reef model.

10.3.2 Deterministic model for population dynamics

For walkers moving with the simple random walk model described in Section 10.3.1 we

have derived in Chapters 4 and 5, asymptotic equations for the average position, H(t), and

the average spread about the mean position in each direction, σ2x(t) and σ2

y(t) respectively,

that are dependent on the parameters s, λ, κ and dτ .

A simple deterministic model is to assume that all the population move with the average

absolute velocity Hy(t)/t = VA(t). With the large time scale involved in this system it is

reasonable to approximate the solutions in (4.93) and (5.93) as being linear for all time so

that

VA = VF , (10.1)

where VF = s(Af1 + Bf1) for the sinusoidal model (see (4.72) and (4.73)), and VF is

determined numerically for the linear model. The equations in (4.93) and (5.93) have

extra terms (decaying in time and constants) due to the assumption that we initially have

an equal spread of directions around the unit circle. However, in general the equation in

(10.1) produces results with a relative error of < 1% when compared to (4.93) and (5.93),

due to the large time scales involved in this model.

From (10.1), the time taken to reach the reef is given by

tR =R

VF. (10.2)

To model the effects of natural death, predation etc., we assume the simplest possible

mortality model — deaths occur as a Poisson process with rate µ. Thus, the probability

that an individual survives to a certain time tR is simply exp(−µtR), and the probability

of surviving to reach the reef is given by

PR(VF , 0) = exp

(

−µRVF

)

, (10.3)


where the notation PR(VF , 0) denotes the probability of reaching the reef for a fish larva

moving with absolute velocity in the y-direction given by VF and with variance per unit

time γ2, where γ = 0 for this deterministic model.

10.3.3 Stochastic model for population dynamics

As in the deterministic model we assume that the average absolute velocity Hy(t)/t can

be approximated by the linear function VF because of the large time scale used. Similarly,

the long-time equations for the spread about the mean position are linear in time (see

Chapters 4, 5 & 6), and it is reasonable to assume that the equations for the spread about

the mean position in each direction, σ2x(t) and σ2

y(t), can be approximated by the linear

functions ς2t and γ2t respectively. To calculate the time at which each walker reaches a

specified position in the y direction, we only consider the mean position in the y direction,

VF t, and the spread in the y direction, γ2t.

From Section 1.1.3, (1.30) gives the stopping time for a population of walkers with an ab-

sorbing barrier at y = R, and whose movement is governed by the drift diffusion equation:

∂g

∂t= −u∂g

∂y+v2

2

∂2g

∂y2y > 0, (10.4)

where g(y, t) is the probability density function (p.d.f.), ut gives the average position and

v2t gives the variance about the average position. In the absence of a barrier, such a

population is normally distributed in space with p.d.f. N(ut, v2t), see Section 1.1.2.

Our velocity jump process models are not the same as random diffusive processes as we

have extensively discussed in previous chapters. The advantage of the diffusion equation

with drift, (10.4), is that it is easily solved with various boundary conditions, including

an absorbing barrier (Grimmett & Stirzaker, 2001), whereas this is not possible with

the velocity jump process as we do not know the underlying spatial distribution. From

Chapter 9, we know that the long-time spatial distribution of the velocity jump process

is approximately Normal if the reorientation parameters are small. To make comparisons

to simulated results, we assume that for small reorientation parameters our velocity jump

process can be approximated by (10.4) with u = VF and v2 = γ2. We do not expect an

exact fit between this model and simulated results as the velocity jump process used in

the simulations is not the same as simple diffusion with drift. We will see later that even

this approximate model that includes stochastic effects is a better fit to simulated results

than the deterministic model in Section 10.3.2.

From Section 1.1.3 and Grimmett & Stirzaker (2001), the p.d.f. for the stopping time, tR,

for a population with movement governed by (10.4) and with a barrier at y = R is given

by

ftR =R

√

2πγ2t3exp

(

−(R− VF t)2

2γ2t

)

, (10.5)


which takes a similar form to the stopping time p.d.f. derived by Pitchford et al. (2003)

when modelling the growth of foraging fish larvae as a Brownian process. To calculate

the probability of survival it is necessary to take into account this distribution of stopping

times rather than just assuming all the fish larvae reach the reef at the same time. The

probability of an individual reaching the reef is given by

PR(VF , γ) =

∫

∞

0ftRe

−µt dt

= exp

[

RVFγ2

(

1 −√

1 +2µγ2

V 2F

)]

, (10.6)

which is again similar to the solution obtained by Pitchford et al. (2003). From Pitchford

et al. (2003), PR(VF , γ) > PR(VF , 0) for any γ > 0 — the variance in the model always

results in a higher survival probability even in this simple model. The relative survival

probability is given by

RSP ≡ PR(VF , γ)

PR(VF , 0)= exp

[

Rµ2γ2

2V 3F

+O

(

Rµ3γ4

V 5F

)]

, (10.7)

and as in Pitchford et al. (2003), the beneficial effects of the variance are felt most strongly

in a high death rate, high spatial spread, low absolute velocity regime. Fish larvae are

certainly susceptible to a very high death rate mainly due to predation. In the open sea, a

typical survival rate is O(1%) (Chambers and Trippel, 1997) while Pitchford & Brindley

(2001) use a death rate per day of ∼ 10%, so we may expect variance in the movement of

the larvae to have an important effect on the survival probability.

10.3.4 Survival probabilities for the simple reef model

From the experimental data in the literature given in Section 10.2.4 a realistic value for the

average speed of fish larvae is 0.1 ms−1, while the distance of the reef can be anything from

a few metres to several kilometres — we shall use a value of R = 200 m for the following

simulations. Estimates for the turning rate λ and reorientation parameters κ and dτ are

not available in the literature so we have chosen values that seem sensible and are not

too extreme, these values being λ = 1/2, κ = 2 and dτ = 0.1. With these parameters,

and using the sinusoidal reorientation model, VF = 0.0114 and γ2 = 0.0640, so that we

have a small absolute velocity. The expected arrival time at the reef for the deterministic

model is given by tR ≈ 17500s (just under 5 hours), while the stochastic model gives a

distribution of arrival times peaked around tR.

To illustrate the main result in Section 10.3.3 (that variance is important and the deter-

ministic model will underestimate the survival probability), we have looked at a range of

values for the death rate µ, between 0.0001 s−1 and 0.0004 s−1. This death rate may seem

quite low compared to the death rates discussed previously, but considering the large time

taken to reach the reef it is actually quite high and the probability of survival will be seen

to be O(5%).


Simulations were completed with 100,000 walkers all moving with the random walk model

described in Section 10.3.1, and subject to a Poisson process death rate with values of the

parameter µ between 0.0001 s−1 and 0.0004 s−1. Figure 10.2(a) and (b) show the survival

probability against µ for the deterministic model (10.3), the stochastic model (10.6) and

the simulation results PR(sim), with (a) 0.0001 ≤ µ ≤ 0.0002, and (b) 0.0002 ≤ µ ≤0.0004. We have displayed two separate plots to highlight the relative difference between

the deterministic, stochastic and simulated results at very small survival probabilities.

(a) PR(VF , γ) v µ (b) PR(VF , γ) v µ

Figure 10.2: Plots showing (a) survival probability PR(VF , γ) against death rate for (a)

0.0001 ≤ µ ≤ 0.0002, and (b) 0.0002 ≤ µ ≤ 0.0004. Legend: deterministic model (—),

stochastic model (- -), simulation model (+).

From Figures 10.2(a) and (b), one can see that, although the simulation data is fairly

noisy, the simulated survival probability, PR(sim), is always larger than the deterministic

survival probability, PR(VF , 0). Allowing for simulation noise, there is a reasonable fit

between PR(sim) and PR(VF , γ), the stochastic survival probability, even though PR(VF , γ)

is only a diffusion approximation to the velocity jump process used in the simulations. The

behaviour in Figures 10.2(a) and (b) is very similar to that shown in Figure 1 of Pitchford

et al. (2003).

Figure 10.3 shows the relative survival probability for the theoretical and simulated models,

(see (10.7)), when compared to the deterministic survival probability, PR(VF , 0).

Although Figure 10.3 shows a lot of simulation noise, the general behaviour is clear — as

the death rate µ increases (and consequently PR(VF , 0) decreases), the relative survival

probabilities of both the simulated and stochastic models become larger when compared

to the deterministic model.


Figure 10.3: Plots of relative survival probability RSP against PR(VF , 0) from theoretical

(—) and simulation (+) results.

The simulations were completed using the sinusoidal reorientation model — results using

the linear reorientation model are qualitatively similar, although the linear model gives a

higher survival probability with the same values for the reorientation parameters (recall

that the linear model always gives a higher absolute displacement for the same parameter

values). Similarly, if reorientation parameters that produce a larger absolute displacement

are used then the survival probability is larger and the relative survival probability is

smaller, but the qualitative results are similar. Similar results might also be expected if

the death rate is made smaller but the distance to the reef is made larger.

When dealing with systems with very small survival probabilities, such as this simple

example of fish larvae attempting to recruit to a reef or the model of Pitchford et al. (2003)

with foraging fish, it seems clear that the variability in the model cannot be discounted

as deterministic models will underestimate the true survival probability.

From the literature (Leis et al. (1996), Stobutzki & Bellwood (1988) etc.), it seems likely

that fish larvae move away from the reef during the day (possibly to maintain their position

in the open water), and attempt to reach the reef at night. If we assume that a fish larva

can only recruit during the hours of darkness, then there will be a 6-10 hour time limit for

recruitment to the reef to take place. Any fish that are still in the open water after this time

limit are assumed to die. To model this in our theory and simulations we can introduce

a ‘cut-off’ time, T , and assume that PR(t > T ) = 0. This has important implications —

if our deterministic model predicts that R/VF = tR > T then we now have an expected

survival probability of zero, P (VF , 0) = 0, and the deterministic model is useless. The


stochastic model could still be used although we now have different boundary conditions

and the probability of survival will be smaller.

10.4 Optimal swimming behaviour for fish larvae attempt-

ing to recruit to a reef

In the previous section we noted that the survival probability will increase if the expected

average displacement towards the reef increases. In this section we will investigate the

effect of changing the reorientation parameters on the survival probability. From Section

4.2.4, we consider the parameter dτ as the ‘sensing’ ability and the parameter κ as the

‘orientating/swimming’ ability (the ability to overcome the inherent randomness in the

environment). We have seen in Sections 8.2.1 and 8.2.2 that the optimal values of the

parameters (to produce the maximum average displacement) are κ → ∞ and dτ = 1

(linear model) and 1 ≤ dτ ≤ π2 (sinusoidal model), and we might expect the largest

survival probabilities for these optimal parameter values.

We have looked at a simple infinite reef model in the previous section and the results are

not repeated here. Similarly to the simulation results presented in Chapter 8, the models

and parameter values considered in this section do not fit for the asymptotic solutions

from Chapters 4 and 5, and so we only look at simulation results.

10.4.1 Model 2: simple circular reef model

We assume exactly the same environment as in Section 10.3.1, except that instead of an

infinite linear reef we now have a small circular reef of radius r, centred at the point (0, C),

see Figure 10.4. We also introduce a cut-off time, T , after which any fish larvae that have

not reached the reef are assumed to be dead.

In a similar manner to the model described in Section 8.4, the fish larvae always attempt

to reorientate to the centre of the reef, so that the preferred direction is not constant but

dependent on the spatial position.

Simulations of 1000 fish larvae moving towards a circular reef have been completed for the

sinusoidal and linear reorientation models for various values of the reorientation parameters

κ and dτ . In all the simulations we use the following parameter values: the cut-off time

T = 10 hours (36, 000 s), the speed s = 0.1 ms−1, the turning frequency λ = 0.5 s−1, the

distance to the centre of the reef C = 500 m, the radius of the reef r = 10 m, and the death

rate µ = ln(0.5)/36, 000 s−1, so that the probability of surviving the effects of predation,

natural death etc. for 10 hours is Psurv(10hrs) = 0.5 (at which point any surviving fish

larvae that have not reached the reef are killed off). Up to the cut-off point T , this is a

low death rate compared to the previous section.


Figure 10.4: Simple circular reef model.

With these parameters, the theoretical minimum time to reach the reef is given by tmin =

4900 s, corresponding to moving directly from the origin to the edge of the reef in a straight

line (i.e. VF = s = 0.1). Such models that do not include random fluctuations of the

individual movement have been considered by Armsworth (2000, 2001). The probability

of reaching the reef and surviving in this extreme case is PR(VF = 0.1) = 0.9100. This

gives an expected upper bound for the survival probabilities of the subsequent simulations

that do include the effects of random reorientations in the individual movement.


The plots in Figure 10.5 show how the simulated survival probability for reef model 2 with

sinusoidal reorientation changes as (a) κ increases for 4 values of dτ , (b) dτ increases for

4 values of κ.

In Figure 10.5(a) the survival probability PR(dτ , κ) > 0 as long as dτ > 0 and κ is large

enough, while in Figure 10.5(b) PR(dτ , κ) = 0 for κ < 0.4 and all values of dτ , indicating

that the survival probability is likely to be less than 1/1000 for these values.



linear reorientation changes as (a) κ increases for 4 values of dτ , (b) dτ increases for 4

values of κ.

As in the sinusoidal model, in Figure 10.6(a) the survival probability PR(dτ , κ) > 0 as long

as dτ > 0 and κ is large enough, while in Figure 10.6(b) PR(dτ , κ) = 0 for κ < 0.4 and

all values of dτ , indicating that the survival probability is likely to be less than 1/1000 for

these values.


(a) PR(dτ , κ) v κ (b) PR(dτ , κ) v dτ

Figure 10.5: Plots showing survival probability PR(dτ , κ) for sinusoidal reorientation and

Model 2 against (a) κ, for dτ = 0.1 (—), dτ = 0.3 (- -), dτ = 0.5 (· · ·), and dτ = 1.0 (- · -);

(b) dτ , for κ = 0.4 (—), κ = 1.0 (- -), κ = 2.0 (· · ·), and κ = 4.0 (- · -).


Figure 10.6: Plots showing survival probability PR(dτ , κ) for linear reorientation and

Model 2 against (a) κ, for dτ = 0.1 (—), dτ = 0.3 (- -), dτ = 0.5 (· · ·), and dτ = 1.0

(- · -); (b) dτ , for κ = 0.4 (—), κ = 1.0 (- -), κ = 2.0 (· · ·), and κ = 4.0 (- · -).


10.4.1.3 Comments on survival probabilities for Model 2

Figures 10.5(a) and 10.6(a) show similar behaviour — for each value of dτ there is a critical

value of κ, such that for all κ < κcrit, PR(dτ , κ) = 0 in our simulations. Compare these

results to the plots in Figure 8.1 — the highest survival probability corresponds to the

highest average displacement in the preferred direction, as might be expected. However,

the increase in the average displacement in Figure 8.1 is approximately linear with κ for

small κ. The plots in 10.5(a) and 10.6(a) show a sudden jump from a survival probability

close to zero to a survival probability that is close to the theoretical maximum. As κ

increases further the survival probability seems to tend asymptotically to the theoretical

maximum.

Figures 10.5(b) and 10.6(b) also show similar behaviour — there is a sudden jump in the

survival probability for a critical value of dτ . Comparing to the plots in Figure 8.5, the

largest survival probability corresponds to the optimal value of dτ , however the increase

in the average displacement with dτ is a lot smoother and there is no sudden jump as we

see with the survival probability plots. The maximum survival probability seems to be

independent of dτ and critically dependent on κ — even if the sensing ability is optimal

dτ = dopt, if there is too much turbulence or the fish larvae cannot orientate to the preferred

direction very well (small κ) then the survival probability will always be low.

If the reef is further away, or the fish have a slower speed, then the survival probabilities

decrease but the qualitative behaviour is similar and plots are omitted.

It seems that in general, the fish larvae either have a very low (or zero) survival probability

or a reasonably high survival probability with not much middle ground. The values of the

reorientation parameters are critical to determining the survival probability.

It is not unreasonable to think that our fish larvae would evolve to be just good enough

swimmers to give a high survival probability but would not be any better than necessary

(i.e. they would be just at the top of the switch in survival probabilities observed above).

If this were the case, a slight change in the environment would result in a catastrophic

change in survival probability that results in massive decrease in the number of fish larvae

managing to recruit.

10.4.2 Model 3: simple current model

We assume exactly the same environment as in Section 10.4.1, except that we now intro-

duce a cross-current of fixed magnitude U , see Figure 10.7.

A cross-current would have no effect on the simple infinite linear reef model as the current

only increases the x position and fish larvae would still reach the reef at the same time

(albeit at a different point on the reef). However, the cross-current will certainly have an

effect when the fish larvae are trying to reach a small circular reef. It is not of particular


Figure 10.7: Circular reef with a constant current.

interest to look at a current that produces a drift directly towards the reef relative to the

origin of the fish larvae, as this just increases the absolute velocity VF , while a current that

produces a drift away from the reef will decrease the absolute velocity VF , see Armsworth

(2000).

Simulations of 1000 fish larvae moving towards a circular reef with a constant cross-current

have been completed for the sinusoidal and linear reorientation models for various values

of the reorientation parameters κ and dτ . In all the simulations we have used the same

parameter values as in Section 10.4.1, and the current speed U = 0.05 ms−1, which is not

unrealistic when compared to measured current speeds near reefs, see Frith et al. (1986).

With these parameters, the theoretical minimum time to reach the reef is found by

Pythagoras Theorem, where we have a triangle of sides 490 m, 0.05tmin m and hypotenuse

0.1tmin m. Thus, the optimal orientation is to move slightly into the current, so that the

drift takes one to the reef, see Armsworth (2001). Solving this gives tmin = 5658 s. The

probability of reaching the reef and surviving in this case is PR = 0.8968, which gives an

expected upper bound that is only slightly smaller than in Section 10.4.1. It is clear that

by increasing the magnitude of the current U , the minimum time to reach the reef tmin

will increase and consequently the survival probability will decrease. In fact as U → s the

theoretical survival probability tends to zero, PR → 0.



sinusoidal reorientation changes as (a) κ increases for 4 values of dτ , (b) dτ increases for

4 values of κ.



Figure 10.8: Plots showing survival probability PR(dτ , κ) for sinusoidal reorientation and

Model 3 against (a) κ, for dτ = 0.2 (—), dτ = 0.5 (- -), dτ = 1.0 (· · ·), and dτ = 1.5 (- · -);

(b) dτ , for κ = 1.8 (—), κ = 2.0 (- -), κ = 3.0 (· · ·), and κ = 5.0 (- · -).

In Figure 10.8(a) the survival probability PR(dτ , κ) > 0 as long as dτ > 0.1 and κ is large

enough, while in Figure 10.8(b) PR(dτ , κ) = 0 for κ < 1.8 and all values of dτ . Thus, the

true survival probability is likely to be less than 1/1000 for κ < 1.8, which is larger than

the equivalent value in the non-current model (κ < 0.4). Unlike the non-current model, for

this range of κ values the survival probability is likely to be less than 1/1000 for dτ < 0.2.



linear reorientation changes as (a) κ increases for 4 values of dτ , (b) dτ increases for 4

values of κ.

In Figure 10.9(a) the survival probability PR(dτ , κ) > 0 as long as dτ > 0 and κ is large

enough (unlike the sinusoidal model with current for these values of κ), while in Figure

10.9(b) PR(dτ , κ) = 0 for κ < 1.4 and all values of dτ . Thus, the true survival probability

is likely to be less than 1/1000 for κ < 1.4 which is larger than the equivalent value in the

non-current model (κ < 0.4), but smaller than the critical value for the sinusoidal model

with current (κ < 1.8).

10.4.2.3 The effect of the current on the survival probability


dτ = 0.8 changes as U increases for (a) sinusoidal reorientation, (b) linear reorientation.

As expected, the survival probability tends to zero as the current speed increases, PR → 0

as U → s, but the behaviour is not smooth and features the same sudden jump as discussed



Figure 10.9: Plots showing survival probability PR(dτ , κ) for linear reorientation and

Model 3 against (a) κ, for dτ = 0.1 (—), dτ = 0.3 (- -), dτ = 0.5 (· · ·), and dτ = 1.0

(- · -); (b) dτ , for κ = 1.4 (—), κ = 2.0 (- -), κ = 3.0 (· · ·), and κ = 5.0 (- · -).

(a) Sinusoidal PR(U, κ) v U (b) Linear PR(U, κ) v U

Figure 10.10: Plots showing survival probability PR(U, κ) v U with dτ = 0.8 for (a)

sinusoidal reorientation and (b) linear reorientation. Legend: κ = 1.0 (—), κ = 2.0 (- -),

κ = 3.0 (· · ·), and κ = 5.0 (- · -).


previously. For fixed reorientation parameters there is a critical current magnitude, such

that U > Ucrit results in a zero survival probability in the simulations. For the same

parameter values, it is clear that the survival probability for the linear model is greater

than that for the sinusoidal model.

10.4.2.4 Comments on survival probabilities for Model 3

The plots in Figures 10.8 and 10.9 show very similar qualitative behaviour to Figures 10.5

and 10.6 and similar conclusions and comments to those made in Section 10.4.1.3 apply.

The survival probabilities have significantly decreased when compared to the results from

Model 2, and the critical values of the reorientation parameters are now larger — the fish

larvae need to be better at sensing and orientating towards the reef because in general the

current is taking them away from it.

For fixed reorientation parameters, the survival probability is also critically dependent on

the current magnitude, U . For small U , the survival probability is close to the upper

bound predicted, with a slow decrease in survival probability as U increases. However,

once U > Ucrit there is a massive drop in the survival probability which becomes close to

zero. If the current is faster than the speed of movement, U > s, then the fish larvae have

zero survival probability in this reef environment.

10.4.3 Further models

It is straightforward to adapt our simulation model to include even more realistic effects

such as spatially dependent parameters as in Sections 8.3.1 and 8.3.2 — it makes sense for

the larvae’s sensing ability (dτ ) to increase as it gets closer to the reef, while an increase

in κ could correspond to the larvae moving away from the turbulent open water and into

the calmer waters near the reef so that we have an inhomogeneous environment.

If we model the sensing ability as spatially dependent it is necessary to consider the

orientation cues to the reef that the larvae use, see Montgomery et al. (2001). If the cue

is sound (McCauley & Cato (1998) and (2000), Stobutzki & Bellwood (1998) etc.), then

although the fluid dynamics and turbulence of the water could effect the dispersal of the

sound wave, it is not unreasonable to model the signal as spreading out in all directions

from the centre of the reef. If the cues are chemical, then currents and turbulence will

have a much more obvious effect on the dispersal of the signal — the orientation cue

will be different depending on whether a larvae is upstream or downstream from the reef

(Armsworth, 2000) It is likely that fish larvae use a number of different orientation cues

at different scales — sound at large distances, and then chemical or even visual cues at

smaller distances, see Montgomery et al. (2001).

We have completed several simulations looking at the spatial dependence of the orienta-

tion cues, and consequently the reorientation parameters κ and dτ . In general, there is


little qualitative difference to the results in Sections 10.4.1 and 10.4.2 and results are not

presented. For example, the survival probability will decrease if dτ is spatially dependent

with a limiting value dopt, when compared to a fixed value of dτ = dopt for all spatial

positions. The overall survival probability is still highly sensitive to the value of κ that is

used.

One extension to the model that may be worth considering is a variable death rate µ

depending on the spatial position and also the depth of the fish larvae (which may require a

three dimensional simulation model). There is evidence that fish larvae can sense predators

and will move away and or change their depth to avoid being eaten, see Leis & Carson-

Ewart (1999). There is also the possibility that fish larvae use the fact that currents

are different magnitudes (and possibly even different directions) at different depths, to

maximise their chances of reaching the reef (Armsworth, 2001).

What seems certain is that passive advection of the fish larvae through pure diffusion or

favourable currents is unlikely to result in a non-zero survival probability. By developing

basic reorientation and swimming abilities a reef fish larva can dramatically increase its

chances of surviving to reach the reef and recruit into the adult population.

10.5 Conclusions

By adding individual-based mortality to the theoretical and simulation random walk mod-

els developed in previous chapters we have shown how they can be applied to a useful

application — the survival probability of pelagic reef fish larvae. We have used data from

the literature to estimate parameter values such as swimming speed and current speed.

There does not appear to be any data in the literature that we can use to estimate our

reorientation parameters, so we have made what we consider sensible estimates.

For a simple ‘infinite’ linear reef environment model with a high death rate we have very

similar results to Pitchford et al. (2003), in that a simple deterministic model (where all

the fish larvae are assumed to have the same absolute velocity) will always underestimate

the survival probability obtained from stochastic models and simulation results. The

variability in the movement of the fish larvae is important when the survival probability

is very low.

More complicated reef environment models have been studied using simulations — it is

possible to introduce a circular reef of fixed size and also a cross-current of constant mag-

nitude. Using these simulation models we have investigated the effect that the individual

sensing ability (dτ parameter) and orientating ability (κ parameter) of the fish larvae has

on the survival probability of a population attempting to recruit. There appears to be

a critical dependence on the reorientation parameters with either a very low (or zero)

survival probability, or a relatively high survival probability and no middle ground. This


may be significant if fish larvae have evolved to be adapted to a particular environment

— if the environment changes and the fish larvae subsequently are less able to sense or

orientate to the reef then there will be huge decrease in the survival probability. Deter-

ministic models such as Armsworth (2000) and (2001) do not predict this high sensitivity

on the swimming abilities of the fish larvae.

More complicated models such as spatially dependent reorientation parameters or diffusive

signals have been considered, but initial simulation results do not show any significantly

different behaviour to the simpler models. Worthwhile extensions may be to investigate

a variable death rate that could account for the fish larvae avoiding predators by moving

away or changing their depth, or a variable current to take into account the fish larvae

altering their depth to find more favourable currents. Both these extensions may be better

suited to three-dimensional simulations.


• Our theoretical and simulation random walk models can be used to model the move-

ment and recruitment of fish larvae returning to reefs.

• In general, deterministic models underestimate the survival probability if there is

variability in the system. The variability will have more effect on the survival prob-

ability if there is a high death rate, a large spatial spread and a small absolute

velocity.

• The survival probability is highly sensitive to the reorientation parameters κ (which

we consider the orientating ability), and dτ (which we consider the sensing ability).

A change in the environment could have produce a large decrease in the survival

probability if the fish larvae can not change their swimming abilities.

• The parameter κ is highly critical to the survival probability even if dτ is optimal

— it is no use being able to sense the direction of the reef perfectly if the fish larvae

cannot orientate to move in that direction.

• More complex reef environment models that we have simulated produce similar re-

sults, but further three dimensional simulations that take into account the depth of

the fish, variable currents, and a variable death rate may be worthwhile if there is a

need to study a particular environment.

Chapter 11

Concluding remarks

11.1 Main results

In this thesis we have extended the method of calculating the moments of the underlying

spatial distribution of a two-dimensional velocity jump process (Othmer et al., 1988) to

include realistic reorientation models as suggested by Hill & Hader (1997), where bias

is introduced by the dependence of the mean turning angle on the absolute angle of the

previous direction of movement. Simple diffusive models based on position jump processes

allow for infinite propagation and cannot take account of the dependence of the mean

turning angle on the previous direction of movement, whereas a velocity jump process

can take this into account as the random walk is in the velocity and not the position.

The main problem encountered with the velocity jump process is that it is not possible

to derive an equation for the underlying spatial distribution in two dimensions, although

in one dimension one can derive the telegraph equation (Goldstein, 1952; Kac, 1974).

Othmer et al. got around this problem by deriving equations for the moments of the

underlying spatial distribution directly. The reorientation models used by Othmer et al.

are somewhat artificial, but do result in a simple closed system of differential equations for

the moments of the underlying spatial distribution. Hill & Hader’s reorientation models

are more realistic and have reorientation parameters (κ and dτ ) that can be directly

measured by simple experiments. However, to arrive at a closed system of differential

equations for the higher order moments using these more realistic reorientation models, it

is necessary to make several assumptions and the final solutions for the required statistics

are asymptotic expansions only valid for small values of the parameter dτ .

Using numerical simulations of velocity jump processes with Hill & Hader’s reorientation

models, we show that the asymptotic solutions are a good fit to simulated data over a

wide range of parameter values, and only start to break down when both reorientation

parameters are large. Using the same simulation model it is also possible to investigate ve-

locity jump processes at parameter values for which the asymptotic solutions are not valid.

We have looked at large reorientation parameters and spatially dependent parameters, as

310

CHAPTER 11: Concluding remarks 311

well as the effect of a changing preferred direction. The same simulation model could be

used to investigate further extensions of the velocity jump process where it may not be

possible to derive theoretical solutions or approximations. Extensions to the model that

would be biologically relevant include variable speed and/or turning frequency, waiting

times between steps, temporally dependent parameters, nonhomogeneity in the environ-

ment, interactions between individual walkers, and allowing movement in three dimensions

rather than restricting the model to two dimensions.

With the same simulation model, we have investigated how valid the method used by Hill

& Hader to calculate the reorientation parameters from observed data is, when the original

random walk is not continuous but a velocity jump process. Their method is valid but

for velocity jump processes with large times between turns it tends to underestimate the

true values of the reorientation parameters. The closer the velocity jump process is to a

continuous random walk, the better the estimates of the reorientation parameters are. If

the sampling length used is too large the method will also underestimate the reorientation

parameters, and comparing simulation results to the results of Hill & Hader suggests that

their sampling length was too large, although their smallest sampling length used was an

experimental constraint.

Various authors have tried to determine the relation between the root of the mean squared

displacement (MSD) and the mean dispersal distance (MDD), either by resolving simula-

tions (Byers, 2000, 2001) or finding a direct analytic relation (Bovet & Benhamou, 1988;

McCulloch & Cain, 1989). We have suggested a model for the correction factor between

the root of the MSD and the MDD for a correlated and unbiased random walk that is

valid for all time. The model is much simpler than previous ones in the literature (Byers,

2000, 2001), but fits simulated data well. With simulations we have studied the MDD of a

biased and correlated random walk, and although we have not found a simple equation for

the correction factor, we have shown for certain cases that the spatial distribution cannot

be Normal, unlike the case with the long-time distribution of an unbiased and correlated

random walk.

Finally, we have demonstrated a direct application of our velocity jump process model

to the directed movement and subsequent recruitment of reef fish larvae. With a simple

reef environment, we use a diffusion approximation of our velocity jump process model

to compare to simulated results, and show that deterministic models underestimate the

survival probability if there is a high death rate. In more complex reef environments, sim-

ulations are used to investigate the optimal swimming behaviour — although the sensing

ability has an effect on the survival probability, it seems to be the swimming and orientat-

ing ability that determines the maximum survival probability. The survival probability is

found to be highly sensitive to critical parameter values, suggesting that in a homogeneous

population, fluctuations in the environment could have a catastrophic effect on the size

of the recruited population. More complex reef models can be simulated, for example by


adding spatially dependent parameters but this has little effect on the main conclusions

previously made. Realistic effects mentioned in the literature that we have not considered

and that may effect the survival probability include a variable death rate, the effects of

the larvae altering depth, and variable currents and tides. All could be readily modelled

with more sophisticated simulations.

We believe we have presented a fundamental framework for modelling the movement of

animals and micro-organisms using random walks where bias is introduced through the

dependence of the mean turning angle on the previous direction of movement, as observed

in Hill & Hader’s experiments. Modelling the random walk as a velocity jump process with

this realistic reorientation model has its limitations — the asymptotic equations derived

for the spatial statistics of interest are only valid for small reorientation parameter values.

However, this is not a major problem as experimental observations show that realistic

values of the parameters are small. If the asymptotic equations are not valid then simu-

lations can still be used to investigate the behaviour in various situations. Experimental

methods as suggested by Hill & Hader can be used to find the reorientation parameters

for any population of walkers and these can then be used in the theoretical or simulation

models.

11.2 Possible future research

The asymptotic equations we have derived are valid only for a homogenous environment.

In the simple model in Chapter 10, we had to approximate the walk as a diffusive process to

use the absorbing barrier result that corresponds to fish larvae reaching the reef. However,

in situations such as this model where the probability of survival is very small and the

tails of the spatial distribution are very important, it does not seem sensible to use the

diffusion equation. We have discussed previously how the diffusion equation allows for

infinite propagation and it is not a realistic model for the tails of the spatial distribution.

The velocity jump process is much more realistic but much harder to manipulate and use

in different environments as we do not know the underlying spatial distribution. It is

possible to run simulations of such problems to generate results, but a theory for the tails

of distributions in biased and correlated random walks would be a significant advance.

Research into models using the velocity jump process and transport equations (similar

to those used for our analysis) to model directed motion and chemotaxis is ongoing, see

review by Hillen (2002). In general, the systems of equations derived using a transport

equation are not closed and there is no standard moment-closure method. We believe

we are the first to use a moment-closure method with a velocity jump process that has

bias introduced through the dependence of the mean turning angle on the absolute angle

(sinusoidal and linear reorientation models).


The simple velocity jump process model with sinusoidal or linear reorientation is applicable

to any population of animals or micro-organisms moving with directed motion, and can

be used at many spatial scales. There is no data in the literature on the turning behaviour

of fish larvae but we demonstrated in Chapter 10 that this can have a critical effect on he

survival probability. Experiments on tracking fish larvae may be difficult to implement in

situ, but if data can be collected it would be straightforward to analyse using the method

of Hill & Hader to estimate the reorientation parameters, which can then be used in our

theoretical and simulation models. Similar experiments could in theory be carried out on

any organisms that have a directed motion, although at very small spatial scales it becomes

hard not to influence the movement through the experimental set-up (as discussed by Hill

& Hader). Finally, as evidence of the value of this theory, it is interesting to note that

other researchers are now using simulations based on Hill & Hader’s work to study, for

example, the motion of endothelial cells in angiogenesis leading to the development of

tumours (Plank & Sleeman, 2003).

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