Modeling Collective Motion in Animal Groups: from Mathematical … · 2019. 1. 23. · Modeling...

Modeling Collective Motion inAnimal Groups: from

Mathematical Analysis to FieldData

by

Ryan J. Lukeman

B.A., St. Francis Xavier University, 2003M.Sc., Dalhousie University, 2005

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in

The Faculty of Graduate Studies

(Mathematics)

THE UNIVERSITY OF BRITISH COLUMBIA

(Vancouver)

May 2009

c© Ryan J. Lukeman 2009

Abstract

Animals moving together cohesively is a commonly observed phenomenon inbiology, with bird flocks and fish schools as familiar examples. Mathematicalmodels have been developed in order to understand the mechanisms thatlead to such coordinated motion. The Lagrangian framework of modeling,wherein individuals within the group are modeled as point particles with po-sition and velocity, permits construction of inter-individual interactions via‘social forces’ of attraction, repulsion and alignment. Although such modelshave been studied extensively via numerical simulation, analytical conclu-sions have been difficult to obtain, owing to the large size of the associatedsystem of differential equations. In this thesis, I contribute to the modelingof collective motion in two ways. First, I develop a simplified model of mo-tion and, by focusing on simple, regular solutions, am able to connect groupproperties to individual characteristics in a concrete manner via derivationsof existence and stability conditions for a number of solution types. I showthat existence of particular solutions depends on the attraction-repulsionfunction, while stability depends on the derivative of this function.

Second, to establish validity and motivate construction of specific mod-els for collective motion, actual data is required (though scarce). I describework gathering and analyzing dynamic data on group motion of surf scot-ers, a type of diving duck. This data represents, to our knowledge, thelargest animal group size (by almost an order of magnitude) for which thetrajectory of each group member is reconstructed. By constructing spatialdistributions of neighbour density and mean deviation, I show that frontalneighbour preference and angular deviation are important features in suchgroups. I show that the observed spatial distribution of neighbors can beobtained in a model incorporating a topological frontal interaction, and Ifind an optimal parameter set to match simulated data to empirical data.

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Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . xviii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Biological Background . . . . . . . . . . . . . . . . . . . . . . 11.2 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . 21.3 Survey of the Literature : Models . . . . . . . . . . . . . . . . 2

1.3.1 Eulerian Models . . . . . . . . . . . . . . . . . . . . . 21.3.2 Lagrangian Models . . . . . . . . . . . . . . . . . . . . 31.3.3 Unifying Lagrangian Approaches . . . . . . . . . . . . 7

1.4 The Need for Empirical Data . . . . . . . . . . . . . . . . . . 71.5 Survey of the Literature: Empirical Studies . . . . . . . . . . 8

1.5.1 Small Groups, No Tracking . . . . . . . . . . . . . . . 81.5.2 Small Groups with Tracking . . . . . . . . . . . . . . . 91.5.3 Large Groups, No Tracking . . . . . . . . . . . . . . . 101.5.4 What Empirical Data is Needed . . . . . . . . . . . . 11

1.6 Contributions of this Thesis . . . . . . . . . . . . . . . . . . . 111.6.1 Model Analysis . . . . . . . . . . . . . . . . . . . . . . 121.6.2 Empirical Data . . . . . . . . . . . . . . . . . . . . . . 13

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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Minimal Mechanisms for School Formation in Self-PropelledParticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 The Model of Interacting Self-Propelled Particles . . . . . . . 24

2.2.1 Assumptions and Equations . . . . . . . . . . . . . . . 242.2.2 Classification of Schools . . . . . . . . . . . . . . . . . 27

2.3 Schools in One Dimensional Space . . . . . . . . . . . . . . . 292.3.1 Schools Formed by Following One Immediate Neigh-

bour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.2 Schools Formed by Interactions with Two Nearest Neigh-

bours. . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.3 Examples of Some Schooling Forces . . . . . . . . . . 342.3.4 The Stability of a School Solution . . . . . . . . . . . 362.3.5 Numerical Simulations in One Dimension . . . . . . . 41

2.4 The Soldier Formation in 2D Space . . . . . . . . . . . . . . . 432.4.1 Existence Conditions for the Soldier Formation . . . . 452.4.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . 462.4.3 Numerical Simulation of the Soldier Formation . . . . 50

2.5 Schools in the Form of 2D Arrays . . . . . . . . . . . . . . . . 532.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3 A Conceptual Model for Milling Formations in BiologicalAggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2 The Model of Interacting Self-Propelled Particles . . . . . . . 66

3.2.1 Relationship to Previous Models . . . . . . . . . . . . 673.2.2 The Milling Formation . . . . . . . . . . . . . . . . . . 68

3.3 Existence Conditions . . . . . . . . . . . . . . . . . . . . . . . 693.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 72

3.4.1 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 763.5 Numerical Investigation . . . . . . . . . . . . . . . . . . . . . 77

3.5.1 Numerical Simulations . . . . . . . . . . . . . . . . . . 783.5.2 Moving Mill Formations . . . . . . . . . . . . . . . . . 84

3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4 A Field Study of Collective Behaviour in Surf Scoters: Em-pirical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.1.1 Difficulties in Obtaining Data . . . . . . . . . . . . . . 974.1.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . 984.1.3 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . 984.1.4 Challenges Avoided in This Study . . . . . . . . . . . 99

4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.2.1 Location and Materials . . . . . . . . . . . . . . . . . 994.2.2 Surf scoter behaviour in field study . . . . . . . . . . . 1004.2.3 Experimental Technique . . . . . . . . . . . . . . . . . 1004.2.4 Calibration and Testing . . . . . . . . . . . . . . . . . 1004.2.5 Postprocessing Images . . . . . . . . . . . . . . . . . . 1074.2.6 Extracting Positions . . . . . . . . . . . . . . . . . . . 1074.2.7 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . 1084.2.8 Edge effects . . . . . . . . . . . . . . . . . . . . . . . . 1104.2.9 Processed Events . . . . . . . . . . . . . . . . . . . . . 1134.2.10 Body Alignment Versus Velocity . . . . . . . . . . . . 113

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.3.1 Basic statistics . . . . . . . . . . . . . . . . . . . . . . 1164.3.2 Nearest-Neighbour Distance Distributions . . . . . . . 117

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5 Ducks in a Row: Inferring Interaction Mechanisms fromField Data of Surf Scoters . . . . . . . . . . . . . . . . . . . . . 1245.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.1.1 Specific Goals of this Work . . . . . . . . . . . . . . . 1255.1.2 Using Spatial Distributions . . . . . . . . . . . . . . . 125

5.2 Empirical Methods . . . . . . . . . . . . . . . . . . . . . . . . 1255.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . 126

5.3.1 Relative Location of Neighbors . . . . . . . . . . . . . 1265.3.2 Relative Deviation of Neighbors . . . . . . . . . . . . 129

5.4 Building a Model . . . . . . . . . . . . . . . . . . . . . . . . . 1315.4.1 Model Framework . . . . . . . . . . . . . . . . . . . . 133

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5.5 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.5.1 Fixed Parameters . . . . . . . . . . . . . . . . . . . . 1415.5.2 Free Parameters . . . . . . . . . . . . . . . . . . . . . 1435.5.3 Parameter Effects on Radial Distributions . . . . . . . 144

5.6 An Optimal Parameter Set . . . . . . . . . . . . . . . . . . . 1445.6.1 Goodness-of-fit Measure . . . . . . . . . . . . . . . . . 1445.6.2 Optimization Process . . . . . . . . . . . . . . . . . . 1455.6.3 Optimization Results . . . . . . . . . . . . . . . . . . 1475.6.4 Detailed Parameter Exploration Near Optimal Set . . 147

5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576.2 Analysis of Perfect School Solutions . . . . . . . . . . . . . . 157

6.2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 1576.2.2 Perfect School Analysis in Context . . . . . . . . . . . 158

6.3 Empirical Data and Modeling . . . . . . . . . . . . . . . . . . 1596.3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 1596.3.2 Empirical Data and Modeling in Context . . . . . . . 161

6.4 Summary: Main Contributions . . . . . . . . . . . . . . . . . 162

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Appendices

A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166A.1 The Stability Analysis in 1D . . . . . . . . . . . . . . . . . . 166A.2 Stability Analysis for the Soldier Formation in 2D . . . . . . 168

B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171B.1 Transformation to Relative Coordinates . . . . . . . . . . . . 171B.2 Derivation of the Linearized Perturbed System . . . . . . . . 174B.3 Eigenvalue Equation . . . . . . . . . . . . . . . . . . . . . . . 176B.4 Derivation of Inequality (3.37) . . . . . . . . . . . . . . . . . 177

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C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179C.1 Data Sequences: Details . . . . . . . . . . . . . . . . . . . . . 179C.2 Data Analysis: Details on Currents . . . . . . . . . . . . . . . 179

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List of Tables

4.1 Basic statistics: data sequences . . . . . . . . . . . . . . . . . 1174.2 Average speed and nearest neighbor distances for the 14 sep-

arate sequences of data that were analyzed. Units are body-lengths (BL). Speed and NND was averaged over all individ-uals in all frames. The last column is the standard deviationof NND. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.1 Model : Summary of Interaction Forces . . . . . . . . . . . . . 1425.2 Summary of Parameters . . . . . . . . . . . . . . . . . . . . . 1435.3 Using Model V together with an optimization routine, an opti-

mal parameter set is found that best matches simulated neigh-bor distributions to observed neighbor distributions. Parame-ters are dimensionless unless units are given. . . . . . . . . . . 143

C.1 Data Sequences: Details . . . . . . . . . . . . . . . . . . . . . 180C.2 Current statistics: by sequence . . . . . . . . . . . . . . . . . . 181

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List of Figures

2.1 A schematic diagram of model vectors in 2 dimensions. . . . . 262.2 A schematic diagram of particles in 1 dimension, showing

our subscripting convention. Grey arrows indicate distance-dependent interaction forces. . . . . . . . . . . . . . . . . . . . 29

2.3 An example of a distant-dependent force in the form of a Hillfunction with a decay past xf , to depict a finite sensing range. 35

2.4 Four cases of neighbour interaction: [a] neighbour j travelsfaster than i in the same direction, (vji > 0), [b] neighbour jtravels slower than i in the same direction, (vji < 0), [c] neigh-bour j diverging from i, (vji > 0), [d] neighbour j convergingto i, (vji < 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5 [a] An example of interaction forces where the net force haspositive slope at d. Here, g+(x) = 2[3x2/(x2 + 42) − 1], andg−(x) = 3x/(x + 4) − 1. [b] An example of interaction forceswhere the net force has negative slope at d. Here, g+(x) =(1/2)[3x3/(x3 + 23) − 1], and g−(x) = 3x/(x+ 8) − 1. . . . . . 37

2.6 1D simulations showing position in a moving frame moving atvelocity v as indicated. Interaction forces are as in [a] Fig.2.5.a, and [b] Fig. 2.5.b. Self propulsion terms are [a.i] al =0.5, [a.ii] al = 0.7, [b.i] al = 0.9, [b.ii] al = 0.8, while a = 0.1 inall cases. In [a], higher school speed resulted in an increase inNND from 3.26 in [a.i] to 3.49 in [a.ii]. The opposite occurredin [b], where higher school speed caused a decrease in d from3.73 in [b.i] to 3.175 in [b.ii]. . . . . . . . . . . . . . . . . . . . 41

2.7 An example of an approximate soldier formation in Atlanticbluefin tuna, courtesy of Dr. M. Lutcavage, Large PelagicsResearch Center, UNH. . . . . . . . . . . . . . . . . . . . . . . 43

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List of Figures

2.8 A schematic diagram of individuals in soldier formation withfrontal nearest-neighbour detection. Grey arrows indicate thedirection of schooling force, while black arrows indicate direc-tion of motion. . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.9 Numerical simulations in 2D for [a] 8 particles with ~al =[0.1, 0.1], ~a = [0,−0.2] and [b] 6 particles with ~al = [0.1, 0.1],~a = [0, 0]. Trajectories are plotted through time. Note in [a]that the solution line connecting individuals is oriented in thedirection of ~al − ~a = [0.1, 0.3], while the school velocity atsteady state is ~v = [0.2, 0.2]. In [b], both the solution line andschool velocity are in the same direction, with ~al = [0.1, 0.1],~v = [0.2, 0.2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.10 Numerical simulations in 2D, for [a] 10 particles and [b] 25 par-ticles with ~al = ~a = [0.1, 0.1], and g+(x) = 0.5(exp(−x/50) −2 exp(−x/1)). Trajectories are plotted through time, and greyarrows indicate interactions between nearest neighbours. Notethat individuals have constant spacing between nearest neigh-bours, but are not restricted to a particular relative angle withnearest neighbours. . . . . . . . . . . . . . . . . . . . . . . . . 52

2.11 Numerical simulations in 2D with g+(x) = 1−x/2, which cor-responds to (rather non-physical) short-range attraction andlong-range repulsion. Trajectories are plotted in time for [a]4 particles evolved to t = 40, and [b] 6 particles evolved tot = 300. Individuals group in pairs (though due to overlap-ping, some pairs appear as a single particle), and these pairsfollow unpredictable trajectories. . . . . . . . . . . . . . . . . 52

2.12 Numerical simulation of two nearest-neighbour interactions.Trajectories are plotted through time. Note that the particles,initially perturbed, evolve to a regular tiling of the plane withindividual distance d = 0.707. . . . . . . . . . . . . . . . . . . 55

3.1 An example of milling in Atlantic bluefin tuna, courtesy of Dr.M. Lutcavage, Large Pelagics Research Center, UNH. . . . . . 65

3.2 A schematic diagram of the milling formation. Black arrowsindicate direction of motion (tangential to the circle), whilegrey arrows indicate direction of schooling force. . . . . . . . . 69

3.3 A schematic diagram showing the decomposition of the inter-action force into tangential and centripetal components. . . . 70

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List of Figures

3.4 A schematic diagram of particles in the mill formation show-ing angles introduced in the text, and relative position andvelocity vectors. . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.5 Equation (3.17) represented graphically for γ = 0.5, n = 5,and g(x) = A exp(−x/a) − B exp(−x/b). Horizontal axis:inter-individual distance d, vertical axis: distance-dependentforce magnitude. A = 1.5, a = 10, B = 3. In panel [a],b = 1.5, and milling is possible whereas in panel [b], b = 2 andno milling solution exists. . . . . . . . . . . . . . . . . . . . . 72

3.6 A plot of the left-hand and right-hand sides of (3.17) for arange of n values, from n = 4 to n = 12. Note that nointersection occurs for n = 4, but two occur for n = 5. . . . . 73

3.7 Numerical solution to (3.34) over a range of values of g′(d)for four particles (n = 4) and γ = 0.5. The real parts ofeigenvalues are plotted for i = 0, . . . , 3. The asterisks indicatebranches of Re(λ) with multiplicity equal to 2. Note that theregion over which all eigenvalues have non-positive real partsis approximately [-0.26,0.25]. . . . . . . . . . . . . . . . . . . . 79

3.8 As in Figure 3.7, but for five particles (n = 5). The realpart of eigenvalues are plotted for i = 0, . . . , 4. The regionover which all eigenvalues have non-positive real parts is nowapproximately [-0.11,0.19]. . . . . . . . . . . . . . . . . . . . 80

3.9 As in Figure 3.7, but for six particles (n = 6). The real partof eigenvalues are plotted for i = 0, . . . , 5. The region overwhich all eigenvalues have non-positive real parts is now ap-proximately [-0.08,0.17]. . . . . . . . . . . . . . . . . . . . . . 81

3.10 A plot of the condition for mill solutions to exist [a1] or not[b1] given by Equation (3.17), and corresponding simulatedparticle tracks [a2] and [b2], with γ = 0.5. Note that in thecase of an intersection, a stable mill forms at d ≈ 2 (whereasd ≈ 0.5 is unstable). However, when no intersection exists, themill stops rotating and stationary particles are then arrangedwith d such that g(d) = 0 (final direction is outward dueto small repulsive forces felt just before the particles stop).Arrows indicate direction facing at the end of the simulation.The interaction function used is as in Figure 3.5 with A = 0.5,a = 5, B = 1, and [a] b = 0.5, [b], b = 0.8. . . . . . . . . . . . 83

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List of Figures

3.11 [a1] An interaction function with g′(d) lying outside the sta-bility region for both intersections (implying no stable millformation). Simulations using this function are shown in [a2].Note that particles initially form a mill-like solution which isdestroyed as time evolves. [b1] An interaction function withonly one intersection, whose slope is too large (g′(x) > s forall x). Simulations in [b2] show that the radius of the millincreases in time. . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.12 The irregular periodic mill formation. [a] Individual distanced versus time for one particle in a system of 5 particles. Forthe 4 times denoted in [a], snapshots of particles are shownin [b], showing the variation in d between particles 1 and 2.The interaction function used was g(x) = 1 − 0.11x, so thatg′(d) = −0.11 lies on the stability boundary. . . . . . . . . . . 86

3.13 A plot of trajectories in time for six particles with~a = (0.1, 0.1)T .Note that the entire mill moves in the direction of ~a. . . . . . 88

3.14 A plot of trajectories in time for five particles with [a] ~a =(0.223, 0.446)T , and [b] ~a = (0.224, 0.448)T . Note that for asmall parameter change, the system behaviour is fundamen-tally different. In these simulations, g(x) is as in Figure 3.5,with A = 1.5, a = 10, B = 3, and b = 1.6. The correspondingexistence condition is shown in the inset to [b]. . . . . . . . . . 89

4.1 A schematic diagram of the experimental setup. θ is the cam-era axis angle, while φ is the angle-of-view of the camera lens.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.2 A schematic diagram of the setup with angles used in verticaltransformation. φ represents the angle corresponding to realdistance y (in pixels). L is the real vertical extent of the image(in pixels). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.3 The horizontal calibration matrix, associating each (x, y) im-age coordinate with the horizontal real pixel value. . . . . . . 104

4.4 The calibration grid used to test the image transformation.The pixel location of the upper-right corner of each grid vertexwas marked and transformed in MATLAB (see Fig. 4.5). . . . 105

4.5 42 Grid points marked in Fig. 4.4 plotted as ‘x’ marks, withreconstructed positions as ‘o’ marks. . . . . . . . . . . . . . . 106

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List of Figures

4.6 An example of one image from a time series of images collectedin the field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.7 Processing an image to obtain positions. . . . . . . . . . . . . 1094.8 An example frame with reconstructed velocities (grey), show-

ing partially incorrect and missing velocities. In reality, indi-viduals are highly polarized in this frame. . . . . . . . . . . . 111

4.9 The example frame of Fig. 4.8 showing velocities (grey) ob-tained by re-tracking the event. . . . . . . . . . . . . . . . . . 111

4.10 Example trajectories for 4 groups. Starting positions are plot-ted in green, final positions in red. . . . . . . . . . . . . . . . . 112

4.11 In the first frame, the individual at the origin has nearestneighbours filling each quadrant, and so is not an edge indi-vidual. In the second frame, the individual has no nearestneighbours (of the first 8) in the third quadrant, and thus isconsidered an edge individual. . . . . . . . . . . . . . . . . . . 113

4.12 A plot of a group with edge individuals (as determined by thealgorithm outlined in the text) in red, with all other individ-uals in black. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.13 Current vector x-value cx vs. waste tracer drift speed, fromraw images, with linear least-squares fit. . . . . . . . . . . . . 116

4.14 Nearest-neighbour distributions for the first 8 nearest neigh-bours. Successive means are also plotted () above distributions.119

4.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.16 Nearest neighbour distributions of the first neighbour, with

fitted probability density function q(d) (dashed) overlaid. . . . 120

5.1 A typical analyzed frame: input image (a) is processed, trans-formed to real space, and individual positions are reconstructed(b). Positions in successive frames are linked using particle-tracking software, giving velocities (b). . . . . . . . . . . . . . 127

5.2 Trajectories for 4 sequences, showing surf scoters approachingthe dock (y = 0) in a highly polarized fashion. Starting po-sitions are plotted in green, final positions in red. Units aregiven in pixels, where 1 BL = 46 pixels. . . . . . . . . . . . . . 128

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List of Figures

5.3 Neighbour distributions from empirical data, normalized tohave maximal value 1. Neighbor positions are calculated rela-tive to the heading of a focal individual (in the direction [0 1]in the plot, indicated by the white graphic). A frontal bias isseen in neighbor positioning. . . . . . . . . . . . . . . . . . . . 130

5.4 Average density of neighbours in a circular shell at the pre-ferred distance, as a function of angle. A distinct region ofhigher density is centered at 90. . . . . . . . . . . . . . . . . 131

5.5 Spatial distribution of heading deviation, from empirical data.Radial bands are plotted at 1 BL, 2 BL, and 3 BL for reference.The region where deviation is 0 corresponds to the repulsionzone where no individuals are found. . . . . . . . . . . . . . . 132

5.6 A schematic diagram partitioning local space around a focalindividual into sectors of location preference, high tendency toalign, and high tendency to deviate rapidly (move away from)the focal individual. Schematic was assembled according totrends shown in the data in Figs. 5.3 and 5.5. . . . . . . . . . 133

5.7 Attraction-repulsion function g(x) is negative in repulsion re-gion R, zero in alignment region AL, and positive in attractionregion ATT. Magnitudes of attraction and repulsion are con-trolled by weighting parameters ωatt and ωrep. . . . . . . . . . 135

5.8 Model I: (a) A schematic diagram representing the interactionzones, here being simply repulsion. (b) Spatial distributionsof neighbors, relative to a focal individual oriented in the di-rection [0 1] (as indicated by the white graphic at the origin),where density has been normalized to have maximum valueequal to 1. White dashed lines are superimposed at radial dis-tances 1, 2 and 3 BL. (c) Angular density distributions at thepreferred distance are plotted, following the convention of Fig.5.4. I compare distributions for data (green) and the model(blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.9 As in Fig. 5.8, but for Model II. . . . . . . . . . . . . . . . . . 1385.10 As in Fig. 5.8, but for Model III. . . . . . . . . . . . . . . . . 1395.11 As in Fig. 5.8, but for Model IV. . . . . . . . . . . . . . . . . 1405.12 Attraction-repulsion function gf(x) is negative in repulsion re-

gion R, and positive and constant beyond R (up to a limit ofratt). The magnitude of gf(x) is controlled by the weightingparameter ωfront. . . . . . . . . . . . . . . . . . . . . . . . . . 141

xiv

List of Figures

5.13 Average radial neighbor density, from data. . . . . . . . . . . . 1455.14 Effect of parameter variation on radial neighbor distribution

in basic model simulations. Unless varied as indicated, param-eters are ωrep = 5, ωatt = 1, ωal = 1, ωξ = 0.15. 50 Simulationsof 100 individuals were performed to t = 100, and average den-sities over all simulations (after quasi-equilibrium was reached,t = 50 to t = 100) were calculated. . . . . . . . . . . . . . . . 146

5.15 A comparison of spatial neighbor density for [a] the best-fitsimulations of the most appropriate model, Model V , and[b] data (repeating Fig. 5.3). Essential features of the dataare observed in simulated data, including the spatial extent ofneighbors, and the frontal bias. . . . . . . . . . . . . . . . . . 148

5.16 A comparison of average angular neighbor density at the pre-ferred distance for data (green), and simulation of Model Vwith optimal parameters (blue). . . . . . . . . . . . . . . . . . 149

5.17 Error as each parameter is varied about the optimal value.Each parameter is scaled in the plot so that optimal val-ues coincide (blue dotted line). Parameter values exploredwere as follows: ωrep = 5, 7.5, 10, 12.5, 15, ωatt = 0.5, 1, 1.5, 2,ωal = 0.25, 0.5, 0.75, 1, ωfront = 0.05, 0.1, 0.15, 0.2, and ωξ =0.275, 0.3, 0.325, 0.35, 0.375. In each case, the optimal value isa minimum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.18 An example of a follow-the-leader formation observed in surfscoters approaching a dock to forage on mussels. Here, inter-actions to the front are dominant. . . . . . . . . . . . . . . . . 152

B.1 A schematic diagram of particles at the threshold of mill break-ing. The angle between particles is π/2, beyond which particlei no longer senses particle i + 1. ~vm

i indicates the velocity ofparticle i in the absence of autonomous self-propulsion. . . . 177

xv

Acknowledgements

First and foremost, I wish to express my sincere gratitude for the efforts ofmy co-supervisors, Leah Edelstein-Keshet, and Yue-Xian Li. Their guidanceand input were essential for the completion of this thesis. Perhaps more im-portantly, they were instrumental in creating opportunities for me to developmy academic career, and they supported this development with much timeand encouragement, and meticulous care.

I would like to thank the Math Biology faculty, who have given me valu-able feedback throughout my years at UBC, and who stand as a perfectexample of a friendly and supportive academic community.

I would like to thank the IAM faculty, for their excellent teaching, and forcreating a welcoming, exciting academic environment. I also wish to thankthe Mathematics Department, and in particular the staff, for all the helpalong the way.

To my fellow IAM graduate students, thanks for the friendly environmentyou helped create, and for all the entertainment that served as a counterbal-ance to work.

Finally, to my family, and especially my wife, thank you for your encour-agement, support, and love at every step of the way. To Nova Scotia wego!

xvi

Dedication

For my beautiful wife, Sionnach.

xvii

Statement of Co-Authorship

Aspects of this thesis resulted from close collaboration with my supervisors,Yue-Xian Li and Leah Edelstein-Keshet. To facilitate details of this collab-oration, I divide the work into two subdivisions: first, the analytical workresulting in Chapters 2 and 3, and the field data collection and analysisresulting in Chapters 4 and 5.

In the first subdivision, the research program was identified and designed(in preliminary form) by my advisors. I carried out the analysis (shared withadvisors in Chapter 2, independently in Chapter 3), and coded, performed,and interpreted numerical simulations. I shared manuscript preparation forthe publication associated with Chapter 2, and was the primary author forthe publication associated with Chapter 3.

In the second subdivision, I was responsible for the identification anddesign of the research (in consultation with my advisors). I carried out theresearch, including experimental design, data gathering, data processing andanalysis, and model implementation and analysis. I prepared the manuscripts(to be published) associated with Chapters 4 and 5.

xviii

Chapter 1

Introduction

1.1 Biological Background

Across many species, animals often aggregate into cohesive, ordered groups.Particularly fascinating are the aggregations in which coordination and pat-tern are evident at the group level, seemingly at odds with the many indi-viduals comprising the group. Familiar examples range from schools of fish,herds of ungulates, flocks of birds, and swarms of insects; clearly, biologi-cal aggregates form at many different scales. The unifying property of suchaggregations is the emergence of order at the group level through interac-tions among individuals within the group. Through this mechanism, com-plex group structure can arise from the many relatively simple interactionsthat simultaneously take place among group members. Understanding themechanisms of interaction between individuals that permit cohesive grouppatterns forms the central focus of this thesis.

Aggregative behaviour in animals is generally viewed as a behaviour thatoptimizes individual fitness in the face of selective pressures acting on the in-dividuals. The particular motivations and pressures vary according to speciesand situation, but most commonly, membership within a group confers ben-efits through reduction of predatory risk or facilitation in foraging. However,individuals within groups also experience costs associated with the aggre-gate, primarily through increased competition for resources. The size of thegroup is determined through the competing influences of selective cost andadvantage conferred by the aggregate. Considerable work has been done toaddress these questions of why animals aggregate but this is not the focusof this thesis. Here instead, I address the ‘how’ of aggregations, to focuson what particular mechanisms of inter-individual interaction lead to whatgroup-level properties. In this way, we seek to understand the emergent or-der that is observed in large groups, in terms of individual properties andbehaviours.

1

1.2. Mathematical Modeling

1.2 Mathematical Modeling

Mathematical modeling is the primary tool for exploring the connection be-tween individual properties and group properties. Using models, researchersare able to test sets of interaction mechanisms and visualize the resultinggroup behaviour, generally not predictable directly from the rules alone. Al-though particular implementations may differ dramatically, interactions aregenerally modeled as a combination of any or all of the following three in-fluences: repulsion away from individuals very close, attraction to individ-uals far away, and alignment with nearby neighbours. Lagrangian models[3, 13, 24, 25, 30, 36, 47, 56, 60], based on tracking the positions and veloci-ties of individuals contrast with Eulerian models [1, 18, 19, 20, 22, 29, 31, 34,39, 53, 54] which describes the local flux of individuals via population den-sity [54]. Lagrangian models are thus comprised of high-dimensional coupledordinary differential equations (ODEs), whereas Eulerian models are formu-lated in terms of advection-diffusion partial differential equations (PDEs).Eulerian models are most suited to very large populations (insects, bacte-ria, etc.), while Lagrangian models are best suited for smaller groups withdistinguishable individuals.

Eulerian models are derived from stochastic versions of Lagrangian mod-els (under suitable approximations). Eulerian models are more easily an-alyzed in many cases using tools of partial differential equations, whereasLagrangian models provide more information about individuals, at a cost ofanalytical difficulty due to their high dimension [21].

1.3 Survey of the Literature : Models

1.3.1 Eulerian Models

Interactions in Eulerian models can be either local (e.g., [17, 31]) or non-local (e.g., [18, 19, 34, 53, 54]). Local interactions represent responses toonly immediate neighbours; in the continuum, interactions at some spatiallocation depend only on the density at that single point in space [17]. Inconstrast, non-local interactions represent responses to neighbours withinsome region (of non-zero radius), which leads to integro-differential equations.

2

1.3. Survey of the Literature : Models

A typical 1-D non-local Eulerian model is given by

∂ρ

∂t=

∂

∂x

(

D∂ρ

∂x− v(ρ)ρ

)

,

where x is the spatial coordinate, ρ is density of organisms, D the diffusioncoefficient, and v(ρ) the velocity [54]. To incorporate non-local effects, v istypically modeled as a spatial convolution. The details of interaction choicesfor a given model are given by the formulation of this convolution, in par-ticular the type of weighting function used. Further modifications include adensity-dependent diffusion coefficient D(ρ) [1] instead of constant D [29].Eulerian models have been used to analyze a variety of solutions, includingtraveling band solutions [17], vortex-type solutions [53] stationary clump so-lutions [54], and numerically generate an array of more complex solutionssuch as zig-zag movement and ‘breathers’ [18]. However, the abstractionof individuals to a local density obscures important information regardingindividual properties within groups, and analysis is almost entirely limitedto one-dimensional cases. In this thesis, we focus entirely on the Lagrangianmodeling perspective, which maintains the integrity of individuals within thegroup.

1.3.2 Lagrangian Models

Lagrangian models have the benefit of being very intuitive: individual in-teractions of attraction, repulsion, and alignment are encoded explicitly asforces acting between individuals, together with some autonomous forces (i.e,drag, self-propulsion). Many Lagrangian models were developed to simulatebehaviour of particular animal groups. Sakai [47] and Suzuki and Sakai [49]constructed a simulation model based on equations of motion, with forces ofthrust, drag, attraction/repulsion, and alignment. By varying the relativestrength of noise to interaction forces, these studies showed the existenceof different solution types : amoeba-like groups, doughnut (milling) shapedgroups, and rectilinear motion (polarized groups). Although examination ofthe model was limited, this work established a model framework for collectivemotion in animal groups. For example, self-propulsion and drag forces usedin the model of Chapter 2 and 3 are based on thrust and drag from [47] and[49].

Aoki [2, 3, 4] simulated the movement of a fish school by modeling speedand direction of individuals as stochastic variables, with interactions with one

3


neighbour, chosen probabilistically (as a function of distance). Aoki’s modelwas zonal, in that it partitioned local space around an individual into radialregions of repulsion, alignment, attraction, and searching response. In [2],individual properties were shown to mediate positioning within the group.Numerous zonal models based on the same principle of interactions restrictedto distinct spatial regions followed, including the model I introduce in Chap-ter 5 for flocks of ducks swimming collectively. Huth and Wissel [25, 26]compared the decision-based algorithm of [3] versus an averaging of inter-actions with some number of neighbours, showing the averaging mechanismto lead to more robust schools, and realistic polarity, whereas decision-basedschools were less organized. Huth and Wissel’s model was zonal (following[3]), and included a ‘blind angle’ behind an individual where no interactiontakes place.

From different motivations (computer animation), Reynolds [46] presenteda model of flocking for ‘boids’, (artificial birds) based on attraction, repul-sion and alignment with neighbours. Though more complicated forces wereincluded (e.g., banking) for realism, the model showed how realistic flockingbehaviour can result from purely local interactions, without external controlor ‘leaders’.

Niwa [36, 37, 38] developed a stochastic differential equation model forfish schooling, composed of a locomotory force (derived from swimming en-ergetics), a ‘grouping force’ of attraction/repulsion, and alignment. Groupproperties were investigated as the level of noise relative to interactions wasvaried. Niwa showed that the group exhibits transitions between behavioursas noise levels change, from amoeba-like, disorganized groups, to polarizedgroups moving rectilinearly (similar to behaviors described in [3]). Niwa alsoshowed that by tuning randomness to a particular level, noise can facilitateschooling by reducing the time of onset of schooling.

Warburton and Lazarus [60] investigated the relationship between theshape of interaction force (attraction/repulsion) on group properties, show-ing that although stable groups form across most functions with short-rangerepulsion and long-range attraction, the shape of these response curves affectsnearest-neighbour distance, group shape, and level of cohesion. Althoughlimited to generic interaction curves and conclusions via simulation, [60] es-tablished the importance of interaction function shape on group properties,a link I study analytically in Chapters 2 and 3.

Mogilner et. al [35] considered specific types of interaction functions (i.e.,exponential, inverse power) and analytically derived conditions on function

4


parameters to guarantee stable, cohesive well-spaced groups. In contrastto this thesis, in [35] individuals are assumed to be coupled all-to-all (i.e.,global sensing of neighbors) and particular functional forms were used toobtain analytical conclusions.

Gueron et. al. [24] developed a zonal model for animal groups based ondistance-dependent interactions. Between zones of repulsion and attraction,the model included a neutral zone where neighbours imposed no forcing. Byinvestigating the level of cohesion permitted by varying widths of the neutralzone, the authors argued that neutral zones of certain width permit cohesion,but reduce the amount of acceleration and deceleration based on neighbourposition, which would confer an energetic benefit. Vabo and Nottestad [55]presented a cellular automata model to investigate group dynamics in thepresence of a predator, and was able to generate behaviours similar to thoseseen in nature, including ‘vacuole’, ‘fusion’, and ‘fountain’ formations.

Viscido et. al. [58] investigated the dependence of group dynamics onnumber of interacting neighbours. They found that when the number of in-fluencing neighbours is similar to total population size, static, milling-typesolutions occur, whereas when population size is much larger than numberof influencing neighbours, mobile, polarized groups form. In [59], a varietyof metrics were proposed to investigate the parameter-dependence of groupproperties, which suggested that a neutral zone, relative magnitude of align-ment, and number of influential neighbours are crucial to determining thegroup structure. In Chapter 5, I investigate relative magnitudes of alignment,attraction, and repulsion that best reproduce features of observed groups.

Couzin et. al. [13], using a fixed-speed zonal model (repulsion, align-ment, attraction), investigated group behaviour as zone widths are changed.By changing width of the alignment zone, group structure changed fromdisorganized, to milling, to polarized. Interestingly, the authors showed thatthese transitions exhibited hysteresis: the previous history of group structureinfluences collective behaviour as individual interactions change. Stated an-other way, the current geometry of the group, in conjunction with individualproperties, determines group response to parameter changes. In Chapters 2and 3 of this thesis, I show co-existence of two different solutions over certainparameter regimes; whether a group is found in one (polarized) or the other(milling) depends on the geometry of the group. Couzin et. al [12] used asimilar model as in [13], but included some proportion of ‘informed individ-uals’, who had some desired direction of travel together with an alignmentforce. They showed that as group size increases, the proportion of individuals

5


needed to guide the entire group decreases.Lagrangian Models in Physics. Though the Lagrangian models aboveare typically biologically inspired, there have also been significant contribu-tions from the physics literature. Vicsek et. al. [15, 56] used a fixed-speedmodel of particles interacting via heading matching of neighbours (withoutattraction/repulsion interactions) showed transitions in group behaviour. Inparticular, solutions where no net transport occurs (quasi-stationary) switchto finite-transport solutions through symmetry-breaking of the rotationalsymmetry. These changes in group structure were invoked by tuning particledensity and level of noise in the system. Levine and Rappel [30] showedvortex-type solutions (mills) that self-organized without any external in-fluence or chemotaxis, and then derived an advection-diffusion continuumanalogue to the model via coarse-grain averaging. D’Orsogna et. al. [16]categorized different solution types via approximate analogy with canonicaldissipative systems. This permitted a statistical mechanical analysis of themodel, and parameters of the Morse-type interaction function were used toderive conditions for H-stable (well-spaced) versus catastrophic increasingdensity with increasing population) solutions. Chuang et. al. [11] derived a2D continuum analogue to [16], and performed a linear stability analysis onstationary solutions.Lagrangian Models in Engineering. From the engineering perspective,work has been done in developing decentralized control rules for systems ofagents subject to various movement constraints, with application to, e.g.,cooperative robotics [9], and unmanned aerial vehicles [28]. Tanner et. al[50, 51] developed control laws on interacting agents that give rise to tightformations and collision avoidance. Sepulchre et. al. [48] proposed a designmethodology to stabilize parallel and circular motion of particles moving atunit speed in the case of each agent interacting with all other agents. Marshallet. al. [33] studied cyclic pursuit from a control perspective, analyzing theparticular solution of fixed-speed agents following one another in a circularmanner. Existence and (linear) stability conditions were derived based oncontrol inputs. The solution analyzed in [33] is similar to the milling solutionI study in Chapter 3, although individuals are subject to different constraintssuch as fixed velocity, which affects the anaylsis significantly.

6

1.4. The Need for Empirical Data

1.3.3 Unifying Lagrangian Approaches

Although all the models described above share the common property thatindividuals interact with a combination of repulsion, attraction, and align-ment forcing, clearly there is a wide array of specific implementations. Thevariety of models stems not only from species-specific choices, but also inad-hoc choices made by modelers where no data exists to guide such choices.In an attempt to provide a unifying framework to such modeling pursuits,Parrish et. al. [40] have organized a number of previous Lagrangian mod-els according to a number of model features. Below, model properties arecategorized in a similar way to [40], with some modification.

• Population size: the number of individuals simulated by the model.

• Speed: models vary according to whether speed is constant (e.g. [13,25, 26, 55]), varying according to equations of motion (e.g., [16, 30, 36,37, 38]), or drawn from a probability distribution (e.g., [3]).

• Interaction (spatial dependence): Interactions can be zonal, (e.g., [3,12, 13, 24, 25, 26]) or based solely on linear distance (e.g., [16, 30].Zonal models provide a framework for imparting a hierarchy of decisionsin the interaction, (e.g., aligning and attracting only if no individualsare within the repulsion zone).

• Neighbor scaling: given a set of influencing neighbours, relative weight-ing of influence imparted by each neighbour can be constant or distance-weighted.

• Rule size: models vary according to how many neighbours are ‘sensed’by a given individual, ranging from 1 (nearest-neighbour interaction)to every individual (all-to-all coupling).

1.4 The Need for Empirical Data

Clearly, given the variety of choices above, there are many possible models.In order to choose between implementations, (in)validate models, and chooserealistic parameters, empirical data of group motion is required. However,gathering data on moving groups of animals is a typically difficult under-taking, whereas a mathematical model with arbitrary choices of interaction

7

1.5. Survey of the Literature: Empirical Studies

can be implemented with ease. Obtaining empirical data in the field is com-plicated by the typically three-dimensional nature of animal aggregations(resulting in occlusion of interior individuals), the difficulties of calibratingmeasurement equipment at varying locations, high speeds of movement, andthe transient behaviour (both in space and time) of animals in the fieldin general. Overcoming these difficulties by moving to a controlled labora-tory setting places restrictions on the spatial extent and size of groups, and(depending on the species and experimental setup) can create difficulty inobtaining natural behaviours within an artificial environment. Furthermore,for many species (e.g., flocking birds), a laboratory setting is simply notfeasible.

1.5 Survey of the Literature: Empirical

Studies

1.5.1 Small Groups, No Tracking

Despite the difficulties mentioned above, significant efforts have been madeto record positions and movements of individuals within groups. Early workfocused on recovering individual positions in relatively small groups, butdid not link individuals across frames to construct trajectories (and thushad no dynamic component). Cullen et. al. [14] used stereo photographyto reconstruct 3D positions of 10 fish over 11 photographs, showing thatpilchards tend to prefer neighbours at diagonal positions, both in bearing andelevation. Major and Dill [32], also using stereo photography, reconstructed3D positions of flocks of dunlin and starlings, in flocks of 25-76 individuals.They found that dunlin have a tighter, more cohesive structure than starlings,and that dunlin have a propensity for nearest neighbours behind and belowa focal individual, while few nearest neighbours occupy relative positions infront and below. Budgey [8], motivated by birdstroke tolerance in aircraft,reconstructed positions of starlings, rock doves, lapwings, mixed gulls, andCanada geese within flocks of 21-61 birds using the stereo method in thefield. Nearest-neighbour distances were calculated and a linear relationshipbetween nearest-neighbour distance and wingspan was postulated.

Aoki and Inagaki [5] gathered three-dimensional positions of anchovygroups at night via a stereo pair of cameras dragged by a drifting vessel, andshowed that anchovies have a tendency at night to swim at similar depths as

8


neighbours, and also to have neighbours in front of or behind. Furthermore,the nearest-neighbour distance at night was increased as compared to day-time measurements. Ikawa et. al. [27] modified the orthogonal method byadding a third camera to reconstruct positions of 5-10 mosquitoes in a swarm,and calculated nearest-neighbour distance and some speeds for individualsin the swarm.

1.5.2 Small Groups with Tracking

By linking positions of an individual through successive frames in time, atrajectory can be reconstructed, giving dynamic information on interactions.Partridge [41] studied fish (20-30 saithe) schooling in an annular tank, cap-tured by a camera mounted on a rotating gantry, with positions calculatedby the shadow method, where overhead positions and shadows cast by a lightsource are matched to calculate 3-D position. Individual fish were uniquely‘freeze-tagged’ and so individual trajectories could be tracked in successiveframes. The data was used to compare two volume-estimation methods (thetotal volume method versus the individual ‘free-space envelope’ method).Using the same experimental setup as [41],

Partridge et. al. [44] compared an obligate schooler (herring), a stronglyfacultative schooler (saithe), and a weakly facultative schooler (cod), showingdifferences in spacing, angular preference of neighbours, and school shape.Also, saithe were shown to decrease NND as either school speed or schoolsize increased. Partridge [43] further analyzed saithe schools via correlationsbetween heading and speed of individuals and their nearest neighbours as afunction of lag, to investigate response times of a focal fish to the behaviorof its neighbour. This analysis showed significant speed correlation betweena focal fish and its first nearest neighbour, maximal at small response times(less than 0.3 sec), but low mean heading correlation due to high variabilityamong individuals. Partridge [42] studied small groups (2-6) of minnows ina tank using the shadow method to reconstruct positions, and calculated ve-locities by tracking individuals through successive frames. Partridge foundthat two-fish schools behave differently compared to 3-6 fish schools, in termsof neighbour correlations, spacing, and relative positions of neighbours, sug-gesting that results from two individuals cannot be generalized to schools.A natural extension of this observation is to question whether or not smallgroups (10-20 individuals) accurately represent behavior found in large ag-gregations, the answer to which lies in data sets of large groups.

9


Pomeroy and Heppner [45] used orthogonal photography (where imagesare captured simultaneously by orthogonal cameras) to capture the move-ments of flocks of 12-16 rock doves during a turn, finding that rock dovesreposition themselves relative to the flock throughout a turn such that thearc ascribed by each individual varies less than if the group turned as a rigidbody.

More recently, Grunbaum et. al. [23] used pairs of video cameras record-ing movements of giant danios in a tank, and then tracked individuals usingpurpose-built tracking software, to generate long-time three-dimensional tra-jectories of groups of 4-8 fish. By generating probability density functions(PDF) of nearest-neighbour distance, and calculating empirical orthogonalfunctions of the PDF, advective fluxes were calculated for an advection-diffusion equation for nearest-neighbour position. Using the same experi-mental setup, Viscido et. al. [57] compared empirical observations of schoolswith simulation data, concluding that in order to reproduce realistic featuresof schools, the magnitude of alignment forcing in models should be almosttwo orders of magnitude smaller than attraction/repulsion forcing.

Tien et. al. [52] tracked the movements of 40 fish (creek chubs and blac-knose dace) enclosed in a shallow natural pool in a creek via an overheadcamera. The shallow water essentially restricted movement to two dimen-sions. By analyzing movements of fish relative to nearest neighbours, it wasshown (to some degree of correlation) that these fish respond to neighboursin a series of radial zones: a nearby region of mutual repulsion, then a regionof neutrality, followed by a region of attraction, where the attraction responseis made only by the focal fish (i.e, not mutual attraction). In the presenceof a simulated predator, the radial zones decreased in size.

1.5.3 Large Groups, No Tracking

A significant breakthrough in empirical data collection was made by Balleriniet. al. [6] who captured three-dimensional positions of starling flocks of up to3000 individuals at 10 frames per second, using a trifocal stereo photographymethod. Their (static) analysis of these flocks found a strong anisotropy inthe relative position of nearest neighbours. By quantifying this anisotropy,the number of neighbours with which an individual interacts (on average)was determined. Importantly, it was found that the number of interactingneighbours was not dependent on density, suggesting that individual interac-tion is ruled by topological (i.e., a specific number of neighbours) interaction,

10

1.6. Contributions of this Thesis

as opposed to metric (i.e., all neighbours within a fixed sensing region) in-teraction. In a related work based on the same data set, Cavagna et. al. [10]introduce statistical measures of density as a function of sensing radius of afocal individual. It was shown here that starling flocks have higher densitiesat the edge, and individuals have weak ‘shells’ of neighbours - in betweenrandom and crystalline. As I will show in Chapter 5, similar ‘shells’ of neigh-bors are observed in scoter flocks I study. In another related work, Balleriniet. al. [7] reported group statistics (morphology, orientation, turning dynam-ics, density, nearest-neighbour distance) of the starling flocks. Although thisdata set represented a significant step forward, incomplete reconstructionsof positions (only 80% of individuals were reconstructed in any given frame)prevented any tracking of individuals in time.

1.5.4 What Empirical Data is Needed

Although there have been numerous empirical studies involving a few in-dividuals tracked in time (i.e., having reconstructed trajectories), or manyindividuals without tracking (i.e., data limited to static images), there is avoid of empirical data for tracking many individuals simultaneously. Themain obstacle in obtaining such data is the need for reconstructing all indi-viduals in each time frame in order to track individuals effectively. The 80%position reconstruction rate of [6] precludes trajectory reconstruction acrossusable time scales, as the probability of an n-frame trajectory is 0.8n, whichimplies a trajectory reconstruction success rate of 10% over 10 frames (cor-responding to one second, in [6]). Yet, tracking of only few individuals raisesthe issue of whether or not the number of individuals is sufficient to triggerthe schooling behaviour in groups. Furthermore, in small groups, edge ef-fects (where individuals at the exterior of a group have different statisticalproperties as compared to interior individuals) dominate analysis, and anyspatial description of interactions is limited by the small spatial extent ofsmall groups (especially so in three dimensions).

1.6 Contributions of this Thesis

In this thesis, collective motion in animal groups is investigated in twoways: by developing and applying analytical tools to explore a Lagrangianmodel, where previously, exploration was limited to numerical simulation;

11


and through gathering and analyzing empirical data against which modelscan be tested.

1.6.1 Model Analysis

In the first part of this thesis, we study a general Lagrangian model of n(i = 1, . . . , n) self-propelled particles, of the form

~xi = ~vi, (1.1)

~vi = ~fa + ~fi, (1.2)

where ~x ≡ d~x/dt denotes a derivative with respect to time, ~fa is a force thatis generated autonomously by each individual without the influence of otherindividuals and ~fi is a force that is generated due to the interaction with oth-ers. Because ~fi represents interindividual interactions, Eqs. (1.1)-(1.2) arecoupled, often with relatively complex connection structures. Furthermore,terms in Eq. (2.2) are typically nonlinear. Thus, model equations cannot besolved explicitly, and steady-state existence and stability analyses are typi-cally not possible. For these reasons, Lagrangian models have been studiedprimarily by simulation; i.e., quantifying changes in model output resultingfrom parameter variation, and categorizing different types of behavior exhib-ited by the model. Although such efforts are useful, they lack the clarityafforded by analytical observations, where the relationship between modelparameters and group-level properties is clear-cut.

In this thesis, we consider a special class of solutions, termed perfectschools : geometrically simple organizations of individuals with fixed spac-ing. For these solutions, the interindividual coupling and equilibrium statescan be written explicitly, leading to existence conditions. Then, the simpli-fied coupling in perfect school solutions is exploited to obtain a matrix systemof Eqs. (2.1)-(2.2) where the coefficient matrix is structured. This structurepermits a linear stability analysis on the system for particular perfect schoolsolutions. These analytical results give clear expressions for how group struc-ture and stability depend on model inputs (i.e., individual properties). Thiscomplete characterization of model outputs from inputs is missing in simu-lation studies, which can only draw inferences from limited parameter spaceexploration. In particular, I derive existence conditions dependent on self-propulsion, drag, and interaction function strength. I then derive conditionson stability of perfect school solutions in terms of the interaction function,and its derivative.

12


The complexity of Lagrangian models lies in the many interactions amongindividual units. Though interactions may be relatively simple, a complexweb of interactions creates feedbacks in the system that make emergent groupproperties difficult to predict. Herein lies the strength of an analytical treat-ment: analysis, as in this thesis, eliminates the unpredictability of groupstructure that emerges from model inputs because the analytical results linkemergent group properties to individual properties via straightforward math-ematical relationships.

Although the types of solutions considered in this work are overly sim-ple when compared to real-world animal groups, the analysis neverthelessis useful for understanding qualitative aspects of such animal groups. Forexample, general properties of attraction and repulsion between individualscan be deduced based on observed spacing, velocity, and group geometry. InChapter 2, one-dimensional solutions, and ‘soldier formations’ in two dimen-sions (wherein individuals are organized in a linear formation) are studied. InChapter 3, milling formations (equidistant individuals following one anotheraround a closed circular path) are studied. I show how different solutions canco-exist for sets of parameters, depending on the arrangement of individuals(i.e., the model state).

1.6.2 Empirical Data

In the second part of this thesis, to address the void of dynamic data forrelatively large groups of animals, empirical data is gathered on groups of afew hundred surf scoters (diving ducks) swimming collectively on the surfaceof the water in English Bay, near Vancouver, BC, Canada. This data issignificant, as it represents a 10-fold increase in group size wherein individualsare still reliably tracked. Thus, for the first time (to our knowledge), completetrajectories (and therefore individual velocities) are known for all members ofa relatively large group, allowing for interindividual responses to be measuredin time. Because models of collective motion are naturally dynamic (e.g.,time-dependent ODEs), trajectories provide a necessary and valuable dataset against which models can be formulated and validated.

Data processing tools are created that allow for complete reconstructionof positions of individual ducks in each frame, and trajectories of individualsare reconstructed using existing particle-tracking software. Methods are de-veloped to correct observations for camera angle and currents in the water,and to correct tracking errors caused by deficiencies in the existing software.

13


A complete description of empirical methods as well as basic group statistics(mean velocity, mean spacing, etc.) are reported in Chapter 4. In Chapter5, two-dimensional spatial distributions are used to characterize individualinteractions, and a model that captures some aspects of the empirical datais presented and studied via numerical simulation. As I show in this chapter,to account for observations, we infer that ducks have a hierarchy of rules,have an angular bias in their attraction to neighbors, and exhibit differentavoidance manoeuvres depending on relative angle between individuals. Aconcluding chapter follows in Chapter 6.

14

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20

Chapter 2

Minimal Mechanisms forSchool Formation inSelf-Propelled Particles 1

2.1 Introduction

A fundamental problem in biology is how behaviour and interactions at thelevel of the individual impact the emergent behaviour at the level of thegroup as a whole. In this paper, we examine the connection between individ-ual behaviour and group properties within the context of social organismsthat form a school or a flock. We ask how the parameters associated witheach individual, together with the effective forces that represent their mutualinteractions influence the existence of a school, its geometry, the speed anddirection of movement, as well as the stability of the school structure.

Our perspective in this paper is based on the Lagrangian viewpoint, thatis, we follow individual particles, rather than densities of organisms. Weformulate equations of motion based on the Newtonian approach, i.e. de-scribing changes in the velocities and positions of the particles under forcesof propulsion and interaction. Such an approach leads to a generalized systemof ordinary differential equations that is given below.

~xi = ~vi, (2.1)

~vi = ~fa + ~fi, (2.2)

where ~x ≡ d~x/dt denotes a derivative with respect to time. Here, we roughly

classified all forces into two general categories: ~fa that is generated au-tonomously by each individual without the influence of other individuals

1A version of this chapter has been published as ‘Y.-X. Li, R. Lukeman, and L.Edelstein-Keshet, 2008, Minimal mechanisms for school formation in self-propelled parti-

cles, Physica D, 237(5), p 699-720

21

2.1. Introduction

and ~fi that is generated due to the interaction with others. Depending onthe specific choice of these forces, the model can have very different proper-ties. For example, in [1, 5, 12, 13, 14], ~fa = (α− β|~vi|2)~vi was studied. Thismodel was originally obtained by minimizing the specific energy cost of aswimming fish [26]. In [8], ~fa = −β~vi which is simply a drag force that helpsto stabilize the motion of a particle. In both these models, the movement ofan individual is purposeless in the absence of the interaction with other par-ticles. It either stays motionless or keeps moving at a constant speed alongits initial direction of motion. In the present study however, there is somepreferred direction, and each individual knows where to go and whether it isat a leading position in the group.

The interaction force ~fi that is directly responsible for many importantcharacteristics of the emergent aggregate pattern plays a key role in defininga distinct model. Pairwise interaction is the most commonly studied force,although in [8], a localized mean field interaction was used to describe theinfluence of nearest neighbours on the movement of a particle. Interactionswith infinite range and all-to-all coupling have been considered in most stud-ies cited above as well as the study of non-Newtonian particles moving inviscous medium [11]. An important question considered in these studies ishow specific features of the aggregate change as the number of particles is in-creased. Whether the nearest neighbour distance (NND) collapses to zero inthe limit of an infinitely large group of particles has been shown to influencestrongly the possible school patterns that can occur in the system [1, 5, 11].In the present study, our focus is very different from these studies. We arenot concerned with the effects of an infinite size of the group but rather focuson groups with finite number of particles and with short range interactions.In most cases, we only consider nearest neighbour interactions. Our maingoals are to find analytic insights, based on simplifications and assumptionsthat render the problem amenable to analysis. These insights allow us topinpoint the minimal mechanism that determines each specific feature of aschool pattern. This approach is also different from a number of other studiesin which numerical insights were the main focus [2, 3, 4, 6, 7, 17, 20, 25].

In order to arrive at analytic results, we consider classes of special solu-tions that we refer to as perfect schools. These are configurations of particlesin which the spacing is constant and identical, and where individuals in thegroup move at a constant speed. We further subdivide perfect schools intothose in which individuals share a common heading, and those in which theydo not. In this paper, we are primarily concerned with the former. This is

22

2.1. Introduction

the Lagrangian analogue to motion as a traveling wave with uniform interiordensity.

It is well known that aggregates occur if particles mutually attract atlong distances and repel at short distances. When the interaction is all-to-all, the NND can approach zero as the size of the group increases dependingon whether the pairwise interaction satisfies certain conditions [1, 5, 11]. Alarge number of possible transient and static patterns can occur in particlesthat interact with each other in such a way, and it is beyond the scope of thispaper (and likely impossible) to predict all possible patterns. In this paper,we focus mainly on regular arrays, both in 1D and 2D, that are reached afterthe transient state is over. Our approach is to exploit the simplicity of theseregular schools in an attempt to address the following questions analytically:(i) Under what conditions on the school forces do such perfect schools occur(existence)? (ii) Given their existence, what further conditions guaranteethat these patterns are not destroyed by random perturbations (stability)?While real schools and flocks are much more plastic and irregular, focusingon these patterns leads to clear-cut results that can be derived analytically.

The relative simplicity of our model and the precise definition of thesespecial solutions allow us to reach several new conclusions that are difficultto obtain by simulations alone. We will show that by looking for perfectschool solutions, we arrive at conditions linking self-propulsion forces withinteraction forces evaluated at the NND. Stability depends on the slope ofthe interaction force as a function of NND. The structure of the interactionmatrix that determines how many neighbours interact with each individualis crucial for achieving explicit stability analysis. At the same time, in orderto extend our results to cases that are too challenging to analyze fully, or tomore complex cases, we complement our study with numerical simulations.

Our main goal in this paper is to understand which factors lead to whichof the properties of a group. As will be shown, the questions posed above canbe answered conclusively only when analytical solutions of the schools andexplicit expressions for determining their stability can be achieved. Many re-alistic school structures observed in nature are irregular under the influenceof noise from various sources. The range of structures that we explore, how-ever, belong to a class of simple and regular patterns that allow us to achieveanalytical results and predictions. These structures are often observed in na-ture (albeit more irregular), and based on our theory presented in this paper,are the most likely realized patterns given simple rules of nearest neighbourinteraction.

23

2.2. The Model of Interacting Self-Propelled Particles

The questions posed above can be answered conclusively in the case of thespecial solutions that form the class of interest. This would not be true if wewere to consider all possible (dynamic and steady state) group configurationsthat satisfy our model equations. For this reason, we do not here consider avariety of more complicated and more realistic group structures that occursin nature. Nevertheless, the range of structures we do explore are consistentwith groups of autonomous robots, or formations of vehicles on a highwayor in military environments, topics that have emerged as recent areas ofapplication of such theory. Understanding the possible behaviour of suchartificial autonomous self-propelled agents forms an additional motivationfor this paper.

In Section 2, we introduce the basic assumptions of the model we usein this study and the differential equation system that describes the motionof the particles. In Section 3, we study schools in one dimensional spaceanalytically and numerically. The simplicity of these school solutions allowsus to develop a basic theoretical framework. We also introduce the conceptof stressed and stress-free schools. In Section 4, a soldier formation solu-tion in two dimensions is studied analytically using a similar approach as inSection 2. Numerical simulations in 2D are then used to verify analyticalresults on soldier formations. In Section 5, we discuss some numerical resultsthat demonstrate the formation of other types of 2D arrays. Results aresummarized and discussed in Section 6.

2.2 The Model of Interacting Self-Propelled

Particles

2.2.1 Assumptions and Equations

The following basic assumptions apply to the schools that we study in thispaper.

(A1) All particles are identical and obey the same set of rules.

(A2) Each particle is capable of sensing if it is located in the interior or atthe edge (front/rear) of the school.

(A3) The force experienced by one particle from a given neighbour is com-posed of a distance-dependent term and a velocity-dependent term.

24


Only pairwise interactions are considered.

(A4) The forces exerted by different neighbours are additive.

(A5) The magnitude of the forces between a pair of particles depends on thedistance between the pair and their relative velocities.

(A6) The distance-dependent force between a pair is a vector parallel to thevector connecting the pair while the velocity-dependent force is parallelto the difference between the velocities of the pair.

Consider a group of n self-propelled particles, each identified by a sub-script i = 1, 2, . . . , n, with position and velocity denoted by ~xi and ~vi, re-spectively. We assume particles are polarized, that is, each has a front and aback. For simplicity, we assume that the body alignment of the ith particleis identical to its direction of motion, i.e. the direction of ~vi. The movementof the ith particle is governed by the classical Newton’s Law of Motion whichyields

~xi = ~vi, (2.3)

~vi = ~ai − γ~vi + ~fi, (2.4)

where the mass of each individual is scaled so that mi = 1 for all particles.~ai is an autonomous self-propulsion force generated by the ith particle whichmay depend on environmental influences and on the location of the particlein the school. This is the term that makes this model different from somepreviously studied models [1, 5, 11, 12]. The term γ~vi is a damping forcewith a constant drag coefficient γ (> 0) which assures that the velocity isbounded. Eq. (2.4) implies that if a particle stops propelling, its velocity

decays to zero at the rate γ. ~fi is the interaction or schooling force thatmoves the ith particle relative to its neighbour(s). If one particle describedby (2.3)-(2.4) is moving in D dimensional space (D = 1, 2, 3), the number offirst-order equations for each particle is 2D. For a group of n such particles,there are 2Dn first-order equations.

In the following analysis, ~ai is typically a constant vector which may bedifferent for a leader, follower or individual in the tail (i.e., the last particle)of the group. Based on the model assumptions (A3-A6), the schooling forceis given by

~fi ≡ ~fxi + ~f v

i =∑

j

g±(|~xji|)~xji

|~xji|+∑

j

h±(|~vji|)~vji

|~vji|, (2.5)

25


where~xji = ~xj − ~xi; ~vji = ~vj − ~vi.

~fxi and ~f v

i are the position and velocity dependent forces, respectively. Indexj sums over all the particles that influence the movement of particle i. g+

and h+ (g− and h−) are chosen when the influence comes from the front(behind), i.e. if ~xji · ~vi > 0 (< 0). g+ 6= g− and h+ 6= h− imply that theforces from particles in front and those from particles behind are different orasymmetric. g±(x) are defined for x ≥ 0, i.e., are functions of the absolutedistance between two particles. To produce a nonzero spacing distance, g±(x)should typically be positive for large x and negative for small x, indicative ofshort-range repulsion and long-range attraction. This feature is universal inalmost all models of aggregate formation. (See review of attraction-repulsionmodels in [11].) In most results that we studied in this paper, g±(x) possessessuch a feature, unless specified otherwise. Furthermore, the forces that eachparticle can generate should be bounded.

Based on these definitions, g+(·) > 0 (< 0) means that particle j whichis in front of particle i exerts an attractive (repulsive) force on the latter;while h+(·) > 0 (< 0) implies that the velocity-dependent force makes thevelocity of particle i converge to (diverge from) that of particle j (see Fig.2.1). Note that the position-dependent and velocity-dependent componentsof the schooling force can point to different directions and possess differentmagnitudes. All constants and functions that appear in (2.3)-(2.5) are modelinputs that are assumed to be given.

Figure 2.1: A schematic diagram of model vectors in 2 dimensions.

Different choices of the functions g± and h± can generate large numbersof school formations that are qualitatively different. Further classification ofthe subtypes of these models is possible based on specific selection of these

26


functions. We do not intend to take on such a task in this paper. However,examples that we present in the following sections clearly demonstrate thecrucial importance of the form of these functions in determining the schoolpatterns that emerge.

Using the classical Newton’s Law of Motion to describe a system of self-propelled particles is an approach adopted in numerous previous studies[8, 15, 16] although, as we shall discuss later, differences in the dynamicsthat govern the autonomous self-propulsion and in the range of pairwise in-teractions can lead to significant diffferences in the school patterns that canemerge.

2.2.2 Classification of Schools

Having given the definition of perfect schools in Section 1, we note that suchschools can be further divided into two types: those in which each individualhas an identically constant direction, denoted as Type I schools, and those inwhich the individuals change direction in time, denoted as Type II schools.Type I schools are the main focus of this paper. Type II schools are typicallymore difficult to analyze. A good example is a mill formation, in whichparticles rotate around an invariant circle (see, e.g., [10]). Here, headings arechanging continuously as particles rotate, and each individual has a differentheading at any given time. Other Type II schools include groups collectivelyperforming a turning manoeuvre. Note that in both Types I and II perfectschools, the group moves collectively as a rigid body under transformation(e.g., translation, rotation, etc).

For a perfect school, we can define the center of mass and the schoolvelocity by

~X(t) =1

n

n∑

i=1

~xi(t), ~V (t) =1

n

n∑

i=1

~vi. (2.6)

The combined autonomous self-propelling force and schooling force are:

~A =1

n

n∑

i=1

~ai, ~F =1

n

n∑

i=1

~fi. (2.7)

Using these definitions, we can write down the equations of motion of theschool as a whole.

~X = ~V , (2.8)

~V = ~A + ~F − γ~V . (2.9)

27


In the special case ~vi = ~V , where ~V is a constant vector and identical for alli (i.e., a Type I school), the equations of motion for each individual must beidentical to those of the whole school.

We distinguish between two types of individuals: those denoted as leaders,and their followers. In general, we define a leader particle as an individualthat does not see any other particles in its frontal visual field, where thesign of ~vi · ~xji distinguishes the frontal and rear visual fields. All othersare denoted as followers. We associate distinct self-propelling forces ~al and ~awith the leaders and the followers respectively (where superscripts are labels,not exponents). In one-dimensional schools, we consider the special case ofinteractions both to the front and rear of an individual. In this case, weadditionally distinguish between a tail particle (i.e., an individual that doesnot see any particles in its rear visual field) and other interior followers, andassociate a distinct self-propelling force ~at with the tail particle.

In our model of school formation, we consider as inputs the leader andfollower autonomous self-propulsions ~al and ~a, and the interaction functionsg± and h±. Model outputs are the individual distance d between particles,the school velocity ~v and (where applicable) the autonomous self-propulsionof the tail at.

The perfect schools that we study in this paper are idealizations of someobserved social aggregates in nature. Fish schools in the form of surfacesheets or linear soldier formations are often encountered. Single-file trailfollowing occurs in ants, caterpillars, as well as birds; linear and ∨-shapedflocks are observed in formations of many bird species.

The analytical methods we adopt here are very similar to those usedin the study of synchronized clustered states in neuronal networks [9]. Wesubstitute a specific school solution of interest into the system of nonlin-ear differential equations and determine the conditions for the existence ofsuch a solution. Then, we introduce a small perturbation in the equationsnear this school solution and linearize the system. We find the eigenvaluesof the linearized system, and we determine conditions that make their realparts negative. We then verify these existence and stability conditions usingnumerical simulations of the system, and explore patterns that arise fromarbitrary initial conditions.

28

2.3. Schools in One Dimensional Space

2.3 Schools in One Dimensional Space

Aggregates of animals moving on a single trail are examples of schools inone dimensional space (1D). Vehicles moving on a single-lane highway canform similar aggregates. Consider a group of n particles moving in 1D. Theequations of motion are now reduced to the scalar equations:

xi = vi, (2.10)

vi = ai − γvi + fi, (2.11)

where i = 1, 2, . . . , n. For convenience, we associate the label 1 with theleading particle, and sequentially label the remaining n−1 particles (see Fig.2.2). Then a1 ≡ al and ai ≡ a for i = 2, . . . , n− 1, and an ≡ at.

Figure 2.2: A schematic diagram of particles in 1 dimension, showing oursubscripting convention. Grey arrows indicate distance-dependent interac-tion forces.

Once a perfect school in 1D is formed, only two quantities are requiredto completely characterize the school: its velocity, v, and nearest-neighbourdistance, d. These will be determined by the choice of specific interactionforces and parameter values. According to Eq. (2.5), the schooling force isgiven by

fi =∑

j

g±(|xji|)xji

|xji|+∑

j

h±(|vji|)vji

|vji|, (2.12)

where xji = xj − xi is the distance between neighbours and vji = vj − vi isthe relative velocity of particles i and j, and j sums over all particles thatinfluence the movement of particle i.

We change into a coordinate system that is moving at the school speedv. Let pi and qi be the position and velocity of the ith particle in this movingcoordinate. Using the relation pi = xi − vt and qi = vi − v, we obtain thedynamic equations in the moving frame:

pi = qi, (2.13)

qi = ai − γ(qi + v) + fi. (2.14)

29


In the moving frame, a perfect school solution corresponds to a stationarydistribution of the particles in space. Thus, every perfect school must be asteady state solution of (2.13)–(2.14); i.e., pi = 0, qi = 0, for which

qi = 0, (2.15)

ai − γv + fi = 0, (2.16)

for all i. For such a steady state,

fi =∑

j

g±(|j−i|d) sgn(j−i)+∑

j

h±(0)0 =∑

j

g±(|j−i|d) sgn(j−i), (2.17)

where j sums over all particles that influence particle i and where

sgn(j − i) =j − i

|j − i| .

Eq. (2.17) implies that for a steady-state solution in which interaction con-nections are fixed, all the schooling forces are functions of d only.

2.3.1 Schools Formed by Following One ImmediateNeighbour

The simplest case of a perfect school in one dimension occurs when eachparticle follows only one nearest neighbour at constant distance d in front ofit. Then the leader feels no schooling force, i.e., f1 = 0, so al − γv = 0 whileall followers including the last particle experience the same schooling forcefi = g+(d). Because the tail particle interacts in the same way as interiorparticles, we let a = at, so an ≡ a. Evaluating Eq. (2.16) at steady statefor the leader gives v = al/γ. Substituting this expression for v into Eq.(2.16) and evaluating at steady state for a follower gives g+(d) = al −a. Thevalue of d is determined by the number of intersection points between thefunction g+(x) and the horizontal line at the value al − a. There could bezero, one, two, or more intersection points depending on the functional formof g+(x), implying that the number of school formations that exist could bezero, one or more. As we shall illustrate later, not all solutions are stable.The stability depends on the slope of g+(x) at each intersection point.

Note that if all particles have identical autonomous self-propulsion, i.e.ai = a, then fi = g+(d) = 0 when the school is formed, implying that

30


all schooling forces vanish at the steady state perfect school. Only whenthe particles deviate from their ‘preferred’ positions in the school would theschooling forces become nonzero. Then the effect of these forces brings theparticles back to their correct positions (if stable) or amplifies the deviation(if unstable). A school in which schooling forces vanish at steady state willbe denoted a stress-free school. However, if the leader has a distinct self-propulsion, i.e. al 6= a, then fi = al − a 6= 0 which means that the schoolingforces do not vanish when the school is formed. We shall call such a schoola stressed school.

2.3.2 Schools Formed by Interactions with Two

Nearest Neighbours.

Consider a 1D school with particles labeled as in Fig. 2.2, such that a1 = al,ai = a for i = 2, . . . , n − 1 and an = at. Now, we assume that each particleinteracts with its nearest neighbours ahead and behind it, giving the followinginteraction forces:

f1 = −g−(|x2 − x1|) + h−(|v2 − v1|) sgn(v2 − v1), (2.18)

fi = g+(|xi−1 − xi|) − g−(|xi+1 − xi|) + h+(|vi−1 − vi|) sgn(vi−1 − vi)

+ h−(|vi+1 − vi|) sgn(vi+1 − vi), i = 2, · · · , n− 1, (2.19)

fn = g+(|xn−1 − xn|) + h+(|vn−1 − vn|) sgn(vn−1 − vn). (2.20)

When a school is formed, vi = v and xi−1 − xi = d (and thus xi+1 − xi =−d) for all i, which implies that

f1 = −g−(d), (2.21)

fi = g+(d) − g−(d), (i = 2, · · · , n− 1), (2.22)

fn = g+(d). (2.23)

Next, we substitute these forces into the steady-state equation (2.14) whichgives

γv = al − g−(d), (2.24)

γv = a + g+(d) − g−(d), (2.25)

γv = at + g+(d), (2.26)

31


where an = at denotes the autonomous self-propulsion of the tail particle.Solving these equations for v, al, and at we obtain

v =1

γ[al + at − a], (2.27)

al = a+ g+(d), (2.28)

at = a− g−(d), (2.29)

which leads to the following conclusions:

1. The speed of the school depends explicitly only on the self-propulsionforces al, at and a and γ, among which al and a are parameters withfixed values. Eq. (2.28) determines the value of d as well as the numberof possible school solutions based on the given form of g+(d). Once thevalue of d is determined, Eq. (2.29) determines the value of at, whichis the self-propulsion force the tailing particle must adopt in order tokeep up with the school.

2. Individuals at the front or rear of the group have fewer interactions.Thus, to keep a fixed nearest-neighbour distance, some compensationfor missing forces is needed. Otherwise, it would be impossible tomaintain a perfect school.

We note that in many biological examples, which deviate significantlyfrom our perfect school assumptions, there is crowding at the edges of a herdor swarm. Examples of this type occur in herds of wildebeest shown in [19]and in locust hopper bands [24]. Both examples show a gradual increase ofdensity of individuals towards a front edge, and sharp boundary at that edge.

In the case of interactions with both nearest neighbours, a stress-freesteady-state solution implies that g+(d) = g−(d) = 0. From Eqs. (2.27)–(2.29), the stress-free condition implies that at = al = a, and v = a/γat steady state; that is, each individual must have the same autonomousself-propulsion, and the school velocity is dependent only on the commonautonomous self-propulsion and the friction coefficient.

Alternatively, for a stressed solution with nearest-neighbour distance dand v > 0, (2.27)–(2.29) imply that

al > g−(d). (2.30)

The school velocity is given by v = [al − g−(d)]/γ, and thus (2.30) assuresthat v > 0. Further, Eq. (2.30) means that the leading particle must generate

32


a sufficiently large autonomous self-propelling force to overcome attraction(or effective drag) due to the particle behind it, while the difference betweenthe two forces determines the school speed. Eq. (2.28) means that the differ-ence between al and a together with the functional form of g+(x) determinethe number, and values of allowable group spacing d. For fixed al and a, theautonomous self-propulsion of the tail particle must be adjusted to satisfyEq. (2.29). We note that a stressed school can occur even if al and a havedifferent signs; i.e., if the self-propulsion of the leading particle and interiorparticles are in opposite directions. In this case, the schooling forces over-power the intrinsic motion of the interior particles. In this sense, interiorindividuals are all followers, with the tail particle a special type of followerwhose autonomous self-propulsion is distinct from that of the other follow-ers. We note that Eqs. (2.28)–(2.29) place a further restriction on g±(x):that al − a ≤ maxx(g+(x)), and that a− at ≤ maxx(g−(x)) to guarantee theexistence of at least one perfect school solution.

As one example, suppose that a = 0 for all interior particles i.e., indi-viduals base responses on interactions alone, instead of intrinsic autonomousforces. In such a school, to keep a nearest-neighbour distance d, at = −g−(d)for the tail particle. The spacing distance d is determined from (2.28), i.e.,al = g+(d). Then for a monotonic increasing g+(x), increasing al results in alarger nearest-neighbour distance d. However, because steady-state velocityis v = [al − g−(d)]/γ, increasing al does not necessarily increase v. Instead,Eq. (2.27) implies that

v =1

γ(g+(d) − g−(d)) . (2.31)

Eq. (2.31) illustrates the relationship between school velocity and spac-ing. Specifically, if the slope of g+(x) − g−(x) is positive, the distance dbetween neighbours in the school increases as school velocity v increases.On the other hand, if the slope of g+(x) − g−(x) is negative, d decreases asv increases. Examples of these two different cases are shown in Fig. 2.5.This result is immediately applicable to experimental observations. For ex-ample, if the NND d increases as the school moves with a higher speed,there is reason to believe that the case in Fig. 2.5.a might be occurring.It is almost impossible to determine the functional form of g±(x) for re-alistic animals, but by observing the relationship between the changes inschool speed and the NND, we can comment on the nature of the slope ofg+(x) − g−(x). Note that a = 0 is not required for this result; in the case

33


a 6= 0, v = a/γ + (1/γ) (g+(d) − g−(d)) , which is identical to Eq. (2.31),except for a shift in velocity of a/γ.

2.3.3 Examples of Some Schooling Forces

Distance-dependent forces. The theory developed so far applies to anyfunctions g±(x), h±(x). However, requiring realistic schooling forces restrictsour choice of functions. For distance-dependent forces, we take as a specificexample Hill functions, i.e., saturating monotonic increasing functions of x:

g(x) = c

[

(1 + km/dm0 )xm

xm + km− 1

]

, (m ≥ 1), x ≤ xf . (2.32)

m > 0 and d0 > 0 represent the steepness and x-intercept of g(x), respec-tively. Note that g(0) = −c and c[1+km/dm

0 ] are, respectively, the minimumand the saturation values of g(x). Eq. (2.32) defines a four-parameter familyof functions from which g+(x) and g−(x) might be chosen. A decay beyondsome finite distance can represent a limited sensing range (0 ≤ x ≤ xf ) ofthe individuals. Fig. 2.3 shows a typical distance-dependent schooling forcebased on a Hill function. Other typical forces used include exponential forcesof the form

g(x) = A exp(−x/a) −R exp(−x/r), (2.33)

or inverse power forces

g(x) =A

xa− R

xr,

where A,R, a, and r are parameters (see [11]).Velocity-dependent forces. In one dimension, the velocity-dependentforces h±(v) have a simple interpretation. In this case, all individuals havethe same absolute direction (though they may differ by a sign, dependingon whether movement is along the axis in the positive or negative direc-tion). Then the velocity-dependent force given by Eq. (2.12) summed overinfluencing neighbours,

h±(|vji|)vji

|vji|,

can be represented as a single function of vji, since in 1D the direction termvji/|vji| is simply sgn(vji), i.e., ±1. We thus define

h±(v) = h±(|v|) v|v| ,

34


Figure 2.3: An example of a distant-dependent force in the form of a Hillfunction with a decay past xf , to depict a finite sensing range.

where h(v) is defined for all v ∈ R. There are a number of cases to considerwhen determining reasonable forms of h(v). From Fig. 2.4, the influence of a

Figure 2.4: Four cases of neighbour interaction: [a] neighbour j travels fasterthan i in the same direction, (vji > 0), [b] neighbour j travels slower than iin the same direction, (vji < 0), [c] neighbour j diverging from i, (vji > 0),[d] neighbour j converging to i, (vji < 0).

neighbour j on the velocity vi of individual i should cause acceleration whenvji > 0, but deceleration when vji < 0. We assume that there is no influence

35


when vi = vj , i.e., h(0) = 0. We will later show that under the assumptionh(0) = 0, h(v) has no effect on existence of a perfect school solution. Thus, areasonable form of h(v) is a monotonically increasing function with h(0) = 0.This type of assumption is incorporated in models for swarming in 2D and3D, discussed in [12, 15, 16, 17, 20] as “arrayal forces”, since these forces actto array individuals in parallel. As an example, the function we use in thispaper is

h(v) = C tanh(v/ρ),

where C > 0 and ρ are parameters governing the amplitude and steepness ofh(v), respectively. If ρ < 0, h(v) is monotonically decreasing; we include thisgenerality, but we will later show that only the monotonically increasing case(ρ > 0 in this example) enhances stability of the perfect school solution. Asin the distance-dependent functions, particular choices of parameters leadto different shapes of functions for h+(v) and h−(v). In [8], h(v) = αv(α > 0 is a constant) in the case when only nearest neighbour interactionsare considered.

The number of reasonable choices of the functions g±(x) and h± that yieldqualitatively identical results as the expressions we adopted in this study isinfinite. Any information that leads extra restrictions of these functionswould be valuable for making our choices more realistic.

2.3.4 The Stability of a School Solution

We next explore the stability of a 1D perfect school. To do so, we considera perfect type I school, moving with speed v and nearest-neighbour distanced: vs

i = v and xsi = x0

1 + vt − (i − 1)d (where x01 is the position of the

leading particle at t = 0). This corresponds to the steady-state solutionps

i = x01 − (i− 1)d and qs

i = 0 in the moving frame coordinates.Using the moving-frame analogues of Eqs. (2.18)–(2.20) and (2.12), and

the expression for v from Eq. (2.27) in Eqs. (2.13)-(2.14), we obtain themoving-frame equations of motion for nearest-neighbour interactions aheadand behind individuals:

dpi

dt= qi, (2.34)

dq1dt

= al − g−(|p2 − p1|) + h−(|q2 − q1|) sgn(q2 − q1)

36


Figure 2.5: [a] An example of interaction forces where the net force haspositive slope at d. Here, g+(x) = 2[3x2/(x2 +42)− 1], and g−(x) = 3x/(x+4)− 1. [b] An example of interaction forces where the net force has negativeslope at d. Here, g+(x) = (1/2)[3x3/(x3+23)−1], and g−(x) = 3x/(x+8)−1.

−γq1 − (al + at − a), (2.35)

dqidt

= a+ g+(|pi−1 − pi|) − g−(|pi+1 − pi|)+h+(|qi−1 − qi|) sgn(qi−1 − qi)

+h−(|qi+1 − qi|) sgn(qi+1 − qi) − γqi − (al + at − a), (2.36)

dqndt

= at + g+(|pn−1 − pn|) + h+(|qn−1 − qn|) sgn(qn−1 − qn)

−γqn − (al + at − a). (2.37)

37


To investigate the linear stability of the school solution, we introduce asmall perturbation to the equilibrium values of the velocity and the positionof each particle in the moving frame:

pi(t) = psi + δi(t), (2.38)

qi(t) = qsi + ωi(t), (2.39)

where psi = 0, and qs

i = 0 for all i. Substituting Eqs. (2.38) and (2.39) intoEqs. (2.34)–(2.37), and expanding nonlinear terms in a Taylor series up tofirst order, we obtain

dδidt

= ωi, (2.40)

dω1

dt= −g−(d) − g′−(d)[δ1 − δ2] − h′−(0)[ω1 − ω2]

−γω1 + (a− at), (2.41)

dωi

dt= g+(d) − g−(d) + g′+(d)[δi−1 − δi] − g′−(d)[δi − δi+1]

+h′+(0)[ωi−1 − ωi] − h′−(0)[ωi − ωi+1]

−γωi + (2a− al − at), (2.42)

dωn

dt= g+(d) + g′+(d)[δn−1 − δn] + h′+(0)[ωn−1

−ωn] − γωn + (a− al), (2.43)

where i = 1, · · · , n in Eq. (2.40) and i = 2, · · · , n − 1 in Eq. (2.42) andwhere we have used h±(0) = 0. Using Eqs. (2.28)–(2.29), we can reducethese equations to the following simplified form governing perturbations:

dδidt

= ωi, (2.44)

dω1

dt= −g′−(d)[δ1 − δ2] − h′−(0)[ω1 − ω2] − γω1, (2.45)

dωi

dt= g′+(d)[δi−1 − δi] − g′−(d)[δi − δi+1]

+h′+(0)[ωi−1 − ωi] − h′−(0)[ωi − ωi+1] − γωi, (2.46)

dωn

dt= g′+(d)[δn−1 − δn] + h′+(0)[ωn−1 − ωn] − γωn, (2.47)

38


where i = 1, · · · , n in (2.44) and i = 2, · · · , n− 1 in Eq. (2.46). The eigenval-ues of the coefficient matrix associated with this linear system give stabilityinformation. It is not straightforward to obtain explicit expressions for theeigenvalues in the general case of Eqs. (2.44)–(2.47). However, when wesimplify the system further for the case of following only one immediateneighbour in front, we obtain the system

dδidt

= ωi, (2.48)

dω1

dt= −γω1, (2.49)

dωi

dt= g′+(d)[δi−1 − δi] + h′+(0)[ωi−1 − ωi] − γωi, (2.50)

dωn

dt= g′+(d)[δn−1 − δn] + h′+(0)[ωn−1 − ωn] − γωn. (2.51)

When written in matrix form, the coefficient matrix corresponding to Eqs.(2.48)–(2.51) is a 2 × 2 block matrix with n × n blocks. To obtain theeigenvalues of this matrix we first simplify the determinant equation usingthe following result from [18]: given the block matrix

[

A BC D

]

,

where A,B,C, and D are m×m matrices, if A and B commute, then

det

[

A BC D

]

= det [DA− CB] . (2.52)

This result allows us to reduce the dimension of the matrix in the determinantformula for the eigenvalues. Assuming only forward (or, similarly, backward)sensing allows a further reduction to an upper (lower) triangular matrix, forwhich the determinant can be obtained from the product of the diagonalentries. The details of this calculation are given in Appendix A, where it isshown that the associated 2n eigenvalues are

λ1 = 0, (2.53)

λ2 = −γ, (2.54)

λ± = −(h′+(0) + γ)/2 ±√

[(h′+(0) + γ)/2]2 − g′+(d), (2.55)

39


where eigenvalues λ+ and λ− each have multiplicity n−1. The zero eigenvaluein Eq. (2.53) characterizes the translational invariance of the solution, whileλ2 in Eq. (2.54) is always negative for γ > 0. However, in the absence ofdamping (γ = 0), the multiplicity of the zero eigenvalue is 2, indicating thatthe school solution is neutrally unstable. From Eq. (2.55), we obtain thestability conditions (guaranteeing λ± < 0)

h′+(0) + γ > 0, (2.56)

g′+(d) > 0. (2.57)

Therefore, the contribution of the speed-dependent force is stabilizingif h′+(0) > 0, and destabilizing if h′+(0) < 0. However, even if the speed-dependence is destabilizing (i.e. if h′+(0) < 0), the solution can still be stableif the damping, γ, is large enough to compensate in Eq. (2.56). This suggeststhat a non-zero damping term is indispensable for ensuring stability of aperfect school solution. We have already shown that conditions for existenceof the perfect school are independent of velocity-dependent interaction forces;condition (2.56) also illustrates that h+(v) is not essential for stability of theschool solution. If h+(v) = 0, i.e., there is no velocity-dependent force at all,

then λ± = −γ/2 ±√

(γ/2)2 − g′+(d). Then, assuming Eq. (2.57) holds, theschool solution remains stable if γ > 0.

On the other hand, the distance-dependent force is crucial, both for exis-tence (as previously shown) and for stability of a perfect school solution. Ifg+(x) = 0, λ± = 0, −(h′+(0) + γ). Recalling that the multiplicity of λ+ isn − 1, the school solution becomes neutrally unstable in the absence of theforce g+(x), as the multiplicity of the zero eigenvalue is then n. Also note thatthe system can exhibit an oscillatory approach to steady state in the presenceof complex conjugate eigenvalues; i.e., when [(h′+(0) + γ)/2]2 < g′+(d).

The results on school formation in 1D reveal that damping and the pres-ence of a distance-dependent schooling force are key factors for the existenceof a stable school pattern. Velocity-dependent schooling forces can make theschool solution more or less stable depending on the sign of h′+(v), as shownin Eq. (2.56). We have shown that the school speed is determined by theautonomous self-propelling forces of particles at different locations withinthe school and is independent of the schooling forces. We also showed thatschool formation occurs even when each particle follows only its neighbourimmediately ahead. This ‘follower’ school presents one minimal model of per-fect school formation in 1D, and its simplicity has permitted an analytical

40


description of the existence and stability of such schools.

2.3.5 Numerical Simulations in One Dimension

Figure 2.6: 1D simulations showing position in a moving frame moving atvelocity v as indicated. Interaction forces are as in [a] Fig. 2.5.a, and [b] Fig.2.5.b. Self propulsion terms are [a.i] al = 0.5, [a.ii] al = 0.7, [b.i] al = 0.9,[b.ii] al = 0.8, while a = 0.1 in all cases. In [a], higher school speed resulted inan increase in NND from 3.26 in [a.i] to 3.49 in [a.ii]. The opposite occurredin [b], where higher school speed caused a decrease in d from 3.73 in [b.i] to3.175 in [b.ii].

We simulate the one-dimensional model (2.10)–(2.11) in MATLAB usingthe solver ODE45, which integrates a system of ODEs with an embedded

41


Runge-Kutta method. The algorithm is based on the Dormand-Prince 4(5)formulae, which uses a 4th and 5th order pair of methods. We show thecase of distance-dependent interactions only; i.e., h± = 0. Data is plotted inthe moving frame, relative to the average velocity at t = tend. Particles areinitialized randomly in the interval [0,11]. The model inputs g±(x), al anda are chosen to satisfy Eq. (2.28), which in turn determines d. When d isknown, at follows from Eq. (2.29), giving the self-propulsion that the lastparticle must produce to keep up with the school. The absolute magnitudesof model inputs were chosen sufficiently small to avoid large changes in modelvariables at each time step. Relative magnitudes of model inputs were cho-sen so that motion was not dominated by any of the forces of autonomouspropulsion, interaction, or drag.

We test two distinct examples of interaction forces to illustrate our results.In case 1 (Fig. 2.6.a) we use forces as in Fig. 2.5.a, whereas in case 2 (Fig.2.6.b), the forces are as in Fig. 2.5.b. Increasing al in both cases generates alarger distance d, but, as noted in our analytic results, we expect velocity toincrease in case 1 and decrease in case 2 as d is increased. These results areborne out by simulations.

Comparing results presented here to those obtained in other models inwhich all-to-all interactions were considered [1, 5, 8, 11], there are severalimportant differences. First, the NND is determined by the self-propellingforces al, a and the interaction force function g+(x) (see Eqs.(2.28-2.29). Itis independent of the number of particles in the group. This is because onlynearest-neighbour interactions are considered here. Increasing the number ofparticles does not increase the forces each particle experiences. Second, the“edge effect” that was typically observed in the other models is eliminatedby making the edge particles behave differently based on their location inthe group. Perfect schools cannot occur if the edge particles are not capableof sensing their locations in the group and behaving accordingly. Third, theinteraction functions that we use here are monotonic functions that typicallyviolate the so called H-stability conditions studied in [1, 5]. H-stability ischaracterized by the conditions under which the NND approaches a finitenon-zero value as the number of particles approaches infinity in an all-to-allcoupled system. The conditions do not apply to the results presented here asonly nearest-neighbour coupling is considered. Fourth, in a perturbed perfectschool solution, the rate at which the system evolves toward the stable schoolpattern is governed by the real parts of the eigenvalues of the linearizedsystem given by Eqs.(2.44-2.47). These eigenvalues are determined by the

42

2.4. The Soldier Formation in 2D Space

Figure 2.7: An example of an approximate soldier formation in Atlanticbluefin tuna, courtesy of Dr. M. Lutcavage, Large Pelagics Research Center,UNH.

parameter γ and the slopes of the interactions forces g′±(d) and h′±(0). Theseeigenvalues are explicitly given by Eqs.(2.53-2.55) when each particle followsonly the immediate neighbour to its front. The rate at which a perfect schoolsolution is approached from an arbitrary initial condition is determined byboth the linear and nonlinear effects of interaction forces.

2.4 The Soldier Formation in 2D Space

Among the numerous types of schooling behaviour found in higher dimen-sions in nature is the soldier formation. Such schools are characterized by aroughly linear organization of individuals with a common heading in some(nonaxial) direction. A school of this type formed by bluefin tuna is shownin Fig. 2.7. In the following treatment, we consider an idealization of soldierformations found in nature, such that all individuals have the same headingand velocity, and whose positions lie on a straight line (see Fig. 2.8). Thesimple linear structure of such a school facilitates our analysis.

We apply the frontal nearest-neighbour interaction rule (i.e., g− = 0) toa soldier formation of n particles, labeled so that 1 is the leader, and theremaining n−1 are followers. Under the chosen interaction regime, ~x1 is the

43


Figure 2.8: A schematic diagram of individuals in soldier formation withfrontal nearest-neighbour detection. Grey arrows indicate the direction ofschooling force, while black arrows indicate direction of motion.

nearest neighbour of ~x2, ~x2 is the nearest neighbour of ~x3 and so on. Notethat the coupling of particles is not predetermined, but instead is determinedby the group geometry. We neglect velocity-dependent interactions (i.e.,h± = 0). The corresponding equations of motion for the system are

d~x1

dt= ~v1, (2.58)

d~v1

dt= ~al − γ~v1, (2.59)

andd~xi

dt= ~vi, (2.60)

d~vi

dt= ~a− γ~vi + ~fi, (2.61)

for i = 2, . . . , n. The schooling force ~fi depends on the relative position ofthe ith individual with respect to its nearest neighbour, (i− 1), and thus hasarguments

~fi = ~fi(~xi−1 − ~xi).

Because we assume that individuals sense only what is in front of them, wehave ~f1 = ~0 for the leader.

44


2.4.1 Existence Conditions for the Soldier Formation

In this section, Eqs. (2.58)–(2.61) are used to obtain existence conditions forthe soldier formation. In the soldier formation d~vi/dt = 0, and so, we considerthe formation as a ‘moving school’ steady state for the system (henceforthreferred to simply as steady state). We note that this steady state differsfrom a full steady state, wherein also d~xi/dt = 0, i.e., where the aggregate isstationary, rather than moving.

The equation of motion (2.59) for the leader at steady state implies

~v =~al

γ, (2.62)

where ~v is the steady-state velocity of the leader. Note that because everyindividual moves at the same velocity in a steady-state school solution, ~vis also the velocity of all individuals in that school. Substituting this intothe steady-state equation of motion for the ith individual gives the existencecondition for a soldier formation:

~f si = ~al − ~a, i = 2, . . . , n. (2.63)

We express ~f si as a unit direction vector (oriented along the axis of the

school at steady-state) and a magnitude as in (2.5), giving ~f si = g+(d)~u,

where ~u = ~xi−1,i/|~xi−1,i|, and d = |~xi−1,i| is the distance between individualsat steady state.

We consider two cases: first, if ~al 6= ~a, then by (2.63), g+(d)~u = ~al − ~a,i.e., the direction of axis of the school, ~u, must be parallel to ~al − ~a. Similarto the case of a 1D school, the magnitude of ~al − ~a determines the value(s)of g+(d) that guarantee the existence of such a school formation, i.e.,

g+(d) = |~al − ~a|. (2.64)

For a monotonic function g+, this uniquely determines the inter-individualdistance d along the axis of the school at steady state. This is an exampleof a stressed school in which interaction forces do not vanish at steady state.Once the parameters ~al and ~a are fixed and g+(x) is specified, ~u, d, ~v, andthe bearing angle are all determined.

On the other hand, if ~al = ~a, then at steady state, ~fi = ~al − ~a = ~0, i.e.,the school is stress-free and there is no restriction on the bearing angle of theaxis of the school.

45


2.4.2 Stability Analysis

In the soldier formation, the cohesiveness of the group can be examined interms of the stability of the system to small perturbations about the steadystate; i.e., the local stability. The stability analysis of the soldier formationin 2D follows closely the analysis of 1D schools.

The geometry of a soldier formation allows some simplification in analysisby a judicious choice of orientation of axes. Specifically, we align our coor-dinate system so that the x-axis lies along the line of individuals. Then they−components of the particle positions are zero at the steady-state soldierformation solution. Note, however, that unlike 1D motion, the individualshere move (in general) in some direction other than the x-axis. Thus wewrite

~xs1 = (0 , 0), ~xs

2 = (d , 0), . . . , ~xsi = ((i− 1)d , 0) , . . . , ~xs

n = ((n− 1)d , 0).

Henceforth, we display only calculations on the ith equation, but thisimplicitly represents all i = 1, . . . , n. We transform to a frame of referencemoving with the school velocity at steady state (Eq. (2.62)). We let

~xi = ~pi + ~vt,

~vi = ~qi + ~v.

We substitute these expressions into (2.58)–(2.61) to obtain

d(~p1 + ~vt)

dt= ~q1 + ~v, (2.65)

d(~q1 + ~v)

dt= ~al − γ (~q1 + ~v) , (2.66)

d(~pi + ~vt)

dt= ~qi + ~v, i = 2, . . . , n (2.67)

d(~qi + ~v)

dt= ~a− γ (~qi + ~v) + ~fi, i = 2, . . . , n (2.68)

Simplifying these equations, noting that γ~v = ~al, we obtain the moving frameequations of motion:

d~p1

dt= ~q1, (2.69)

46


d~q1dt

= −γ~q1, (2.70)

d~pi

dt= ~qi, i = 2, . . . , n, (2.71)

d~qidt

= ~a− ~al + ~fi − γ~qi, i = 2, . . . , n (2.72)

The arguments of ~fi are now ~fi = ~fi(~pi−1 − ~pi), though we continue to omitthese arguments unless needed for clarity. To summarize, ~pi represents theposition of individual i in the moving frame, and ~qi represents the velocityof individual i in the moving frame. The soldier formation is a steady-statesolution to this system, and thus at steady-state, ~qi = 0 and ~a−~al + ~fi = 0,which gives the existence condition in (2.63), and ~ps

i = e1(i − 1)d, wheree1 is the unit vector (1 , 0). Next, we perform a stability analysis on ourmoving-coordinate 2D variables ~pi and ~qi. We introduce small perturbationsin position ~δi and velocity ~ωi about the steady state; i.e.,

~pi(t) = ~psi + ~δi(t) = e1(i− 1)d+ ~δi(t),

and~qi(t) = ~qs

i + ~ωi(t) = ~ωi(t),

where e1 is a unit vector in the x direction. Substituting these perturbationsinto Eqs. (2.69) – (2.72) and noting that time derivatives of steady-statevariables are 0, we get

d~δ1dt

= ~ω1, (2.73)

d~ω1

dt= −γω1, (2.74)

d~δidt

= ~ωi, i = 2, . . . , n, (2.75)

d~ωi

dt= (~a− ~al) + ~fi − γωi, i = 2, . . . , n. (2.76)

We proceed by expanding the nonlinear term, ~fi in a Taylor series. We firstrewrite ~fi as

(fix, fiy) = ~fi(~zi) = g+(|~zi|)~zi

|~zi|= g+(|~zi|)

(

zix

|zi|,ziy

|zi|

)

,

47


where〈zix, ziy〉 ≡ ~zi = ~pi−1 − ~pi = −e1d+ ~δi−1 − ~δi, (2.77)

and|~zi| =

√

z2ix + z2

iy.

That is, ~fi is oriented in the direction of the vector connecting two neigh-bouring individuals, with a magnitude dependent on the distance betweenthe two individuals. The Taylor expansion of ~fi is

~fi = ~fi(−e1d) +

∂ ~fi

∂~zi

|~zi=~zsi

· [~zi − ~zsi ] + . . . , (2.78)

where the expression in braces is a Jacobian matrix, i.e., the partial deriva-tives of fix and fiy with respect to zix and ziy coordinates. The Jacobianmatrix in Eq. (2.78) evaluated at steady state is given by

Df ≡

∂ ~fi

∂~zi

|~zi=~zsi

=

[

g′+(d) 00 g+(d)/d

]

.

Details of this calculation are contained in Appendix B. Df is independentof the equation index i, and is thus identical for all individuals i = 2, . . . , n.Substituting Df into the Taylor expansion of ~fi in Eq. (2.78), and notingfrom Eqs. (2.77) and (A.8) (in Appendix) that

~zi − ~zsi = ~δi−1 − ~δi, (2.79)

we now have~fi = ~fi(−e1d) +Df · [~δi−1 − ~δi] + . . . . (2.80)

Furthermore, because at steady state we have ~fi(−e1d) = ~al − ~a, it followsthat for the small perturbation,

~fi ≃ ~al − ~a+Df · [~δi−1 − ~δi].

We now use this linear approximation for ~fi in Eq. (2.76), resulting in

d~ωi

dt= ~a− ~al + (~al − ~a+Df · [~δi−1 − ~δi]) − γ~ωi.

48


Canceling out terms and combining with Eqs. (2.73)–(2.75) gives the follow-ing system:

d~δ1dt

= ~ω1, (2.81)

d~ω1

dt= −γω1, (2.82)

d~δidt

= ~ωi, i = 2, . . . , n (2.83)

d~ωi

dt= Df · [~δi−1 − ~δi] − γ~ωi, i = 2, . . . , n. (2.84)

The matrix expression of this system is a 2 × 2 block matrix with 2n × 2nblocks. The eigenvalue equation is formulated in determinant form, and thecommutativity of the blocks of the associated matrix allows for a simplifica-tion of the determinant using Eq. (2.52). The eigenvalue calculation is thusreduced to calculating the determinant of an n × n block-triangular matrixwith 2 × 2 blocks, which is given by the product of the determinants of thediagonal blocks. The matrix form of Eqs. (2.81)–(2.84) and eigenvalue cal-culations are given in Appendix B. The coefficient matrix associated withEqs. (2.81)–(2.84) has eigenvalues

λ = 0, with multiplicity 2, (2.85)

λ = −γ, with multiplicity 2, (2.86)

λ = −γ2±√

γ2 − 4g′+(d)

2, each with multiplicity n− 1 (2.87)

λ = −γ2±√

γ2 − 4g+(d)/d

2, each with multiplicity n− 1. (2.88)

The first eigenvalues, λ = 0 with multiplicity 2, indicates a translationalinvariance in two-dimensional space. The next distinct eigenvalue, λ = −γis always negative when damping is present, γ > 0. This further indicatesthe importance of a drag term for stability of solutions.

The remaining eigenvalues have negative real parts provided both

g′+(d) > 0, (2.89)

g+(d) > 0. (2.90)

49


Figure 2.9: Numerical simulations in 2D for [a] 8 particles with ~al = [0.1, 0.1],~a = [0,−0.2] and [b] 6 particles with ~al = [0.1, 0.1], ~a = [0, 0]. Trajectories areplotted through time. Note in [a] that the solution line connecting individualsis oriented in the direction of ~al − ~a = [0.1, 0.3], while the school velocity atsteady state is ~v = [0.2, 0.2]. In [b], both the solution line and school velocityare in the same direction, with ~al = [0.1, 0.1], ~v = [0.2, 0.2].

Interpreted in terms of stability of the school solution, Eqs. (2.89)–(2.90)implies that the schooling force acting to maintain the school must be at-tractive at the NND d. Additionally, this attractive force should increase ifindividuals with spacing d try to move further apart.

2.4.3 Numerical Simulation of the Soldier Formation

We perform simulations in two dimensions using single-neighbour interac-tions, with neighbour detection only in the frontal plane (g− = 0). Weinclude no velocity-dependent interactions (h± = 0). We choose an exponen-tial interaction function following Eq. (2.33), as g+(x) = 0.5(exp(−x/50) −2 exp(−x/1)). Eqs. (2.58)–(2.61) are evolved using ODE45 as in the 1Dsimulations. Initial positions are assigned randomly in [0, 4] × [0, 4], and weset γ = 0.5.

We first test two distinct cases of ~al and ~a values. In case 1 (Fig. 2.9.a)~al 6= ~a 6= ~0, while in case 2 (Fig. 2.9.b), ~al 6= ~a = ~0. As noted in our analyticalresults, we expect the soldier formation to move in the direction of ~al, whilethe solution line connecting individuals is oriented in the direction of ~al −~a.In case 1, this corresponds to a school directed non-axially to the solutionline, while in case 2, the solution line and school direction coincide, leading

50


to a trail-following group. The NND in these solutions is determined by Eq.(2.64). For the cases in Fig. 2.9a and 2.9.b, d = 1.7949 and 1.055 respectively.The difference in d between the two cases is due to the difference in thevalue of |~al − ~a| but not the difference in the number of particles. Becauseonly nearest-neighbour interactions are considered here, this distance remainsunchanged as the number of particles is increased. These analytical resultsare confirmed by simulations shown in Fig. 2.9. Although our stability resultsare only local, the simulations suggest that the soldier formation is a globallystable pattern for the specific choice of parameter values in Fig. 2.9a. In afollow-up study [10], we found that milling formation could coexist with thesoldier formation when the force function g+(x) is bi-phasic. In that case,different initial conditions determine which pattern will eventually emerge.Unlike the soldier formation, the existence of a milling solution and the NNDboth depend on the number of particles for a fixed choice of the force function.

In Section 4.1, we mentioned the case when al = a, which implies no re-striction on the bearing angle of individuals. Eq. (2.64) implies that the zeroof g+(x) gives the individual distance d at steady state for such groups. Fig.2.10 shows examples of such configurations. Note that individual distance atsteady state remains constant at d = 0.7, though the relative angles betweenneighbours is determined by initial conditions.

In previous simulations, we used interaction functions that allowed stableschools to arise. However, if we choose g+(x) so that g′+(x) is negative forall x > 0, then we fail to satisfy the stability condition in Eq. (2.89). Weexplore the motion of particles under such conditions in simulations (Fig.2.11). In these simulations, particles tend to organize in moving pairs whichremain together due to close-range attraction. The path traced out by thesepairs appears to be very sensitive to initial conditions, and does not follow apredictable pattern. Further simulations with such ‘stability-breaking’ inter-action forces reveal the presence of numerically stable mill solutions in whichparticles rotate around an invariant circle. These solutions require a closedcurve of interaction connections between particles in order to evolve. Theexistence and stability of such mill solutions form the topic of Chapter 3.

51


Figure 2.10: Numerical simulations in 2D, for [a] 10 particles and [b] 25 par-ticles with ~al = ~a = [0.1, 0.1], and g+(x) = 0.5(exp(−x/50) − 2 exp(−x/1)).Trajectories are plotted through time, and grey arrows indicate interactionsbetween nearest neighbours. Note that individuals have constant spacingbetween nearest neighbours, but are not restricted to a particular relativeangle with nearest neighbours.

Figure 2.11: Numerical simulations in 2D with g+(x) = 1 − x/2, whichcorresponds to (rather non-physical) short-range attraction and long-rangerepulsion. Trajectories are plotted in time for [a] 4 particles evolved to t = 40,and [b] 6 particles evolved to t = 300. Individuals group in pairs (thoughdue to overlapping, some pairs appear as a single particle), and these pairsfollow unpredictable trajectories.

52

2.5. Schools in the Form of 2D Arrays

2.5 Schools in the Form of 2D Arrays

We have thus far been primarily concerned with 1D and quasi-1D forma-tions of particles. However, extending these ideas to schools in the formof regular 2D arrays (i.e., a perfect school in 2D) is both a logical step inthe mathematical analysis, and is more relevant to many observed biologicalaggregates. This extension to 2D brings a number of challenges to our ana-lytical approach. First of all, similar to the case of 1D schools in which thetail particle should be aware of its special position in the group and behavedifferently to maintain the school, particles located at the edges of a 2D arrayshould also behave differently than particles in the interior. Confounding theissue further, the number of edge particles change as the specific shape ofthe array changes. Thus, it is very difficult to study 2D arrays in a generalform since there exist many possible 2D shapes for a group composed of alarge number of particles. For groups of small numbers of particles, someanalytical insights can be made, though these are shape-dependent and notgeneralizable.

However, numerical simulations can provide insight into the mechanismsof forming schools of 2D arrays that tile the plane in a regular fashion. If werestrict our discussion to the models in which each particle only follows oneclosest neighbour, we must apply the restriction ~al = ~a, as otherwise a soldierformation would emerge (assuming the corresponding stability conditions aresatisfied).

For a group of particles that follow only one nearest neighbour in itsfrontal visual field with ~al = ~a 6= ~0, many stress-free patterns can occurdepending on the initial conditions. The NND remains fixed in all thesepatterns since it is determined by the value of d that makes g+(d) = 0 (notethat g+(x) has only a single zero in our simulations). Two examples are shownin Fig. 2.10. For the g+(x) function chosen in these simulations, d = 0.707 inboth panels. Since interactions only occur between nearest neighbours, thisdistance does not change as the number of particles increases. The bearingangle between neighbours is sensitive to initial conditions, such that smallchanges in initial conditions can lead to large changes in the shape of thegroup that evolves. To generate a perfect school in 2D we could initializeparticles in a regular array, and this array would persist in time so longas it is not subject to noise. However, in both irregular and regular arrays,perturbations can alter the configuration after such groups have been formed.This is due to the fact that the magnitude of the interacting force is only

53


distance-dependent, and thus there exists a circular arc of positions around apreceding particle which its follower can occupy while remaining equidistantfrom its neighbour. Then upon undergoing a perturbation, a particle is freeto return to any point on this arc, and not necessarily to its original pointin the undisturbed array. We also note that velocity-dependent forces act toalign the group, but do not contribute to maintaining a fixed bearing angle.

Then the question arises as to what minimal mechanisms can generatea stable 2D perfect school solution. We have found that by increasing boththe number of neighbours sensed by each particle and the sensing region, wecan generate schools that are stable in numerical simulations.

Consider a group of particles, each interacting with two nearest neigh-bours. We no longer restrict interactions to the front of a particle, and in-stead assume individuals can sense neighbours in any direction. By removingthe restriction of sensing in the frontal region, we eliminate the edge issuesdiscussed earlier. We use a single force for all directions of interaction, givenby g(x) = 0.5(exp(−x/50) − 2 exp(−x/1)), as in earlier examples. Particlesare initialized in a hexagonal tiling, where each hexagon contains an individ-ual in the middle, such that the entire school can be broken into groups ofparticles forming equilateral triangles. This tiling is then perturbed, and thisperturbed tiling is taken as the initial condition. Such a system was foundin numerical simulations to evolve to a regular tiling of fixed interindividualdistance d that moves as a rigid body (see Fig. 2.12). Stability analysis forsuch a solution is quite difficult using analytical techniques introduced herefor two reasons: it is difficult to label individuals to generate structure instability matrices, and due to the geometry, individuals can change whichparticle they interact with as a result of a small perturbation. This latterfact actually serves to stabilize the solution by maintaining proper distancebetween individuals even when a direct interaction may not exist, as a per-turbation closer to a neighbour would cause interaction with this neighbourand consequent return to the initial relative position.

Simulations of 2D schools that yield a large variety of different schoolpatterns with different shapes and features have been carried out in thepresence of small noise, using various interaction schemes. Some groupsform perfect schools, while some exhibit variation at the boundary of theschool, both spatially and temporally. A detailed exploration of all of thesenumerically observed patterns lies outside the focus of this paper.

54


Figure 2.12: Numerical simulation of two nearest-neighbour interactions.Trajectories are plotted through time. Note that the particles, initiallyperturbed, evolve to a regular tiling of the plane with individual distanced = 0.707.

55

2.6. Discussion

2.6 Discussion

The formation of cohesive schools is a phenomenon that occurs in a numberof aggregating organisms. Among the most striking and common examplesare flocks of birds and schools of fish. These groups provide motivation for ahost of mathematical models which seek to lend insight into the behavioralalgorithms used to generate school patterns. In contrast to the Eulerianmodeling approach (see, e.g., [21, 22, 23]) which gives mean field proper-ties and is useful in modeling the global dynamics of large populations, weuse a Lagrangian approach. By modeling individual particles, we gain in-formation on spacing and heading that is lost in the Eulerian approach atthe expense of greatly increasing the dimension of the associated system ofODEs. However, in the case of some particular solutions, these large systemsof nonlinear ODEs permit analytical insights. In our study, we are concernedwith relatively basic interactions between particles, and geometrically simplesolutions that such interactions permit.

Our modeling approach takes as inputs the inherent behaviour of individ-uals, as well as a functional description of how they interact with one another.With these inputs, the model can then address [a], whether or not the schoolcan exist, [b], if it can exist, the spacing and school velocity that prevails atsteady state, [c], the geometry of the school; that is, how individuals alignwith other individuals, and how they move in relation to this alignment, and[d], whether or not a school solution is linearly stable under small perturba-tions. The simplicity of our approach allows us to directly connect featuresof existence, stability and school geometry to basic features of interactionfunctions and model parameters. Our results do not qualitatively depend onthe specific choice of the schooling functions g±(x) and h±(x), provided thesefunctions satisfy the criteria established in this paper.

In 1D, we studied schools formed by interactions to the front, and then,interactions both to the front and back. We obtained existence conditionsfor such schools to exist, and analytically obtained school velocity and indi-vidual distance d that prevail at the perfect school solution. We were able torelate how the slope of the “net schooling force function” g+ − g− near d de-termines the relationship between velocity and spacing in a school, and thisrelationship was shown in simulations. We then obtained stability conditionsin the case of interactions to the front, with the result that both h′+(0) + γand g′+(d) be positive to guarantee stability.

Using essentially the same analytical techniques as in 1D, we explored

56

2.6. Discussion

soldier formations of particles organized in a line, moving in some direc-tion non-axial to the line. We obtained existence conditions which providedanalytical descriptions of spacing, heading direction, school speed and thedirection of the line of individuals. We performed a stability analysis onsuch schools, with the result that both g+(d) and g′+(d) must be positive toguarantee stability. Following our analysis, we performed numerical simu-lations of the model in 2D, showing that the soldier formation appears tobe globally stable with respect to random initial conditions when both theexistence conditions and local stability conditions are satisfied, and ~al 6= ~a.In the case of ~al = ~a, we find no restriction on relative angle between neigh-bours, and individuals arrange themselves with constant distance, but withrelative angles dependent on initial conditions. We then explored situationsin which local stability conditions were not met, which leads to individualspairing off and each pair following an unpredictable trajectory. Lastly, wepresented a minimal mechanism to generate perfect schools in 2D such thatparticles form a regular array that moves at a constant speed. This solutionwas shown to be stable in numerical simulations using a typical interactionfunction.

We have shown analytically that one neighbour is sufficient to generatestable perfect schools in 1D, while two neighbours were shown numerically togenerate stable perfect schools. This leads to a conjecture that interactionswith three neighbours should suffice to generate stable perfect schools in 3D.Exploring such questions analytically and computationally forms the basisof our ongoing work.

Although in this paper, we have studied only simple arrangements of par-ticles, our approach clearly demonstrates how the properties of each aspectof the model influences the observed features of the school. Although wehave emphasized simplicity in formulating minimal mechanisms for schoolformation, our results are nevertheless general, in that we have avoided as-suming particular functional forms to obtain results. Also, the simplicity ofour approach underscores the considerable variation in the class of solutions,and the sensitivity of these solutions to how particles interact.

The study of the formation of social aggregates in self-propelled particleshas become increasingly active. A number of recent studies have revealed alarge number of interesting patterns in groups of particles coupled throughpairwise interactions [1, 5, 8, 11]. In these studies, all-to-all interactions areconsidered and the limiting behavior of the group as the number of parti-cles approaches infinity has been shown to be crucial in determining some

57

2.6. Discussion

important features of the aggregate patterns that can occur. The patternsthat emerge in these systems are often not regular and may even be meta-stable or transient states [1]. Results presented in this work represent a raredistinct attempt at focusing on minimal mechanisms that are sufficient forgenerating regular school patterns that are dynamically stable. We showedthat nearest-neighbour interactions are sufficient for generating regular schoolpatterns with some crucial characteristic features that are independent of thenumber of particles in the group. A detailed analysis of the similarity anddifference between the aggregate patterns in all-to-all coupled and nearest-neighbour coupled self-propelling particles could lead to experimental criteriathat allow us to check which one is closer to reality.

Acknowledgements This work was funded by NSERC discovery grantsto Y.X. Li and L. Edelstein-Keshet. R. Lukeman was funded by an NSERCgraduate scholarship, and by a UBC University Graduate Fellowship.

58

Bibliography



[3] A. Czirok, M. Vicsek, and T. Vicsek, Collective motion of organisms inthree dimensions, Physica A 264 (1999), 299–304.

[4] A. Czirok and T. Vicsek, Collective behaviour of interacting self-propelled particles, Physica A 281 (2000), 17–29.



[7] , The simulation of fish schools in comparison with experimentaldata, Ecological Modelling 75/76 (1994), 135–145.


[9] Y.-X. Li, Clustering in neural networks with heterogeneous and asym-metrical coupling strengths, Physica D 80 (3) (2003), 210–234.

[10] R. Lukeman, Y.-X. Li, and L. Edelstein-Keshet, A mathematical modelfor milling formations in biological aggregates, Bull. Math. Biol. 71(2),352–382.

59






[15] A. Okubo, Diffusion and ecological problems: Mathematical models,Springer Verlag, New York, 1980.

[16] A. Okubo, D. Grunbaum, and L. Edelstein-Keshet, The dynamics of an-imal grouping, Diffusion and Ecological Problems: Modern Perspectives(A Okubo and S Levin, eds.), Springer, N.Y., 2001.


[18] J. R. Silvester, Determinants of block matrices, Maths Gazette 84(2000), 460–467.

[19] A. R. E. Sinclair, The african buffalo : a study of resource limitation ofpopulations, University of Chicago Press, Chicago, 1977.


[21] J. Toner and Y. Tu, Long-range order in a two-dimensional xy model:how birds fly together, Phys. Rev. Lett. 75 (23) (1995), 4326–4329.


[23] C. Topaz, A. Bertozzi, and M. Lewis, A nonlocal continuum model forbiological aggregation, Bull. Math. Bio. 68 (7) (2006), 1601–1623.

60


[24] B. P. Uvarov, Locusts and grasshoppers, Imperial Bureau of Entomology,London, 1928.


[26] D. Weihs, Energetic advantages of burst swimming of fish, J. theor. Biol.48 (1974), 215–229.

61

Chapter 3

A Conceptual Model forMilling Formations inBiological Aggregates 2

3.1 Introduction

In biological aggregates, there are many examples of group-level patterns thatemerge from interactions among individuals. Such phenomena have moti-vated a number of modeling approaches aimed at connecting the interactionsof individuals to the emergent collective behaviours. Lagrangian models,based on tracking the positions and velocities of individuals contrast with theEulerian models, formulated in terms of reaction-transport equations. Exam-ples of the Lagrangian approach include [4, 8, 12, 14, 18, 19, 20, 28, 31, 33].In the Lagrangian framework, each individual is described by several ordi-nary differential equations (for components of velocity and position in 1-3spatial dimensions). This means that the complexity and size of the modelincreases proportionately to the size of the group, rendering such modelsnotoriously difficult to understand analytically. Due to this inherent chal-lenge, many previous Lagrangian models are largely dominated by simulationstudies. Interesting patterns are found and classified, but an open questionremains as to which aspects of a given model leads to which emergent featureof the group. Even when exhaustive search of parameter space is undertaken,linking classes of interaction functions to the types of patterns they form isa major challenge.

Motivated by the desire to obtain analytic results that characterize thelink between individual interactions and the patterns formed by groups, wehere explore a minimal model that has the advantage of analytic tractability.

2A version of this chapter has been published as ‘R. Lukeman, Y.-X. Li, and L.Edelstein-Keshet, 2009, A conceptual model for milling in biological aggregates, Bull.Math. Bio., 71(2), p 352-382

62

3.1. Introduction

The goals of our paper are to further investigate a framework that is simpleenough that we can unequivocally assess which properties of interaction func-tions, parameters, and rules lead to existence and stability of group patterns.Based on the challenges described above, such a task can only be done at thispoint on a restricted set of simple cases where we can exploit regularity ofthe pattern, symmetry properties, and limited coupling to achieve analyticalpower. Fortunately, simulations can be used to complement this analysis,and to expand insights to cases that cannot be solved in closed form.

Emergent patterns in nature can help to inform and validate models forgroup behaviour. One particular pattern that emerges in many of thesemodels, and occasionally in nature, is the milling formation (also known asa ‘vortex formation’), in which individuals move in roughly concentric trajec-tories. In some previous models, milling mechanisms have been engineeredin specific confined domains (due to interactions with boundaries), or by in-cluding “rotor chemotaxis” terms, representing a tendency to maintain someangle to the gradient of an attractive force-field. (We compare some previ-ous examples in our Discussion.) Since many competing models can producesimilar patterns, the emergence of a mill (or for that matter, any other pat-tern) can not be used to claim “correctness” of a given model. However, it isof interest to assess which attributes of a given model produce the resultingpattern. In this paper we explore what features of our minimal model wouldbe consistent with existence and stability of mill formations.

In order to gain insight into the existence and stability properties ofmilling formations, we take the following approach: we consider a spatiallysimple solution that still captures the essential behaviour of milling, givenby equally-spaced particles moving around a circle of fixed radius, at fixedangular velocity. These particles interact only via distance-dependent forcesof attraction and repulsion, neglecting velocity-dependent forces commonin similarly motivated models. We exploit the regularity of this idealizedmilling formation to address the following questions analytically: (i) Underwhat conditions on the school forces do such idealized formations occur (ex-istence)? (ii) Given their existence, what further conditions guarantee thatthese patterns are not destroyed by random perturbations (stability)? Underthis approach, we do not seek to realistically model the observed behavioursin nature (which are undoubtedly more complex), but instead seek to under-stand the connection between individual interaction and group stability viaan abstraction of the milling phenomenon. Focusing on a simpler patternwhile using a simple model leads to clear-cut results that can be derived

63

3.1. Introduction

analytically. This lends insight into the more complicated group structuresfound in nature.

Our approach is somewhat similar to [17] who consider similar formationsfrom a multi-agent control system perspective, and derive stable steeringcontrols for particles moving with fixed speed.

In a companion paper [15], a model for school formation was introducedbased on the Lagrangian viewpoint, that is, following individual particles,rather than densities of organisms. Equations of motion were formulatedbased on the Newtonian approach, i.e., describing changes in the velocitiesand positions of the particles under forces of propulsion and interaction. Thisself-propelled particle model was used to study perfect schools, which are con-figurations of particles in which the spacing is constant and identical, andwhere individuals in the group move at constant speed. One such perfectschool in two dimensions, the soldier formation, was studied in [15] usinglinear stability analysis. Analytic stability bounds were found in terms ofthe slope of the interaction force as a function of interindividual distance.Numerical simulations were used to validate analytic conditions obtained,and investigate more complicated perfect schools. In this paper, we inves-tigate milling formations using the same theoretical framework as describedabove.

Compared with linear aggregates [15], the dynamics of the milling for-mation are significantly more complicated. The dependence of stability onthe slope of the interaction function evaluated at interindividual distance isgiven in terms of a complex-coefficient quartic polynomial. We investigatesolutions to this equation numerically, classifying distinct behaviours in eachstability regime. The structure of the interaction matrix that determineshow individuals interact is crucial to the stability analysis; consequently,block circulant matrices and their properties enter heavily in our analysis.

In Section 2, we introduce the basic assumptions of the model and thedifferential equation system that describes the motion of the particles. Wedevelop existence conditions for the mill formation in Section 3, and stabil-ity conditions in Section 4. In Section 5, existence and stability conditionsare investigated numerically, through numerical solution to the eigenvalueequation, and through simulations. A discussion follows in Section 6.

64

3.1. Introduction

Figure 3.1: An example of milling in Atlantic bluefin tuna, courtesy of Dr.M. Lutcavage, Large Pelagics Research Center, UNH.

Milling Formations in Nature

The milling formation is most often observed in schooling fish. This phe-nomenon was described by A. E. Parr in 1927 [24]. Numerous species of fish,including jack, barracuda, and tuna have been observed to mill in nature [4].Photographs of jack milling in [25, 26] are presented as examples of emer-gent properties of fish schools. Milling tuna have been observed in the Gulfof Maine (see Figure 3.1), while barracuda mills abound in underwater pho-tography. Basking sharks have also been observed in mills during courtingbehaviour [10, 35].

Another example of milling is in army ants. In a thorough treatment,Schneirla [29] describes a rare field observation of Eciton praedator exhibitingmilling behavior following a rain-induced separation of a group of ants fromthe main columnar raiding swarm. Although the sensory mechanism andenvironments differ from fishes to ants, in both cases individuals turn towardsa dominant centripetal stimulation [29].

65


The above instances of milling in nature are self-organized in the sensethat the global pattern emerges solely from interactions among individuals,using only local information [2]. Importantly, the occurrence of such millsis relatively rare in many instances, as they emerge (for example in ants) inunusual or extreme cases. (Schneirla [29] reports that ants in such mills followone another to death by exhaustion.) Fortunately, given a wide variety ofinitial conditions, army ants form trunk trail foraging patterns, rather thanmills; but when a closed-loop topology occurs by chance, their mill is a stableand long-lasting movement pattern.

These observations raise an interesting question concerning the mecha-nisms that ensure the occurrence of self-organized milling patterns. The re-alistic mechanisms remain elusive since we know little about the actual rulesof how animals interact. However, a number of models have been proposedin which milling patterns that closely resemble the observed mills have beenshown to occur [8, 14]. Although experimental verifications are required totest whether these models are realistic, they provide insights into a numberof mechanisms that are capable of producing milling.

3.2 The Model of Interacting Self-Propelled

Particles

The following basic assumptions apply to the idealized schools that we studyin this paper.

(A1) All particles are identical and obey the same set of rules.

(A2) Particles are polarized, that is, each has a front and a back, and aparticle senses other particles only in front of it. (This is a simplificationof a classical assumption by [12].)

(A3) The “effective force” experienced by one particle from a given neighbourdepends only on distance.

(A4) A given particle interacts with its nearest neighbour(s). For analytictractability we consider interactions with at most one neighbor.

(A5) The distance-dependent force between a pair of neighbours is a vectorparallel to the line segment connecting the pair.

66


Consider a group of n self-propelled particles (i = 1, 2, . . . , n), with posi-tion ~xi and velocity ~vi. For simplicity, we assume that the body alignmentof the ith particle is identical to its direction of motion, i.e. the direction of~vi. The movement of the ith particle is governed by the classical Newton’slaw of motion with equations

~xi = ~vi, (3.1)

~vi = ~ai + ~fi − γ~vi, (3.2)

where x ≡ dx/dt denotes a derivative with respect to time and the mass ofeach individual is scaled so that mi = 1 for all particles. ~ai is an autonomousself-propulsion force generated by the ith particle that may depend on envi-ronmental influences and on the location of the particle in the school. ~fi isthe interaction or schooling force that moves the ith particle relative to itsneighbour. The term γ~vi is a damping force with a constant drag coefficient(γ > 0) which assures that the velocity is bounded. Equation (3.2) impliesthat if a particle stops propelling, its velocity decays to zero at the rate γ.Particles are modeled in 2-dimensional space, so for a group of n particles,there are 4n first-order equations.

In the following analysis, ~ai is a constant vector, identical for all particles.Based on the model assumptions (A3-A5), the schooling force is given by

~fi = g(|~xj − ~xi|)(~xj − ~xi)

|~xj − ~xi|, (3.3)

where the index j refers to the nearest neighbour detected by particle i andg(x) is a function of the absolute distance between particles (i.e., definedfor x > 0) that gives the magnitude of the interaction force. To produce anonzero spacing, g(x) should typically be positive for large x and negative forsmall x, indicative of short-range repulsion and long-range attraction. Thisfeature is universal in almost all models of aggregate formation. (See reviewof attraction-repulsion models in [18].)

3.2.1 Relationship to Previous Models

The model presented by Equations (3.1)–(3.2) is well-known. It is discussedin [22, 23] and attributed to schooling studied by [28] and [31].

67


We briefly compare to other models for self-propelled particles. For com-parison, we write (3.1)–(3.2) as

~xi = ~vi, (3.4)

~vi = ~faut + ~fint, (3.5)

where ~faut is the force that is generated autonomously by each individual and~fint that is generated due to the interaction with others. In [3, 8, 19, 20, 21],~faut = (α − β|~vi|2)~vi was studied. This model was originally obtained by

minimizing the specific energy cost of a swimming fish [34]. In [14], ~faut =α ~vi

|~vi|−β~vi (in the case of no velocity averaging). In both cases, particles tend

to a preferred characteristic speed in the absence of interaction (in the first

case, v =√

α/β, while in the second, v = α/β). In this paper, faut = −β~vi,and so in the absence of interaction, the speed of particles decays to zero.For the force of interaction, in [3, 8, 14], fint is chosen as an all-to-all coupleddistance-dependent Morse-type interaction. In our analysis, we leave theform of fint general, but choose a nearest-neighbour coupling over all-to-allcoupling for analytical treatment. We note that the idealized mill formationsconsidered here also occur when using the model equations of [3, 8, 14] underour nearest-neighbour interaction regime.

3.2.2 The Milling Formation

In a perfect mill, n particles move continuously around a circle of radiusr0 with constant angular velocity ω0. The particles are equally spaced atdistance d, labeled sequentially, with particle i sensing particle i + 1, andparticle n sensing particle 1. The sector angle defined by adjacent particlesin the mill is θ = 2π/n, where n is the number of particles (see Figure 3.2for labeling conventions).

To simplify the presentation of our analysis, we set ~ai = 0 for all i; thatis, particles experience no self-propelling force in the absence of interactions.Later, we discuss the case for nonzero ~ai. These assumptions lead to

d~xi

dt= ~vi, (3.6)

d~vi

dt= ~fi(~xi+1 − ~xi) − γ~vi, (3.7)

68

3.3. Existence Conditions

Figure 3.2: A schematic diagram of the milling formation. Black arrows indi-cate direction of motion (tangential to the circle), while grey arrows indicatedirection of schooling force.

for i = 1, . . . , n− 1, andd~xn

dt= ~vn, (3.8)

d~vn

dt= ~fn(~x1 − ~xn) − γ~vn. (3.9)

Equations (3.8)–(3.9) link the nth and 1st particles into a (periodic) ringformation.

3.3 Existence Conditions

The milling solution illustrated in Figure 3.2 represents a steady-state equidis-tant distribution of n particles on a ring of radius r0 when observed in theframe that rotates at constant angular velocity ω0. Solving (3.6)–(3.9) forsuch a steady state in the rotating frame yields r0, ω0, and interindividualdistance d that uniquely determines such a milling solution. Here, we use asimple force-balance argument to obtain the same values. First, decomposethe interaction force ~f into a component tangential to the circle, ~ft, and acentripetal component, denoted ~fc (see Figure 3.3). In a steady-state mill

69


Figure 3.3: A schematic diagram showing the decomposition of the interac-tion force into tangential and centripetal components.

solution, there is no linear acceleration, so tangential forces are in balance.Thus, for each particle,

~ft − γ~v = 0. (3.10)

The centripetal force that maintains the motion of a particle of mass m = 1around a circle of radius r0 at angular speed ω0 is

~fc = ~ac = ω20r0~uc,

where ~uc is a radially-directed unit vector. Because |~v| = r0ω0, we have

~fc =|~v|2r0~uc. (3.11)

Denote by φ and ψ the angles subtended to tangent and radius, respectively,as in Figure 3.4. Then θ = 2π/n, φ = θ/2 = π/n, and ψ = π/2−π/n. Usingthe law of cosines and simple trigonometry, we find that

d = 2r0 sin(

π

n

)

, (3.12)

and express force magnitudes as

|ft| = g(d) cos(

π

n

)

,

|fc| = g(d) sin(

π

n

)

.

70


Figure 3.4: A schematic diagram of particles in the mill formation showingangles introduced in the text, and relative position and velocity vectors.

Using (3.10) and (3.11),

g(d) cos(

π

n

)

= γ|~v|, (3.13)

g(d) sin(

π

n

)

=|~v|2r0, (3.14)

and solving for |~v| and r0 gives

|~v| =g(d) cos

(

πn

)

γ, r0 =

g(d) cos2(

πn

)

γ2 sin(

πn

) . (3.15)

Then angular frequency, ω0 is related to speed and mill radius, r0 by

ω0 = |~v|/r0 = γ tan(

π

n

)

. (3.16)

For given g(x), n and γ, (3.15) and (3.16) give the equilibrium radius andangular velocity of the milling formation. For an n-particle system, thesequantities completely characterize the milling solution. Combining (3.12)and (3.15), we obtain the existence condition

g(d) = sd, (3.17)

71

3.4. Stability Analysis

[a] [b]

Figure 3.5: Equation (3.17) represented graphically for γ = 0.5, n = 5,and g(x) = A exp(−x/a) − B exp(−x/b). Horizontal axis: inter-individualdistance d, vertical axis: distance-dependent force magnitude. A = 1.5,a = 10, B = 3. In panel [a], b = 1.5, and milling is possible whereas in panel[b], b = 2 and no milling solution exists.

where s = γ2/2 cos2(

πn

)

. A steady-state mill formation occurs if and only if

there exists a value of d satisfying (3.17) for a given function g(x). Inter-sections of g(d) and the straight line sd are mill formations. In Figure 3.5,intersections indicate potential steady-state values of d for which the millformation can exist in [a], while in [b] absence of intersections indicate thata steady-state mill formation is not possible. Significantly, the slope s of theright-hand side of (3.17) affects whether or not an intersection exists. Fortypical g(x), a shallower slope increases the likelihood of an intersection, sodecreasing the damping coefficient γ or increasing the number of individualsn increases the likelihood of existence of the milling state. The dependenceon n implies the possibility of a mill formation being destroyed when one ormore particles leaves the group. Figure 3.6 shows a plot of Equation (3.17)for various values of n. In this example, for fewer than 5 particles, a millsolution will not exist.

3.4 Stability Analysis

In this section, we investigate the local stability of the milling solution via alinear stability analysis. Our goal is to linearize (3.6)–(3.9) at the milling so-lution and study the eigenvalues that determine the stability of the solution.

72


Figure 3.6: A plot of the left-hand and right-hand sides of (3.17) for a rangeof n values, from n = 4 to n = 12. Note that no intersection occurs for n = 4,but two occur for n = 5.

Here we exploit the simplifications and assumptions of the model, withoutwhich stability matrices become unwieldy. The cyclic nature of particle in-teractions is reflected by a block-circulant structure in the stability matrix,which has the form

A1 A2 · · · An

An A1 · · · An−1...

A2 · · · An A1

,

where the Ai arem×m block matrices. Because an explicit formulation of thedeterminant is known for an mn×mn block-circulant matrix, we can exploitthis structure in calculating eigenvalues. However, here we encounter a chal-lenge: because ~xi+1−~xi terms appear in the stability matrix, and this vectorhas a different direction for each particle, successive rows of blocks are notidentical modulo a shift. To overcome this issue, we transform coordinates.Let

~xi = R(φi) (~xi+1 − ~xi) , (3.18)

~vi = R(φi) (~vi+1 − ~vi) , (3.19)

73


where

R(φi) =

[

cos(φi) sin(φi)− sin(φi) cos(φi)

]

, (3.20)

is a rotation matrix with angle

φi =2πi

n+ ω0t+

π

n+π

2. (3.21)

This transformation is best understood as a composition of a number oftransformations. First, position and velocity differences are taken to simplifythe system of equations. Second, these differences are rotated by the angleφi, transforming the system to a rotating frame. In this rotating frame,position and velocities appear to be fixed at the milling solution. Third, anindex-dependent rotation of 2πi/n is applied to each particle so that steady-state quantities are independent of index (required for the circulant structurein the stability matrix). Last, a constant rotation of π/n + π/2 is appliedso that the steady-state position and velocity vectors lie on the coordinateaxes. We note that applying these three rotations does not affect eigenvaluesbecause the linearized coefficient matrix and its analogue without rotationare similar matrices.

The transformed system of equations governing rotated position and ve-locity differences is

d

dt~xi = ω0k × ~xi + ~vi, (3.22)

d

dt~vi = ω0k × ~vi +R

(−2π

n

)

~xi+1

|~xi+1|g(

|~xi+1|)

− ~xi

|~xi|g(

|~xi|)

− γ~vi, (3.23)

and steady-state values are given by

~xs

i = ~xs=

[

d0

]

, ~vs

i = ~vs

=

[

0ω0d

]

, (3.24)

for i = 1, 2, . . . , n with n + 1 identified with 1 (see Appendix A for details).

The extra rotation R(

−2πn

)

in (3.23) is required to write this equation in

terms of the transformed coordinates. Note that both ~xs

i and ~vs

i are indepen-dent of the index i; i.e., in this coordinate frame, all particles have the same

74


steady-state coordinates ~xs

and ~vs

. Next we introduce perturbations to thesteady state milling solution in the transformed coordinate system; i.e.,

~xi = ~xs+ ~δi(t), (3.25)

~vi = ~vs+ ~ξi(t). (3.26)

We substitute (3.25)–(3.26) into (3.22)–(3.23) and linearize interaction termsin (3.23). Then, taking together equations for all i = 1, . . . , n, we can writethe full perturbed linear system in matrix notation. To facilitate this, weintroduce a number of 2 × 2 matrices. We let

Ω =

[

0 ω0

−ω0 0

]

, D =∂ ~f

∂~xi

|(d,0) =

[

g′(d) 00 g(d)/d

]

,

RD = R(−2π

n

)

[

g′(d) 00 g(d)/d

]

. (3.27)

The full 4n× 4n linear system is then

d

dt

~δ1~δ2...~δn~ξ1~ξ2...~ξn

=

Ω | IΩ | I

. . . | . . .

Ω | I−D RD | Ω − γI

−D RD | . . .. . . | . . .

RD −D | Ω − γI

~δ1~δ2...~δn~ξ1~ξ2...~ξn

,

(3.28)where I is the 2×2 identity matrix. The coefficient matrix is composed of fourn× n block matrices, three of which are block-diagonal, and one of which isblock-circulant. Each component is itself a 2×2 matrix. If we consider caseswith interactions with two or more nearest neighbors, the structure of suchmatrices becomes intractable using our techniques. For the full derivation of(3.28), see Appendix B. We note that ω0 is given explicitly in terms of n andγ in (3.16), and we can rearrange (3.17) to obtain

g(d)

d= s, (3.29)

75


i.e., expressing g(d)/d explicitly in terms of n and γ. Thus, we reduce theparameters in (3.28) to the drag coefficient γ, the number of particles n, andthe slope of the interaction function at steady state, g′(d).

3.4.1 Eigenvalues

We denote the coefficient matrix in (3.28) as C, and solve the eigenvalues ofC via

det (C − λI4n×4n) = 0,

where I4n×4n is the 4n× 4n identity matrix. Now we use the following resultfrom [30]: given the block matrix

[

A1,1 A1,2

A2,1 A2,2

]

,

where A1,1,A1,2,A2,1, and A2,2 arem×mmatrices, if A1,1 and A1,2 commute,then

det

[

A1,1 A1,2

A2,1 A2,2

]

= det [A2,2A1,1 −A2,1A1,2] . (3.30)

We partition C − λI4n×4n into four n × n block matrices and apply theabove theorem, noting that the corresponding upper left and upper rightsubmatrices commute, to obtain the reduced determinant equation:

det (C − λI4n×4n) = det

Λ + D −RD 0 · · · 00 Λ + D −RD 0 · · · 0

. . .

−RD 0 · · · 0 Λ + D

= 0,

(3.31)where 0 is the 2 × 2 zero matrix, and where we have defined

Λ ≡ (Ω − λI)(Ω − (γ + λ)I)

=

[

−λ ω0

−ω0 −λ

] [

−λ− γ ω0

−ω0 −λ− γ

]

=

[

λ(λ+ γ) − ω20 −γω0 − 2λω0

γω0 + 2λω0 λ(λ+ γ) − ω20

]

. (3.32)

Next, we use a diagonalization method for block-circulant matrices from [7].The diagonalization matrix is given by Fourier matrices; for details see [17]

76

3.5. Numerical Investigation

or [7]. The resulting n 2 × 2 diagonal blocks are given by

D1 = Λ + D + RD,

D2 = Λ + D +RD,

D3 = Λ + D +2RD,...

Dn = Λ + D +n−1RD,

where = exp (2πj/n) , where j =√−1. Note that the powers of give

the n roots of unity. Because the determinant of a block-diagonal matrix isthe product of the determinants of the individual blocks, solutions to (3.31)are solutions to

detD1 detD2 · · ·detDn = 0,

i.e., the collection of solutions to

detDi = 0, i = 1, . . . n. (3.33)

Expanding the (i + 1)th equation leads to a 4th-order complex-coefficientpolynomial

p(λ) = 0 (3.34)

to be solved for eigenvalues λ (see Appendix C). Explicit solutions to thisequation are not easily obtained. However, given parameter values for γ, n,and g′(d), roots can be found numerically.

We have thus far avoided the n = 3 case. Because all three particlesare equidistant in a mill formation, a particle can switch nearest neighboursunder a small perturbation, breaking the connection topology. Thus thethree-particle mill formation is inherently unstable to perturbations.

3.5 Numerical Investigation

We are interested in restrictions on the interaction function that guaranteestable mill solutions, i.e., such that linear stability theory points to eigen-values with negative real parts. Note that based on (3.33), the eigenvalueproblem involves the matrices Λ, R and DR. For fixed n and γ, these ma-trices depend only on g(d)/d and g′(d), and because at the milling steadystate, g(d)/d = s, eigenvalues depend only on g′(d). We thus numerically

77


investigate stability with respect to g′(d) by treating this quantity as a bifur-cation parameter, with the remaining parameters, n and γ, fixed. In Figures3.7 - 3.9, we plot the real part of λ, Re(λ), for each of the four solutions toEquation (3.34), for each of i = 0, . . . , n − 1, for a total of 4n eigenvalues.We note that for all values of n that were investigated, three eigenvalueswith Re(λ) = 0 existed: a zero eigenvalue, associated with the rotationalsymmetry, and a pair of purely imaginary conjugates, associated with thetranslational invariance of the center of the mill within a rotating frame.

For Figure 3.7, n = 4. There is a range of values of g′(d) in [-0.26,0.25]for which the real part of all eigenvalues are nonpositive, consistent withstability. In Figures (3.8) and (3.9), n = 5 and n = 6, respectively. Incontrast to the previous case, the stable range of g′(d) shrinks successively,from [−0.11, 0.19] for n = 5, to [−0.08, 0.17] for n = 6. These results suggestthat the mill solution is unstable if the slope of the interaction function atd is either too large or too small (i.e., too steep). Stable mills exist wheng′(d) < 0, evidently due to the circular geometry, as linear formations studiedin [15] required that g′(d) > 0.

In each of these three cases, the upper limit of the stability region coin-cides with the value of g(d)/d at steady state for each n, that is,

g′(d) < s, (3.35)

for stable mills. (Specifically, for γ = 0.5, and n = 4, 5, 6, we get g(d)/d =0.25, 0.191, and 0.166 respectively). In each of these cases, the stabilityboundary is determined by a branch of eigenvalues that crosses from nega-tive to positive real parts at the boundary value of g′(d). It can be shownanalytically that a branch of Re(λ) crosses over the g′(d) axis at g(d)/d byexamining coefficients of p(λ) when g′(d) = g(d)/d. However, it remains tobe proven that this crossover value always represents the upper-bound of thestability region. As n increases, we have found numerically that the lowerbound on the stability region approaches 0.

3.5.1 Numerical Simulations

A self-organized pattern in a dynamical system refers to a structurally stablesolution that occurs as a result of interactions between individuals in a group[8, 14, 15] and does not occur as a result of forcing constraints asserted byother groups (e.g. predators) and/or the environment [9, 11]. Under the same

78


Figure 3.7: Numerical solution to (3.34) over a range of values of g′(d) forfour particles (n = 4) and γ = 0.5. The real parts of eigenvalues are plottedfor i = 0, . . . , 3. The asterisks indicate branches of Re(λ) with multiplicityequal to 2. Note that the region over which all eigenvalues have non-positivereal parts is approximately [-0.26,0.25].

79


Figure 3.8: As in Figure 3.7, but for five particles (n = 5). The real part ofeigenvalues are plotted for i = 0, . . . , 4. The region over which all eigenvalueshave non-positive real parts is now approximately [-0.11,0.19].

80


Figure 3.9: As in Figure 3.7, but for six particles (n = 6). The real part ofeigenvalues are plotted for i = 0, . . . , 5. The region over which all eigenvalueshave non-positive real parts is now approximately [-0.08,0.17].

81


interaction rules and model parameters, a number of stable self-organizedpatterns can occur and coexist. The emergence of one particular patternis closely related to the initial state or configuration of the group. Largedisturbances can also induce a switch from one pattern to another coexistingone. Mills that we studied in this model are obtained in simulations onlywhen rather specific initial conditions were used, whereas most random initialconditions lead to other patterns. Therefore, they are self-organized patternsthat occur under stringent initial configurations. This is consistent withthe fact that milling is not the most commonly observed pattern in animalaggregates (as is the case in the example of milling army ants).

The particular restriction on initial conditions required for the emergenceof a milling solution in (3.1)–(3.2) is that the nearest-neighbour connectiontopology of the group is a simple closed curve. To ensure this geometry,we chose initial positions to be equidistant individuals around a circle offixed radius r, and then perturbed these (x, y) coordinates by values chosenrandomly in [−r/4, r/4]. Initial velocities are tangential to the circle of ra-dius r, with magnitude ω0r. We point out that once this closed connectiontopology (with arbitrary geometry) is established in our simulations, the cir-cular geometry that emerges is due only to local interactions among particles,without reference to any global information, and thus is self-organized [2].

The model equations (3.1)–(3.2) are evolved using ODE45 in MATLAB.Figure 3.10 compares simulations of n = 6 particles in which Equation

(3.17) has, versus does not have solutions. In Figure 3.10.a1, the left-handand right-hand side of (3.17) are plotted, and intersections indicate individ-ual distances d such that a mill formation can exist. However, our stabilityanalysis indicates that the first intersection corresponds to an unstable solu-tion because g′(d) > s. At the second intersection, the slope g′(d) is withinthe range of stability, and as confirmed by Figure 3.10.a2, particles form amill with spacing d ≈ 2, and r ≈ 1.9.

For the interaction function in Figure 3.10.b, no solution exists for (3.17).Corresponding simulations (Figure 3.10.b.ii) show particles initially forminga mill-like pattern; however, the radius of the group decreases until the indi-vidual distance d is such that g(d) = 0. At this stage, no further interactionsoccur, and the particles stop moving. We do not technically consider suchstationary circular patterns to be “mills”. In the case that there is no zeroof the interaction function g, the group collapses to a point.

We now investigate the behaviour of the system at the boundary of stabil-ity, and inside the region of instability. Earlier, we showed numerically that

82


Figure 3.10: A plot of the condition for mill solutions to exist [a1] or not [b1]given by Equation (3.17), and corresponding simulated particle tracks [a2]and [b2], with γ = 0.5. Note that in the case of an intersection, a stable millforms at d ≈ 2 (whereas d ≈ 0.5 is unstable). However, when no intersectionexists, the mill stops rotating and stationary particles are then arranged withd such that g(d) = 0 (final direction is outward due to small repulsive forcesfelt just before the particles stop). Arrows indicate direction facing at theend of the simulation. The interaction function used is as in Figure 3.5 withA = 0.5, a = 5, B = 1, and [a] b = 0.5, [b], b = 0.8.

83


the system is stable over a region of g′(d) which includes g′(d) = 0. Thus,choosing an interaction function whose slope at d is either too positive, ortoo negative, will result in instability in the mill solution. Just how thisinstability manifests itself can be seen in simulations. Figure 3.11.a1 showsan interaction function with two intersections indicating possible individualdistances. However, at the first intersection, g′(d) is too positive and Equa-tion (3.35) is not satisfied (as indicated by the straight line), while at thesecond, g′(d) ≈ −0.10 is too negative. Both values are outside the stabilityregion [-0.07,0.17] noted in Figure 3.9 for n = 6. The corresponding sim-ulation in Figure 3.11.a2 shows particles initially forming a circular group,though oscillations in individual distance grow until the connection topologyis altered and the mill is broken. If g(d) is linear with slope greater thans, we obtain a single intersection as in Figure 3.11.b1. In this case, simula-tions (Figure 3.11.b2) show a circular group forming, though the radius ofthe group grows exponentially in time, again confirming instability. Thusthere are a number of ways in which the stability of the mill formation islost, including radial increase in time, evolution to a stationary group, andbreaking of the connection topology.

The system has yet a different behaviour with g′(d) chosen to lie on thestability boundary. In this case, oscillations in individual distance d develop(as in the connection-breaking instability), but the oscillation magnitude ap-proaches a fixed value and the system evolves to a new steady-state solution,the irregular periodic mill. Despite the periodic oscillation of distance be-tween particles, the sum of the n distances between neighbouring particlesis fixed when the irregular periodic mill is established. A plot of d in timeshowing this periodic behaviour is shown for one particle in an n = 5 systemin Figure 3.12.a, while snapshots of the particles in time are shown in Figure3.12.b.

3.5.2 Moving Mill Formations

In our treatment of mill formations up to this point, we have assumed that~a = 0; i.e., individuals are compelled to move only based on interactionswith neighbours, and have no intrinsic desire to move in any direction. Inthis section, we explore the effect of assigning a non-zero autonomous self-propulsion to each individual.

In Figure 3.13, six particles initialized as usual, each with ~a = (0.1, 0.1)T ,form a mill that moves in the direction of ~a. Thus, autonomous self-propulsion

84


Figure 3.11: [a1] An interaction function with g′(d) lying outside the stabilityregion for both intersections (implying no stable mill formation). Simulationsusing this function are shown in [a2]. Note that particles initially form a mill-like solution which is destroyed as time evolves. [b1] An interaction functionwith only one intersection, whose slope is too large (g′(x) > s for all x).Simulations in [b2] show that the radius of the mill increases in time.

85


Figure 3.12: The irregular periodic mill formation. [a] Individual distance dversus time for one particle in a system of 5 particles. For the 4 times denotedin [a], snapshots of particles are shown in [b], showing the variation in dbetween particles 1 and 2. The interaction function used was g(x) = 1−0.11x,so that g′(d) = −0.11 lies on the stability boundary.

86

3.6. Discussion

in each individual acts to propel the entire mill along the direction of this in-dividual propulsion. However, because the velocity of particles is now jointlycomposed of autonomous self-propulsion and interaction forces, the mill for-mation can be broken if ~a dominates the motion of individuals. Mill breakingoccurs if some particle i no longer senses particle i+ 1; i.e.,

~vsi ·(

~xsi+1 − ~xs

i

)

≤ 0. (3.36)

Substituting known steady-state quantities in (3.36) and using some calculus,the mill-breaking condition reduces to

|~a| ≥ g(d) cos2(

π

n

)

, (3.37)

(see Appendix D for derivation). In Figure 3.14.a, we set ~a = (0.223, 0.446)T ,

so that |~a| = 0.498 < g(d) cos2(

πn

)

= 0.5, and a stable moving mill is formed.

However, in Figure 3.14.b, a small increment in ~a to (0.224, 0.448)T gives|~a| = 0.501 > 0.5, resulting in a fundamental shift in behaviour: the millformation breaks, and particles form a polarized group with a shift to d ≈ 2(see the inset of Figure 3.14.b). This new solution is consistent with thosestudied in [15]. In this example, the increase in the parameter ~a initiated achange in the connection topology of the particles, which in turn instigatedan evolution to a different solution. Indeed, for any given set of parameters,there are a number of types of solutions that can occur; it is the arrangementof particles and headings that determines the solution to which the systemconverges.

3.6 Discussion

Milling formations exist in a number of biological aggregates, and providefascinating examples of self-organized spatial patterns evolving from localinteractions among individuals. Such patterns found occasionally in natureare irregular owing to heterogeneity in the population and environment, andlikely more complex interactions. To reveal the cause-and-effect relationshipbetween interaction properties and the patterns that emerge, we focusedon idealized mill formations of particles rotating in a circular path of fixedradius. Although much simpler than the natural counterpart, this patternpermits mathematical analysis, which enabled us to make clear predictions

87

3.6. Discussion

Figure 3.13: A plot of trajectories in time for six particles with ~a = (0.1, 0.1)T .Note that the entire mill moves in the direction of ~a.

of how model parameters influence the existence and stability of these pat-terns. Interactions in this model are distance-dependent, and so, likely, donot completely characterize the interactions in natural mill formations. Yet,a detailed understanding of distance-dependent interactions should providebuilding blocks for understanding models incorporating more complex inter-actions.

Using a Lagrangian model based on Newton’s equations of motion, wedescribed mill formations in terms of group radius and angular velocity, andderived the following necessary condition for existence of a mill formationwith spacing distance d, where γ is the drag coefficient, and n is the numberof individuals:

g(d) = sd,

where s = γ2/2 cos2 (π/n). Increasing n, and/or decreasing γ increases thelikelihood that the existence condition is satisfied.

A linear stability analysis on the milling formation resulted in block-circulant matrices that were block-diagonalized by Fourier matrices. Thisanalysis could be done only in a limited, and particularly simple set of cases.

88

3.6. Discussion

Figure 3.14: A plot of trajectories in time for five particles with [a] ~a =(0.223, 0.446)T , and [b] ~a = (0.224, 0.448)T . Note that for a small parameterchange, the system behaviour is fundamentally different. In these simula-tions, g(x) is as in Figure 3.5, with A = 1.5, a = 10, B = 3, and b = 1.6.The corresponding existence condition is shown in the inset to [b].

89

3.6. Discussion

Notably, the assumption of interactions with a single nearest neighbor wasessential to limit the bandwidth of these matrices. The resulting determi-nant equation for eigenvalues yields a system of n complex-coefficient quarticpolynomials, whose combined 4n solutions gives the spectrum of the stabil-ity matrix. Through numerical solution of these equations, stability boundswere found for particular cases of n and γ.

The geometry of the mill formation differs from soldier formations stud-ied in [15] only through the nth particle being connected to the first particle.Yet, remarkably, this connection adds considerable complexity to the issue ofstability of the system. The most interesting stability dependence was seen inthe slope of the interaction function at steady-state, g′(d). A series of rangeswere computed numerically for fixed γ and various values of n. We foundnumerically that as n increases, the region of stability decreases, asymptoticto [0, γ2/2]. Interestingly, we found that increasing γ (or decreasing n) in-creases the stability region, but decreases the likelihood of a mill formationexisting.

Unlike linear schools studied in [15], stable solutions can exist wheng′(d) < 0. This is nonintuitive, as g′(d) < 0 means that a given particlei has a weaker attraction as it moves away from its neighbour to the front,i + 1. This would seemingly inhibit restoring the steady-state milling so-lution. However, the circulant connection of particles must compensate forthis weaker attraction by inducing the particle to the back, i− 1, to be morestrongly attracted to particle i, and as this stronger “pull” propagates suc-cessively back around the mill, the attraction of particle i to i+1 is increased,imparting a stabilizing effect.

Numerical simulations validated our calculation of stability regions. Nu-merous routes to instability can occur, including evolution to a motionlesssteady-state, unbounded increase in radius, and breaking the circulant con-nection topology. Simulations of the system at the boundary of stable andunstable regions of g′(d) indicated the existence a new type of solution, theirregular periodic mill, featuring a stable limit cycle in interindividual dis-tances, resulting in a rotating, irregular, dynamic polygon of individuals withside lengths oscillating in time. Inclusion of an autonomous self-propulsionterm, ~a, for each particle was shown to move the entire mill in the directionof ~a, until |~a| ≥ g(d) cos2

(

πn

)

, beyond which milling formations evolved to

polarized groups similar to those studied in [15].The model used in this paper is similar to other established self-propelled

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3.6. Discussion

particle models as noted earlier. Slightly different interaction forces wereused by some studies [3, 8, 14, 19, 20, 21]. In some of these particles tendto a preferred characteristic speed in the absence of interaction. As our re-sults are analytical and not just numeric, we could consider a more generalinteraction function fint than adopted in other papers. However, this comesat the expense of restricting attention to (single) nearest-neighbour couplingover all-to-all coupling. Furthermore, our particles sense others only in theforward direction, an assumption which can be thought of as a generalizationof the ‘blind zone’ assumptions of fish models (e.g., [4, 12]). Our choice ofa 180 blind zone could be reduced while still maintaining our mill stabilityresults, so long as the blind zone width is sufficient to prevent a perturbedindividual at the back being sensed, breaking the connection topology es-sential for maintaining the mill. Both the all-to-all coupling and our choiceof coupling is likely unrealistic for most naturally occurring mills, yet bothframeworks lend insight into the phenomenon.

As outlined in the introduction, many other competing models can giverise to mills. Milling formations emerge in these models through a number ofmechanisms, and an interesting comparison can be made between our millsand those produced by others types of models. In [1, 6], an individual-basedmodel for collective motion of bacteria is used to generate ‘vortices’ via arotor chemotaxis term in the equations of motion. Including such terms inindividual-based ODE’s explicitly enforces rotation: particles moving tan-gentially to a unimodal attractant source distribution or central force fieldwould exhibit rotational motion. Milling can be generated via an exter-nal environmental gradient, as is the case in the individual-based model ofswarming Daphnia in [16]. Milling can also emerge in models where particleshave a non-zero equilibrium speed in a closed domain; such is the case inthe coupled lattice map model of [11] and the spring-dashpot model used in[9]. Meanwhile, several models have exhibited milling solutions using onlyinteractions among individuals; this can occur in all-to-all coupled distance-and velocity-dependent interactions [14], or distance-dependent interactionsonly [3, 14]. In [4], interactions between individuals are restricted to localized‘zones’ of attraction, repulsion and alignment, and yet mill formations emergein certain parameter regimes. From a continuum perspective, in [5, 13, 32]population density models have vortex-type formations as solutions.

The milling formations that emerge in the individual-based models dis-cussed above are often qualitatively similar to such formations observed innature. In [8], conditions are derived that determine the H-stability of milling

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3.6. Discussion

solutions (that is, whether or not the group will collapse to a point as moreand more individuals are added). Also, milling formations have been stableto small amounts of noise in numerous studies. However, the analytical de-termination of conditions under which such formations exist and are stableto perturbations of its members remains unanswered due to the spatial com-plexity of these formations. Passing to the continuum limit often facilitatesanalysis by a reduction in order of the system, but this occurs at the costof individual properties in favor of mean field properties. In [27], a linearstability analysis was performed on a vortex-type solution of a continuummodel, but was inconclusive.

It is not clear if any functional benefit is associated with milling forma-tions in nature, or if these are epiphenomena that result from aggregation inspecial circumstances. In any case, our results on the relationship betweenmodel parameters and observed formations may prove useful in furtheringthe qualitative understanding of these formations. Furthermore, our resultsare useful for the design of artificial schools (i.e., multi-agent systems) wheresuch rotational coordinated motion is desired, as our modeling frameworkand stability results provide a simple and robust method of programmingsuch agents.

Future work will include obtaining analytical stability bounds on g′(d),which have been presented numerically here. An interesting extension of thiswork would be to investigate the effect of a velocity-dependent interactionin addition to the distance-dependent interactions studied here. Such an ex-tension would result in a more realistic model for natural groups. As oneconsiders more complicated formations of particles, the issue of labeling par-ticles and determining their interaction topology may present a considerablechallenge.

92

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[1] E. Ben-Jacob, I. Cohen, A. Czirok, T. Vicsek, and D. Gutnick, Chemo-modulation of cellular movement, collective formation of vortices byswarming bacteria, and colonial development, Physica A 238 (1997),181–197.

[2] S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz,and E. Bonabeau, Self-organization in biological systems, Princeton Uni-versity Press, Princeton, 2001.



[5] Z. Csahok and A. Czirok, Hydrodynamics of bacterial motion, PhysicaA 243 (2008), 304–318.

[6] A. Czirok, E. Ben-Jacob, I. Cohen, and T. Vicsek, Formation of complexbacterial colonies via self-generated vortices, Phys. Rev. E 54 2 (1996),1792–1801.

[7] P. J. Davis, Circulant matrices, Wiley, New York, 1979.


[9] D. Grossman, I. S. Aranson, and E. Ben-Jacob, Emergence of agentswarm migration and vortex formation through inelastic collisions, NewJournal of Physics 023036 (2008).

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[10] C. J. Harvey-Clark, W. T. Stobo, E. Helle, and M Mattson, Putativemating behavior in basking sharks off the Nova Scotia coast, Copeia No.3 (1999), 780–782.

[11] J. Hemmingsson, Modellization of self-propelling particles with a coupledmap lattice model, J. Phys. A 28 (1995), 4245–4250.


[13] V. L. Kulinskii, V. I. Ratushnaya, A. V. Zvelindovsky, and D. Bedeaux,Hydrodynamic model for a system of self-propelling particles with con-servative kinematic constraints, Europhys. Lett. 71 (2) (2005), 207–213.


[15] Y.-X. Li, R. Lukeman, and L. Edelstein-Keshet, Minimal mechanismsfor school formation in self-propelled particles, Phys. D. (2008).

[16] R. Mach and F. Schweitzer, Modeling vortex swarming in daphnia, Bull.Math. Bio. 69(2) (2007), 539–562.

[17] J. A. Marshall, M. E. Broucke, and B. A. Francis, Formations of vehiclesin cyclic pursuit, Automatic Control, IEEE Transactions on 49 (11)(2004), 1963–1974.





[22] A. Okubo, Diffusion and ecological problems: Mathematical models,Springer Verlag, New York, 1980.

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[23] A. Okubo, D. Grunbaum, and L. Edelstein-Keshet, The dynamics of an-imal grouping, Diffusion and Ecological Problems: Modern Perspectives(A Okubo and S Levin, eds.), Springer, N.Y., 2001.

[24] A. E. Parr, A contribution to the theoretical analysis of the schoolingbehaviour of fishes, Occasional Papers of the Bingham OceanographicCollection 1 (1927), 1–32.

[25] J. Parrish and L. Edelstein-Keshet, Complexity, pattern, and evolution-ary trade-offs in animal aggregation, Science 284 (1999), 99–101.

[26] J. Parrish, S. Viscido, and D. Grunbaum, Self-organized fish schools: anexamination of emergent properties, Biol. Bull. 202 (2002), 296–305.

[27] V. I. Ratushnaya, D. Bedeaux, V. L. Kulinskii, and A. V. Zvelindovsky,Stability properties of the collective stationary motion of self-propellingparticles with conservative kinematic constraints, J. Phys. A 40 (2007),2573–2581.


[29] T. C. Schneirla, A unique case of circular milling in ants, considered inrelation to trail following and the general problem of orientation, Amer-ican Museum Novitates 1253 (1944), 1–25.

[30] J. R. Silvester, Determinants of block matrices, Maths Gazette 84(2000), 460–467.




[34] D. Weihs, Energetic advantages of burst swimming of fish, J. theor. Biol.48 (1974), 215–229.

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[35] S. G. Wilson, Basking sharks schooling in the southern Gulf of Maine,Fish. Oceanogr. 13 (2004).

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Chapter 4

A Field Study of CollectiveBehaviour in Surf Scoters:Empirical Methods 3

4.1 Introduction

Understanding mechanisms governing collective motion in animal groups re-quires that empirical data be collected. Empirical data provides informationagainst which model hypotheses can be tested. Unfortunately, data on mov-ing animal groups is quite scarce due to a host of difficulties associated withgathering such data in a reliable manner. My goal in this part of the researchis to obtain real field observations of animal groups, and infer what are therules that individuals follow, and what are effective interaction forces thatgovern their associations. To obtain this data, I develop an experimentalmethod to gather position and velocity data on flocks of surf scoters swim-ming collectively. Data is gathered by photographing flocks from an overheadposition in time series. The methods described in this chapter yield a dataset that offers dynamic trajectory data of individuals within groups almostan order of magnitude larger than in previous work.

4.1.1 Difficulties in Obtaining Data

Obtaining empirical data in the field is complicated by the typically three-dimensional nature of animal aggregations (resulting in occlusion of interiorindividuals), the difficulties of calibrating measurement equipment at vary-ing locations, high speeds of movement, and the transient behaviour (bothin space and time) of animals in the field in general. Overcoming these diffi-culties by moving to a controlled laboratory setting places restrictions on the

3A version of this chapter will be submitted for publication as ‘A field study of collectivemotion in surf scoters’, R. Lukeman, Y.-X. Li and L. Edelstein-Keshet.

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4.1. Introduction

spatial extent and size of groups, and (depending on the species and experi-mental setup) can create difficulty in obtaining natural behaviours within anartificial environment. Furthermore, for many species (e.g., flocking birds),a laboratory setting is not feasible.

4.1.2 Previous Work

Nevertheless, significant efforts have been made to record positions and move-ments of individuals within groups. To reconstruct three dimensional (3D)positions of individuals, researchers have used three primary methods: stereophotography [1, 2, 3, 5, 8], the shadow method, [9, 10, 11, 12] and orthogo-nal photography [6, 7, 13]. Stereo photography uses two parallel cameras,mounted on a beam with some fixed distance of separation. Parallax mea-surements from photographs then allow 3D positions to be reconstructed.The shadow method tracks both the position of individuals, and also theshadow of individuals resulting from shining a light on the group. By follow-ing both the size and position of the shadow in reference to the position of thecamera and animal, 3D positions can be calculated. The orthogonal methodis similar to stereo photography, except cameras are aligned orthogonally asopposed to parallel, and 3D positions are reconstructed via the geometry ofthe two cameras and positions in images.

4.1.3 Tracking

In order to obtain dynamic information on animal group motion, individualsmust be tracked in time; that is, an individual in one frame must be linkedto the same individual in the next frame, and so on, to reconstruct thetrajectory for each group member. The ability to track individuals within agroup becomes more challenging as group size increases. Thus, there havebeen a number of empirical studies on small groups (2-30 individuals) withtracking [6, 9, 10, 12, 13, 15], and a number of studies of larger groups withoutany tracking in time ([8], 25-76 individuals, [3],21-61 individuals, [2], morethan 1000 individuals). In contrast, the data set described here consists oftrajectory data for all individuals within groups of a few hundred individuals.

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4.2. Methods

4.1.4 Challenges Avoided in This Study

This study avoids a number of the challenges inherent in empirical studies ofanimal groups. First, individuals moved on the surface of the water, whichis essentially two-dimensional. There was therefore no need for multi-cameramethods for reconstructing positions. It should be noted that unlike someother studies in two dimensions, individuals were not artificially restrictedas such; here it was an artefact of the environment in which they naturallymove. Second, individuals swimming at the surface moved at relatively slowspeeds as compared to, for example, flying birds or swimming fish. Lowspeeds of movement reduce temporal resolution required for tracking, allow-ing individuals to be successfully tracked using frame rates typically avail-able on consumer cameras. Third, the well-defined spacing of the group, andthe high contrast provided by individuals against the water surface, permitmostly automatic detection of individuals in images. Last, the surf scotergroups in this study were located in an area that provided both a convenientoverhead location to photograph the group without disturbing their naturalbehaviour, and also features that facilitated accurate image calibration for arelatively large field area.

4.2 Methods

4.2.1 Location and Materials

Data was gathered March 1-12, 2008 in waters adjacent to the Vancouver,BC waterfront, in Burrard Inlet. Overwintering surf scoters were observedforaging near dock pilings around Canada Place, a commercial facility. Anelevated promenade around Canada Place provided overhead locations tophotograph surf scoter groups swimming on the water surface. A Nikon D70sDSLR camera was used, attached to a Manfrotto 190XPROB tripod. Imageswere taken in continuous mode, fired at 3 frames per second to generate atime series of surf scoter movements. A Nikon AF-S Nikkor 18-70mm ED lenswas used, fixed at the maximal focal length (70mm). Although the NikonD70s has a maximum resolution of 2000 x 3008 pixels, images were taken inreduced size (1000 x 1504 pixels) to ensure accurate frame rates in continuousmode, and to reduce computation times in data analysis. Aperture was fixedat f4.5, shutter speed varied to optimize exposure (between 1/8000 sec. and1/250 sec.), and auto-focus was used.

99

4.2. Methods

4.2.2 Surf scoter behaviour in field study

Surf scoters are known to collectively forage primarily for mollusks (e.g.,mussels, clams) [14] via synchronous underwater dives. In the field location ofthis study, overwintering scoters gathered in groups of approximately 200-400individuals in open water. Upon initiating a dive sequence, the group swamon the water surface from open water towards the dock area surroundingCanada Place, presumably to forage on mussels on or around the dock pilings.After this period of movement to the dock area, individuals then dove in ahighly synchronous manner to forage. After a period of foraging underwater,individuals re-emerged at the water surface, again in a highly synchronousmanner, and returned back to open water. This process repeated continuallythroughout the observation period.

4.2.3 Experimental Technique

Because the goal of data collection was to track positions of individualsthrough time, it was necessary to fix the camera elevation, angle, and imageframe during data collection events. To this end, the camera was positionedso that the bottom of the viewfinder image aligned with the outer edge ofthe dock (see Fig. 4.1). Because the boardwalk elevation and dock widthwere fixed in the region of data collection, the camera elevation and anglewere fixed, and easily measured. Furthermore, this set-up allowed the cam-era apparatus to be moved along the boardwalk for a given sequence (butnot during), so as to increase the likelihood of encountering individuals inthe image frame. Once a group of surf scoters began moving in for a foragingevent, the camera was positioned as described above, and upon encounteringindividuals in the image frame, the camera was fired continuously at 3 fpsuntil individuals had completely left the frame, either through a dive, or byswimming outside the image region. A similar technique was used to capturedive-return movements, although these events were captured much less fre-quently due to the difficulty of estimating the location of re-emergence froma dive, as compared to a group moving in on the surface.

4.2.4 Calibration and Testing

Ideally, aerial images are taken directly overhead of the object; in this study,the camera axis was not vertical, but instead approached the water surface at

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4.2. Methods

water

dock

camera

17.5m

9.0m

θ φ

Figure 4.1: A schematic diagram of the experimental setup. θ is the cameraaxis angle, while φ is the angle-of-view of the camera lens.

an angle. In this type of photography (oblique aerial photography), positionswithin the camera frame must be transformed to real-world positions usingknown measurements of the apparatus setup and camera specifications. Ad-ditionally, commercial cameras (such as the one used in this study) are notdesigned for precise data collection, and thus, manufacturer’s specificationsmust be tested for accuracy [4].

Fig. 4.1 gives a schematic diagram of the setup. The camera was ata height of 17.5 m, and the dock was 9 m in width. Because the bottomof the image frame is aligned with the outer dock edge, these two distancemeasurements allow us to easily calculate the camera axis angle:

θ = arctan(

9.0

17.5

)

= 0.475 rad = 27.2o.

The angle-of-view, φ can be calculated from the lens focal length f and the

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4.2. Methods

vertical dimension of the image sensor, dv as

2 arctan

(

dv

2f

)

.

The Nikon D70s has an image sensor of 23.7 x 15.6 mm, and the lens usedhad a focal length of 70 mm, and thus, we calculate

2 arctan(

15.6 mm

140 mm

)

= 0.222 rad = 12.7o.

However, because the true focal length of a consumer lens can vary fromthe specified length [4], the angle-of-view was also measured for the cameraand lens used in this study by photographing known lengths from a varietyof camera axis angles. These measurements gave a true angle-of-view of0.235 rad = 13.5o. In our image post-processing, the bottom 28 pixels (of1000) are removed to eliminate the edge of the dock that may appear in theimage due to imperfect alignment in the field. Thus, the image size is now1504 x 972 pixels, and the angle-of-view is φ = 0.972 · 0.235 = 0.229 rad (or13.1o).

To convert positions from an image frame in our setup to real-world po-sitions, two transformations must be made: a vertical transformation dueto the camera axis angle, and a horizontal transformation due to projectiveperspective distortion: lines that are parallel in reality will converge in animage taken at some non-zero camera axis angle. Denote as φ the angle cor-responding to real distance y, according to Fig. 4.2. In the image frame, asφ ranges through [0, φ], pixels range (linearly) through [0, 972]. Using simpletrigonometry, real vertical position y in pixels can be recovered by

y =972

φ

(

tan

(

θ +pφ

972

)

− tan(θ)

)

, (4.1)

where p is the vertical pixel in the image; i.e, p = 972φ/φ. Substitutingp = 972 into the above formula gives L, the real vertical extent of the image(in pixels) as L = 1421.

Images taken in the field in this study differ from directly overhead imagesonly by rotation in one axis. Borrowing from flight dynamics, the pitch ofthe camera is changed, while the roll and yaw is fixed at 0 (controlled by thetripod head). As a result, the horizontal perspective distortion in the imagesis symmetric about the horizontal centre of the image, where there is no

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4.2. Methods

θ

φ

y0

φ

L

Figure 4.2: A schematic diagram of the setup with angles used in verticaltransformation. φ represents the angle corresponding to real distance y (inpixels). L is the real vertical extent of the image (in pixels).

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4.2. Methods

x

yHorizontal transformation matrix

500 1000 1500

200

400

600

800

0

500

1000

1500

Figure 4.3: The horizontal calibration matrix, associating each (x, y) imagecoordinate with the horizontal real pixel value.

perspective distortion. This distortion then increases linearly to the edges.To obtain the horizontal transformation, the camera used in the study wasset up to photograph a grid at camera axis angle θ, to reproduce the fieldconditions. In order to deduce the transformation, we need only know theratio of the real length of the top of the image frame to the bottom of theimage frame, which gives the maximal perspective distortion. This ratio wascalculated for a number of trials using a ruler to measure real length at eachedge of the frame, giving a ratio of 1.197. Using this value, a calibrationmatrix is constructed giving the horizontal map from image pixels to realpixels, shown in Fig. 4.3.

Combining the vertical transformation in (4.1) with the horizontal trans-formation shown in Fig. 4.3, positions in an image are transformed into realpositions in space, in units of pixels. In order to test that these spatial trans-formations accurately reconstruct real positions, we photograph a grid witha camera with camera axis angle θ (Fig. 4.4). Grid locations, equidistant

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4.2. Methods

Figure 4.4: The calibration grid used to test the image transformation. Thepixel location of the upper-right corner of each grid vertex was marked andtransformed in MATLAB (see Fig. 4.5).

in reality, will have perspective distortion in the image. These locations aremarked on the image, and transformed accordingly. As shown in Fig. 4.5,the transformed positions accurately reconstruct the original grid. We notethat there is also a very small amount of pincushion distortion which weassume is negligible.

The other camera specification to be tested was the frame rate whileshooting in continuous mode. The camera specifications give a rate of 3 fps.This was tested by taking 60 consecutive images of a stopwatch in continuousmode, at a shutter speed of 1/250 sec., using the same image resolution asin the field study. The camera proved to be remarkably accurate in thisrespect, as time between capturing images was 1/3 sec. (built-in frame rate)+ 1/250 sec. (shutter speed) with error less than 1%. Due to the relativelylow resolution of the images taken in this study, images could be written tostorage as quickly as they were taken, thus the camera storage buffer wasnever compromised and an accurate frame rate was maintained.

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4.2. Methods

0 500 1000 15000

200

400

600

800

1000

1200

1400

x

y

Transformed grid points

Figure 4.5: 42 Grid points marked in Fig. 4.4 plotted as ‘x’ marks, withreconstructed positions as ‘o’ marks.

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4.2. Methods

Figure 4.6: An example of one image from a time series of images collectedin the field.

4.2.5 Postprocessing Images

Data was stored as a series of JPEG images (see Fig. 4.6 for an exam-ple). These images were then processed in MATLAB to isolate individualducks and estimate the center of mass of each individual. The result is aset of (x, y) coordinate positions characterizing each individual in a frame,for every frame. These positions were then linked frame-by-frame, to createtrajectories. We describe this process below.

4.2.6 Extracting Positions

Image processing was performed in MATLAB. JPEG images are stored as anm x n x 3 matrix; i.e., 3 color layers (red, green, blue) ofm x nmatrices, wherem and n are the horizontal and vertical pixels (1504 and 972 respectively, inthis study). Each color value ranges from 0 to 255, indicating the amountof a particular color at a given pixel location. Consistent color differencesin images gathered in this study were exploited as a first step in isolatingindividuals (in this case, ducks tended to have red pixel values greater thanblue and green). Images were then thresholded to black and white (Fig.

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4.2. Methods

4.7.a). Next, morphological operations were performed on the thresholdedimage, which identified and reinforced objects within the image (Fig. 4.7.b).Then, the thresholded image was overlayed with the original (Fig. 4.7.c),so that the automatic object detection could be manually compared withthe original image to detect any errors. In the example in Fig. 4.7, oneindividual was missed by the algorithm, while a few others are representedby two objects. At this stage, an image editing user interface (purpose-builtin MATLAB for this study) was used to fix any errors in the automaticobject detection algorithm. Operations included joining objects, markingmissed individuals, separating objects that represent two individuals, andmodifying objects that poorly represent the center of an individual. Therewas typically minimal manual editing due to the well-spaced nature of thegroups in this study. Following the editing step, the center of mass of eachobject was calculated, and plotted (Fig. 4.7.d; centers of mass plotted in red).Last, the center-of-mass positions were transformed to real pixel positionsaccording to the perspective transformations described in Section 2.4, andsaved.

4.2.7 Tracking

Following the image processing of images, the data is reduced to a set ofpositions for each frame. These data were exported to ImageJ, an imageediting software package. The ImageJ ‘Particle Tracker’ plugin was usedto associate a position in one frame to the corresponding position in thenext frame. Repeating this process through successive frames generated atrajectory through time for an individual, and results were imported backinto MATLAB, where speed and heading were calculated from the trajectorydata. However, the tracking step presented a number of technical difficulties.

Particle tracking algorithms function best when the maximum distancetraveled between frames is less than half the minimum distance betweenindividuals. The camera used in this study limited our frame rate to 3 fps,which was not fast enough to ensure this bound on distance traveled betweenframes. Also, particle tracking codes are generally written for motion withoutdirectional bias; i.e., assuming a random-walk type of motion. However,motion of individuals in this study was generally highly polarized, resultingin directed motion. As a result of these technical difficulties, trajectories (andthus velocities) were partially incorrect, and incomplete (see Fig. 4.8). Atypical mistake by the algorithm was to associate a position in one frame with

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4.2. Methods

(a) Image after thresholding. (b) Image after morphological oper-ations.

(c) Image after overlay. (d) Image after manual editing andcenter-of-mass marking.

Figure 4.7: Processing an image to obtain positions.

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4.2. Methods

a position in the next frame in the opposite direction of travel (i.e., behind theindividual). Although it is intuitively clear that this is exceedingly unlikelyfor individuals in this study, the tracking algorithm cannot incorporate thisbias. To overcome this difficulty, an algorithm was developed to correcttracking errors, outlined below.

1 Trajectories are calculated with particle-tracking software, filtered toretain realistic sections (i.e., changes in position and heading less thansome threshold), while unrealistic sections are discarded.

2 In a given frame, all individuals associated with a good trajectory areassigned a velocity based on the trajectory.

3 For individuals not associated with a good trajectory, velocity is esti-mated by calculating the mean velocity of neighbours in a local region.

4 The entire sequence is re-tracked, using the estimated velocities as pre-dictions for the following step, and building the trajectory based on theclosest individual to the predicted location.

After applying this algorithm, trajectory reconstruction was virtuallycomplete (less than 1% of individuals remained without association to a real-istic trajectory). Fig. 4.9 shows an example frame with velocities that havebeen obtained after the re-tracking step - accuracy is considerably improvedas compared to Fig. 4.8. It is useful to note that for the polarized behaviourexhibited by scoter flocks in this study, only a fraction of velocities need tobe recovered during the initial tracking step. These first-pass velocities pro-vide a basis for estimating the expected position in the next frame. Then,velocities (and successively, trajectories) can be calculated exactly by find-ing the nearest actual position to the expected position in the next frame.Trajectories calculated after re-tracking were not only more accurate, butalso were considerably longer on average, having fewer interruptions due totracking deficiency. Examples of trajectory results are shown in 4.10, showingconvergence, turning and splitting of groups.

4.2.8 Edge effects

When performing statistical analysis of the data collected in this study, itis desirable in certain cases to exclude contributions from edge individualsdue to skewing of statistical measures. For instance, relative location of

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4.2. Methods

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Figure 4.8: An example frame with reconstructed velocities (grey), showingpartially incorrect and missing velocities. In reality, individuals are highlypolarized in this frame.

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Figure 4.9: The example frame of Fig. 4.8 showing velocities (grey) obtainedby re-tracking the event.

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4.2. Methods

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Figure 4.10: Example trajectories for 4 groups. Starting positions are plottedin green, final positions in red.

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4.2. Methods

Figure 4.11: In the first frame, the individual at the origin has nearest neigh-bours filling each quadrant, and so is not an edge individual. In the secondframe, the individual has no nearest neighbours (of the first 8) in the thirdquadrant, and thus is considered an edge individual.

nearest neighbours all tend to lie on one side of an edge individual. In orderto identify and exclude edge individuals from such statistical measures, thefollowing criteria were used. The location of the first 8 nearest neighboursfor each individual was calculated. Then, any individuals who do not have atleast 1 nearest neighbour in each of the 4 quadrants relative to the individual(see Fig. 4.11) are identified as edge individuals. Also, if the distance tothe nearest neighbour in each quadrant is beyond a threshold distance, the(isolated) individual is also identified as an edge individual. Fig. 4.12 showsa plot indicating edge individuals identified within a group.

4.2.9 Processed Events

We categorize data into sequences, referring to a series of images taken in suc-cession of a particular group movement (approach, return, etc.). A total of 14sequences were processed, with frames per sequence ranging from 22 to 137.828 frames were analyzed in all, and a total of 75269 positions were analyzedover all frames (42599 after eliminating edge individuals). Sequence-specificdetails are listed in Appendix C.1.

4.2.10 Body Alignment Versus Velocity

It is normally reasonable to assume that the velocity of an individual is in thesame direction as the vector describing the body alignment of the individual.

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0 500 1000 15000

500

1000

Figure 4.12: A plot of a group with edge individuals (as determined by thealgorithm outlined in the text) in red, with all other individuals in black.

However, when there are environmental influences such as currents, discrep-ancies can arise between alignment and actual velocity. Raw data in thisstudy has a convenient qualitative indicator of the presence of currents: dur-ing collective motion, individual scoters are observed to occasionally excretewaste. After excretion, the waste can be tracked in successive time frames- the direction and speed of this drift provides quantitative information oncurrents. In this study, currents predominantly were directed parallel to thedock, so we neglect any current components perpendicular to the dock.

To quantify currents, we first measure the left/right drift of waste in a testframe from each event (in body lengths per second), giving a relative measureof the strength of the current for each event. Then, a frame from each eventis analyzed for body alignment - the alignment of each individual is markedmanually, and the average body alignment of all individuals in a frame iscompared to the average velocity for the chosen frame - any discrepancy isassumed to be due to currents.

The current-induced discrepancy is a function of the direction of travel- clearly there is little alignment-velocity directional discrepancy if an indi-vidual is moving in the same direction as the current, as opposed to movingperpendicular to the current. Therefore, the current is quantified as a vec-

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4.2. Methods

tor which sums with the normalized alignment vector to give the normalizedvelocity vector. To this end, let b = (bx, by) be the vector of body align-ment of an individual, let v = (vx, vy) be the (normalized) velocity vector(which contains the effects of any current), and let c = (cx, cy) be the currentvector that accounts for the discrepancy in alignment and velocity, so that(v + c)‖b. Writing in component form gives

[

vx + cxvy

]

=

[

bxby

]

,

where we have set cy = 0 due to the assumption of along-dock currents.Cross multiplying and equating the first component gives

cx =bxvy − byvx

by. (4.2)

For each sequence, one frame is analyzed, and an average alignment vectorb is calculated by manually marking the body alignment vector for eachindividual, and then averaging over all individuals in the frame. Also, thevelocity of each individual is known via the tracking algorithm, from whichan average velocity v for the frame is calculated. These values are used tocalculate cx via Eq. (4.2). In this analysis, we assume currents to be constantthroughout a given sequence (the longest of which lasts approximately 45seconds). Analyzed data was collected over 4 days in 2008: March 1 (29minutes), March 6 (60 minutes), March 7 (107 minutes) and March 12 (50minutes). Sequence-specific details on currents are given in Appendix C.2.

At this stage, alignment data for each event could be calculated from thecx values. However, the waste (tracer) drift in frames is a much more directindicator of currents (i.e., less prone to measurement error), and so instead,a relationship between observed tracer velocity and cx is established via alinear least-squares fit, and this linear function is then used to compute theresultant current vector applied to velocities. In Fig. 4.13, data for tracermeasurements and cx calculations are shown, with the linear fit.

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4.3. Results

0 0.2 0.4 0.6 0.8−0.6

−0.4

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tracer shift, BL/s

curr

ent v

ecto

r

Figure 4.13: Current vector x-value cx vs. waste tracer drift speed, from rawimages, with linear least-squares fit.

4.3 Results

4.3.1 Basic statistics

We report basic statistics on observed surf scoter flocks in this study. Unitsof distance are body-lengths (BL), calculated from an average of individualscoters in images. In the analyzed sequences, surf scoters within groups hada mean nearest-neighbour distance of 1.435 ± 0.251 BL, and moved at anaverage speed of 2.007 BL/sec. Spacing and velocity was relatively similaracross sequences, although sequences featuring the group returning from thedock to open water (S-3,S6b-c,S-12) showed a higher average speed (2.85BL/sec, 2.45 BL/s, 2.67 BL/s, respectively). In S-3, the group was returningfrom a dive and had a considerably lower nearest-neighbour distance (aver-age of 1.15 ± 0.18 BL). It was generally noticed that immediately followingresurfacing from a dive, aggregates were tighter than those approaching thedock. The sequence S-6a featured individuals converging into a group, and soindividuals had not achieved equilibrium spacing, resulting in larger nearest-neighbour spacing (1.71 ± 0.26 BL).

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Table 4.1: Basic statistics: data sequencesSequence avg. speed NND std(NND)

S-1 2.150 1.538 0.1979S-2 2.129 1.477 0.2156S-3 2.847 1.148 0.1836S-4 1.741 1.345 0.2135S-5 1.985 1.323 0.1883S-6a 1.758 1.710 0.2606S-6bc 2.449 1.482 0.2214S-7 1.891 1.623 0.2971S-8 1.376 1.486 0.2335S-9 2.006 1.5126 0.2402S-10 1.900 1.4304 0.2366S-11 2.123 1.3519 0.2037S-12 2.672 1.4463 0.2426S-13 2.080 1.3208 0.1942S-14 2.715 n/a n/atotal: 2.007 1.435 0.2506

Table 4.2: Average speed and nearest neighbor distances for the 14 separatesequences of data that were analyzed. Units are body-lengths (BL). Speedand NND was averaged over all individuals in all frames. The last column isthe standard deviation of NND.

We summarize basic statistics for each event, where distance is given inunits of body lengths, for all individuals, and excluding edge individuals.

4.3.2 Nearest-Neighbour Distance Distributions

For each individual within a group, its nearest neighbours are ordered withrespect to distance, and labeled in increasing order of distance as NN1, NN2,NN3. . .. Such ordered sequences of neighbours are calculated for all individ-uals over all time frames, and frequency distributions of successive nearest

117

4.3. Results

neighbours are constructed. Distributions of successive nearest neighboursare plotted in Fig. 4.14. Distribution means and standard deviations increasenearly linearly with successive neighbours, as seen in Fig. 4.15.

Motivated by the regularity of distributions in Fig. 4.14, a functional formis sought to account for nearest-neighbour distributions observed in scoterflocks. In [16], a probability density function (pdf) for nearest-neighbor dis-tance is derived in the context of biological entities forming a (static) spatialpattern via competition for space (e.g., placement of trees in a forest viacompetition for light and nutrients, etc.) Although the context of the deriva-tion is different, the driving principles are very similar. In the case of scoterflocks, the use of space is not for resources, but instead to maintain a minimaldistance from neighbors so as to avoid collisions. In [16], space usage of anindividual is modeled as a Gaussian distribution, where overlaps with nearbyneighbors present competition for that space, analogous to a repulsion forceacting between individuals in a group. In this framework, a pdf is derived,using methods from information theory, for observed nearest-neighbor dis-tance. A good fit between the derived form of the pdf, and empirical dataon spacing in scoter flocks, suggests that the spacing properties of groups inour data set are part of a more general class of spatial patterns in biologyresulting from interactions among individuals under spatial competition (orequivalently, attractive and repulsive interactions).

The pdf has the form

q(d) = rC exp

(

−β exp (−d2/4w2)

1 − exp (−d2/4w2)− ǫπd2

)

, (4.3)

where w represents the effective size of an individual, ǫ is the number densityof the group (that is, number of individuals per unit of area), β is a fittingparameter, and C is a scale factor such that

∫ ∞

0q(d)dd = 1. (4.4)

To match nearest-neighbor distributions to this derived form, we first normal-ize nearest-neighbor distributions for the observed data so that the distribu-tion curve of observed data integrates to 1. Then, ǫ is estimated by countingindividuals in a fixed area (within the flock) across a number of trials, givingǫ = 0.3358. Next, a curve-fitting algorithm is used to fit β and w, and finallyC is calculated by satisfying Eq. (4.4). We note that the parameter w is

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0 1 2 3 4 50

1000

2000

3000

4000

Distributions of NN1−NN8 distances

NND, in BL

freq

.

Figure 4.14: Nearest-neighbour distributions for the first 8 nearest neigh-bours. Successive means are also plotted () above distributions.

typically derived from the data, as opposed to fitting it, but because bodysizes of individuals in this study are not circular, we leave this parameter tobe fitted. Results of the curve-fitting give optimal parameters β = 2.44 andw = 0.9743. Because units of distance are in body lengths, and given thatbody width is slightly less than body-length, a value of w slightly less than 1is reasonable. Fig. 4.16 shows the observed nearest neighbor distribution to-gether with the fit to q(d). The nearest-neighbour distribution data was alsofitted to a Gaussian distribution, resulting in a poorer fit than Eq. (4.3), asthe right-hand tail of the data distribution is longer than the left-hand tail.Although the functional form of 4.3 was derived only for the first nearest-neighbor, each successive neighbor distribution can be accurately fitted byvarying β and w. However, there is currently no theory to explain successivedistributions of interacting particles (whereas functions have been derivedfor distributions of successive neighbors for random points).

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4.3. Results

0 2 4 6 81

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dard

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iatio

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Figure 4.15:

0 1 2 30

0.5

1

1.5

2Theoretical and observed NND distribution (NN1)

distance, d, in BL

q(d)

Figure 4.16: Nearest neighbour distributions of the first neighbour, withfitted probability density function q(d) (dashed) overlaid.

120

4.4. Conclusion

4.4 Conclusion

Empirical data on animal groups has previously been limited to static anal-yses of large groups, or dynamic analyses of a few tens of individuals. Inthis work, we have extended such empirical studies to dynamic descriptionsof relatively large (a few hundred) individuals. This data was gathered byphotographing in time series surf scoters collectively swimming in watersnear Canada Place, Vancouver, BC. Individuals moved in well-spaced two-dimensional flocks with relatively low velocities, thus overcoming many ofthe traditional obstacles associated with empirical studies of animal groupmotion.

Scoter flocks foraged in predictable locations near a dock with an elevatedarea that allowed for oblique overhead photography. Corrections due to theangle of photography were easily made by measured features of the dock andbuilding where the study took place. Data was gathered with a digital SLRcamera firing at 3 fps, and resulting images were processed to isolate indi-viduals in MATLAB via software designed for this experiment. A trackingalgorithm was developed in conjunction with conventional particle-trackingsoftware to accurately track polarized groups of individuals, and individualtrajectories were reconstructed. Methods were developed to isolate interiorindividuals within a group to facilitate statistical measures of the group, andalso to calculate water currents, and eliminate the effects of these currentson individual motion. 14 data sequences were analyzed in total, representinga total of over 75000 reconstructed positions over 828 time frames. Basicstatistics, including neighbour spacing, mean velocity, and successive neigh-bour distributions were reported.

The basic results presented here are primarily limited to static aspects ofthe data, and do not make use of the individual trajectories that have beenconstructed. Related work featured in Chapter 5 of this thesis, however,analyzes dynamic aspects of the empirical data described here.

121

Bibliography

[1] I. Aoki and T. Inagaki, Photographic observations on the behaviour ofjapanese anchovy engraulis japonica at night in the sea, Mar. Ecol. Prog.Ser. 43 (1988), 213–221.



[4] A. Cavagna, I. Giardina, A. Orlandi, G. Parisi, A. Procaccini, M. Viale,and V. Zdravkovic, The STARFLAG handbook on collective animal be-haviour: Part 1, empirical methods, to appear: Anim. Behav. (2008).



[7] T. Ikawa, H. Okabe, T. Mori, K. Urabe, and T. Ikejoshi, Order andflexibility in the motion of fish schools, Journal of Insect Behavior 7(2)(1994), 237–247.



122



[11] , Internal dynamics and the interrelations of fish in schools, JComp Physiol 144 (1981), 313–325.



[14] J.-P. Savard, D. Bordage, and A. Reed, Surf scoter (melanitta perspicil-lata). in the birds of north america no. 363 (a. poole and f. gill, eds.),The Birds of North America Inc., Philadelphia, PA, 1998.


[16] G. Zou and H. Wu, Nearest-neighbor distribution of interacting biologicalentities, J. Theor. Biol. 172 (1995), 347–353.

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Chapter 5

Ducks in a Row: InferringInteraction Mechanisms fromField Data of Surf Scoters 4

5.1 Introduction

Across many species, animals often aggregate into cohesive, ordered groups.To understand the mechanisms of interaction governing such groups, a rela-tively large number of hypothetical models have been developed [1, 6, 9, 10,11, 15, 19, 21, 23]. However, the variety of different models raises an issue:given a particular collective behavior of interest, one must select the propermodel, and validate the hypotheses in that model. The key to choosing be-tween competing models and validating a given choice is by comparison toempirical data. Unfortunately, there is very little empirical data with whichto make this comparison, due to difficulties in gathering data on large, dy-namic animal groups. The data that has been obtained is limited either bysmall group sizes [8, 16, 18, 20] or an inability to track individuals throughtime [2], resulting in only static data (which limits comparison to a dynamicmodel).

Model hypotheses are typically founded on a series of interaction forces:repulsion away from individuals very close, attraction to individuals far away,and alignment with nearby neighbours. The wide variation in models arisesin the particular implementation of these forces. Here, an attempt is madeto build a model to match empirical data by proposing and testing a seriesof hypotheses.

4A version of this chapter will be submitted for publication as’ Ducks in a Row: Infer-ring Interaction Mechanisms from Field Data of Surf Scoters’, R. Lukeman, Y.-X. Li andL. Edelstein-Keshet.

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5.2. Empirical Methods

5.1.1 Specific Goals of this Work

In this chapter, my aim is to draw inferences from the analysis of data col-lected by methods in Chapter 4. I aim to identify the following aspects ofinteraction.

1. The minimal components, from repulsion, attraction, and alignment,that are necessary to obtain simulation results comparable to observa-tions.

2. Whether there is a hierarchy of interaction rules.

3. The spatial extent of these repulsive, aligning and attractive forces.

4. Angular preferences for neighbors, and how to implement such prefer-ences in a model.

5. The relative weighting of forces that best describes observed data.

5.1.2 Using Spatial Distributions

Because interactions are typically modeled as functions of distance betweenindividuals, the spatial structure of individual behavior relative to neighborspresents a key statistic to decipher these interactions. Obtaining such datareliably on animal groups requires a significant spatial extent of the group(i.e., large group sizes, particularly in three dimensions), as otherwise edgeeffects dominate. Unfortunately, as group size increases, tracking individualtrajectories becomes more difficult [5]. Yet, trajectories are needed to de-scribe individual velocities (and differences among mutual velocities), whichin highly polarized groups provides a useful indicator of behavioral response.

5.2 Empirical Methods

Reconstructing trajectories of animals within large groups in the field iscomplicated by the often three-dimensional nature and associated occlusionthat occurs in imaging, high speeds of movement, and difficulty calibratingequipment for groups whose location is not highly predictable. To overcomethese difficulties, we focused on groups of a few hundred surf scoters, collec-tively foraging near a dock on the waterfront of Vancouver, British Columbia.

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5.3. Empirical Results

Their collective swimming movement on the water surface (essentially two-dimensional) during approach to the dock from open water was captured intime series via oblique overhead photography (Figure 5.1.a). These forag-ing groups moved at relatively slow speeds, were well-spaced, and occupiedpredictable locations having convenient calibration properties. The experi-mental setup thus allowed high quality data to be gathered, permitting re-construction (via image processing software and particle-tracking software)of individual positions and trajectories (and thus velocities) (Figure 5.1.b).

Data was gathered over a two-week period in March, 2008, and 13 sep-arate sequences of group motion were reconstructed. Each sequence wascomprised of between 25 and 137 frames taken at 3 fps, with number ofindividuals per frame ranging from a few individuals to over 200. Overall,over 75000 positions were reconstructed. Unlike [2], individual positions in aframe were completely recovered, which permitted trajectory reconstruction(Figure 5.2). Across all sequences, individuals had a mean nearest-neighbordistance of 1.45 body-lengths (BL) with standard deviation 0.2506 BL. Meanspeed was 2.0 BL/s. For complete details on empirical methods, consultChapter 4.

5.3 Empirical Results

5.3.1 Relative Location of Neighbors

We first investigate spatial distributions of nearest neighbors, in two dimen-sional space around individuals. For a given focal individual i placed at theorigin, the relative position of nearest neighbours j, ~xj−~xi, is plotted relativeto the heading of the focal individual. By summing over neighbours of eachindividual in each analyzed frame, a two-dimensional distribution emerges(normalized to have maximal value 1), indicating preferred locations rela-tive to neighbors. for computational reasons, only the first 40 neighbors areplotted, a convention we carry through on similar plots both for data andsimulations.

In Figure 5.3, a circular region of essentially no neighbors indicates a zoneof repulsion which maintains the well-spaced structure within groups. Movingout radially, a region of high density indicates preferred location of nearestneighbours (with mean nearest-neighbour spacing 1.44 BL). Further, thereis a region directly in front, and directly behind a focal individual, of higher

126


(a) Raw image

x

y

(b) Reconstruction of position and velocity

Figure 5.1: A typical analyzed frame: input image (a) is processed, trans-formed to real space, and individual positions are reconstructed (b). Posi-tions in successive frames are linked using particle-tracking software, givingvelocities (b).

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0 500 1000 15000

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Figure 5.2: Trajectories for 4 sequences, showing surf scoters approachingthe dock (y = 0) in a highly polarized fashion. Starting positions are plottedin green, final positions in red. Units are given in pixels, where 1 BL = 46pixels.

128


density, indicating a preference for neighbors directly in front and behind,as opposed to alongside. We note that such a plot has an inherent circularreflective property due to mutual neighbour pairs, and so it is reasonable tosuggest that the high density of neighbours to the back is a plot artefact (asopposed to an indication of interaction to the back).

Moving out radially again, there is an ‘echo effect’ of neighbor density.This periodic structure is more evident in the frontal direction than sideways,indicating a more structured positioning in the frontal direction. The neigh-bor distribution indicates that individuals have a tendency to ‘follow theleader’, while maintaining a minimal distance in all directions. Interestingly,the group was occasionally observed to approach the dock in true single-fileformation, though such aggregations were not included in distributions.

To further characterize the angular bias in neighbours in the radial regionof preferred spacing, a mean of distribution values is calculated around anannulus centered on the maximal distribution value with radial width 0.5BL. To account for the reflective symmetry in distribution plots, data for[0,−180] is combined with data for [0, 180], where angle 0 corresponds tothe right-side of an individual, and 180 to the left-side. Fig. 5.4 showsmore clearly what is shown in Fig. 5.3 : there is a higher neighbor densitycentered at 90, with an approximate width of 40. Beyond this high-densityregion, fluctuations in neighbor density are small, and likely due to inherentvariability in the system. Although Fig. 5.4 only affirms what is alreadycontained in Fig. 5.3, this data measure will prove useful further on whencomparing the data to simulations.

5.3.2 Relative Deviation of Neighbors

Because trajectories of individuals were constructed, we are able to calculatethe spatial distribution of the relative heading of neighbours. Similar to Fig.5.3, a focal individual i is placed at the origin, and at each relative neighborposition j, the absolute value of the difference in heading, |~vj−~vi| is recorded.Averaging over each individual in each frame, a two-dimensional distributionof heading deviation is constructed, shown in Fig. 5.5.

At radial distances near the mean nearest-neighbor spacing, deviation isminimal, with slightly lower deviation sideways. However, at distances lessthan the preferred spacing, deviation is strongly angle-dependent: directly infront (and behind), deviation is high, while at alongside positions, deviation islow, indicating a collision avoidance mechanism whereby individuals deviate

129


Figure 5.3: Neighbour distributions from empirical data, normalized to havemaximal value 1. Neighbor positions are calculated relative to the headingof a focal individual (in the direction [0 1] in the plot, indicated by the whitegraphic). A frontal bias is seen in neighbor positioning.

130

5.4. Building a Model

0 50 100 1500.5

0.6

0.7

0.8

0.9

1angular neigh. density at preferred distance (data)

relative angle, in degrees

dens

ity

Figure 5.4: Average density of neighbours in a circular shell at the preferreddistance, as a function of angle. A distinct region of higher density is centeredat 90.

to the side if too close to a frontal neighbor, but do not strongly deviate whento close to the side of neighbors. Moving out radially, deviation increasesbeyond the preferred distance, due to a decrease in correlation with neighborheading with distance.

Taken together, Figures 5.3 and 5.5 suggest that the dominant distance-dependent interaction occurs in the frontal direction : individuals modify po-sition to move in line with individuals to the front, while less-than-preferredinter-individual distances to the front result in a strong repulsive reaction viaa deviation to the side. In contrast, alignment with neighbors is strongest tothe side. We summarize this information in Figure 5.6.

5.4 Building a Model

Because real animal groups do not strictly adhere to rules in the way thatsimulated groups do, we cannot hope to exactly match observed trajectoriesof individuals within the group with a model, and further, the more aspectsof the data one attempts to recover in a model, the more complicated the

131


Figure 5.5: Spatial distribution of heading deviation, from empirical data.Radial bands are plotted at 1 BL, 2 BL, and 3 BL for reference. The regionwhere deviation is 0 corresponds to the repulsion zone where no individualsare found.

132


Figure 5.6: A schematic diagram partitioning local space around a focalindividual into sectors of location preference, high tendency to align, andhigh tendency to deviate rapidly (move away from) the focal individual.Schematic was assembled according to trends shown in the data in Figs. 5.3and 5.5.

model becomes. Our goal here, in contrast, is to build as simple a model aspossible to recover one particular aspect of the observed data: the spatialdistribution, (and in particular the frontal bias) of neighbors. Our approachis to compare analyzed data to models that incorporate increasing levels ofcomplexity.

The structure of relative position and alignment in Figures 5.3 and 5.5motivate a zonal modeling framework similar to [6, 10], where space aroundan individual is partitioned into radial bands of repulsion, alignment, andattraction. We note that the relatively fixed density across all data sequencesprecludes any hypothesis testing of a topological versus metric interaction,as in [2].

5.4.1 Model Framework

We formulate an acceleration analogue to the fixed-speed zonal model in [6].n individuals with position ~xi and velocity ~vi (i = 1, . . . , n) are modeled asparticles obeying equations of motion, according to

~xi = ~vi, (5.1)

~vi = ~fi,aut + ~fi,int + ~ξi, (5.2)

133


where ~fi,int corresponds to social forces of interaction with neighbors, ~fi,aut

to autonomous forces due to influences other than interaction, and ~ξi to

(Gaussian) noise. Writing ~ξi = ωξ~ξi, where ~ξi has mean 0 and standard

deviation 1, strength of noise is controlled by the parameter ωξ. We set~fi,aut = ~a − γ~vi, as in [12, 13, 19], where γ is a drag coefficient, and ~a isan intrinsic desired direction of travel (in the context of our data, the desireto approach the dock to forage). An upper limit of 10 is set on number ofinteracting neighbors.

Model I: Repulsion Only

We set ~fi,int = ωrep~fi,rep, where ωrep is a weighting parameter and

~fi,rep =1

nrep

nrep∑

j=1

g(|~xj − ~xi|)~xj − ~xi

|~xj − ~xi|, j ∈ R, (5.3)

where nrep is the number of neighbors sensed in the repulsion zone R (Fig.5.8.a), and g(x) is as in Fig. 5.7. We set rrep = 1.44, the radius of therepulsion zone, corresponding to the mean spacing observed in the empiricaldata.

The neighbor distribution in Fig. 5.8.b shows that repulsion alone cap-tures the well-spaced property of the observed data, with highest neighbordensity corresponding to rrep. However, as one expects, no angle preferenceto neighbors occurs with only repulsion, as the force acts equally in all di-rections, as shown in Fig 5.8.c. Thus, Model I is inadequate. Although notable to fully capture observed neighbor distributions, clearly repulsion is animportant component of the interaction.

Model II: Repulsion and Attraction

We add a long-range attractive force to ~fi,int, so that ~fi,int = ωrep~fi,rep +

ωatt~fi,att, where ωatt is a weighting parameter and

~fi,att =1

natt

natt∑

j=1

g(|~xj − ~xi|)~xj − ~xi

|~xj − ~xi|, j ∈ ATT. (5.4)

Here, ATT is the region of attraction, as in Fig. 5.9.a, with width ratt.To eliminate the artificial case of attraction and repulsion forces canceling,we impose a hierarchy of interaction such that if nrep > 0, then ~fi,att = 0.

134


AL ATTR

x

g(x)

1

-1

Figure 5.7: Attraction-repulsion function g(x) is negative in repulsion regionR, zero in alignment region AL, and positive in attraction region ATT. Mag-nitudes of attraction and repulsion are controlled by weighting parametersωatt and ωrep.

Thus, repulsion is the dominant interaction, and attraction occurs only ifno neighbors are within the repulsion zone. With the inclusion of attraction,groups are tighter, shown by a less diffuse neighbor distribution in Fig. 5.9.b,yet still no angle preference exists for neighbors (Fig. 5.9.c) and so Model IIis inadequate.

Model III: Repulsion, Neutrality, Attraction

Following [9, 20], we include a neutral zone between repulsion and attractionzones (Fig. 5.10.a). Neighbors detected within the neutral zone impart no

interaction force. We maintain the hierarchy that if nrep > 0, then ~fi,att = 0.Inclusion of a neutral zone does not affect neighbor distributions (Fig. 5.10.b-c), as relative location to neighbors still arises from a balance between short-range repulsion, and cohesion imparted by attraction; the only difference isthe distance at which neighbors are detected to generate cohesion. Model IIIis therefore inadequate.

Model IV: Repulsion, Alignment, Attraction

We add an alignment force force to ~fi,int, so that ~fi,int = ωrep~fi,rep+ωatt

~fi,att+

ωal~fi,al, where ωal is a weighting parameter and

~fi,al =1

nal

nal∑

j=1

~vj

|~vj |, j ∈ AL. (5.5)

135


AL is an alignment zone, between zones of attraction and repulsion (i.e.,having replaced the neutral zone of Model III), with width ral. nal is thenumber of detected neighbors in the alignment zone AL. The alignmentforce acts differently than attraction and repulsion in that is a propulsiveforce which increases the speed of an individual. To eliminate the artefact oflarge speed changes depending on whether nal is non-zero or not, we set

~fi,al =~vi

|~vi|

if nal = 0, following [11].Inclusion of alignment imparts a bias in neighbor location, as fewer neigh-

bors are found directly to the side (Fig. 5.11.b). This occurs due to an in-terplay between alignment and noise-induced heading changes. However, thebias is gradual (Fig. 5.11.c), and does not match the frontal preference shownby the data, which has more distinct regions of higher and lower neighbordensity. Exploration of parameter space did not reveal any parameter com-bination that matched the angular preference in data well, and so Model IVis deemed inadequate.

Model V : Repulsion, Alignment, Attraction, Frontal Interaction

In order to obtain a frontal preference for neighbors as in empirical observa-tions (Fig. 5.3-5.4), a further modification is proposed. A fourth force ~fi,front

is added to the interaction force, so that

~fi,int = ωrep~fi,rep + ωatt

~fi,att + ωal~fi,al + ωfront

~fi,front,

where ωfront is a weighting parameter. ~fi,front is an attraction/repulsioninteraction (with strength as in Fig. 5.7) with a single neighbor ~xj,θ in afrontal angular region with angle θ; i.e.,

~fi,front = gf(|~xj,θ − ~xi|)~xj,θ − ~xi

|~xj,θ − ~xi|,

where gf(x) is given in Fig. 5.12. If no such ~xj,θ is detected, ~fi,front = 0.This force differs from other forces in that it is ‘topological’ (following theterminology of [3]) : should a neighbor exist in the frontal θ region (andwithin sensing range ratt), an interaction takes place with the first encoun-tered neighbor, regardless of distance. In this way, individuals within the

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Rep.

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de

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Figure 5.8: Model I: (a) A schematic diagram representing the interactionzones, here being simply repulsion. (b) Spatial distributions of neighbors,relative to a focal individual oriented in the direction [0 1] (as indicated bythe white graphic at the origin), where density has been normalized to havemaximum value equal to 1. White dashed lines are superimposed at radialdistances 1, 2 and 3 BL. (c) Angular density distributions at the preferreddistance are plotted, following the convention of Fig. 5.4. I compare distri-butions for data (green) and the model (blue).

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Figure 5.9: As in Fig. 5.8, but for Model II.

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Rep. Att.N.

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Figure 5.10: As in Fig. 5.8, but for Model III.

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Figure 5.11: As in Fig. 5.8, but for Model IV.

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5.5. Parameters

R

x

gf(x)

1

-1

rrep

Figure 5.12: Attraction-repulsion function gf(x) is negative in repulsion re-gion R, and positive and constant beyond R (up to a limit of ratt). Themagnitude of gf(x) is controlled by the weighting parameter ωfront.

group balance forces of repulsion, alignment, and attraction in all directionswith a follow-the-leader type of force with a frontal individual.

Inclusion of a frontal interaction force, in combination with attraction,repulsion, and alignment, generates spatial distributions and angular biassimilar to those generated from empirical data, and so we deem Model V tobe adequate in capturing the spatial distribution of neighbors. Distributionsare dependent on the relative contribution of forces, which we investigate viaweighting parameters in the next section. Our goal is twofold; to understandthe effect of parameter variation on neighbor distributions, and to obtain anoptimal parameter set that best matches observed data.

5.5 Parameters

Parameters in the full model are categorized into two groups: those thatare fixed (either determined from data, or having negligible effect on modeloutput), and those that are free parameters, fitted to the model. A summaryof parameters is given in Table 5.3.

5.5.1 Fixed Parameters

Interaction Zones

Interaction zones of repulsion, alignment, attraction are controlled by radiirrep, ral, and ratt, respectively. We fix rrep = 1.45, corresponding to the mean

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5.5. Parameters

Table 5.1: Model : Summary of Interaction Forces

force hierarchy formulation

~fi,rep if nrep > 0 ~fi,rep =1

nrep

nrep∑

j=1

g(|~xj − ~xi|)~xj − ~xi

|~xj − ~xi|, j ∈ R

~fi,al if nrep = 0, nal > 0 ~fi,al =1

nal

nal∑

j=1

~vj

|~vj|, j ∈ AL, ~fi,al =

~vi

|~vi|if nal = 0

~fi,att if nrep = 0, natt > 0 ~fi,att =1

natt

natt∑

j=1

g(|~xj − ~xi|)~xj − ~xi

|~xj − ~xi|, j ∈ ATT

~fi,front if ~xj,θ exists ~fi,front = g(|~xj,θ − ~xi|)~xj,θ − ~xi

|~xj,θ − ~xi|

nearest-neighbor distance in the data. In Fig. 5.5, deviation as a functionof radius is minimal at the preferred distance of 1.45 BL, and then increaseswith radius. We estimate ral = 3, twice the distance of minimal deviation,which corresponds to a threshold mean deviation value of approximately 11.Although alignment is clearly not radially symmetric in Fig. 5.5, it is beyondthe scope of this paper to match alignment distributions, and so we assumethat the alignment zone is circular. We choose ratt = 5 as an estimation ofinteraction radius, though model results are not sensitive to this width, dueto the upper limit on interacting neighbors. The angular width of the frontalinteraction zone is chosen to be θ = 60, as the angular region of preferredneighbors in the data extends approximately 30 to the left and right of theheading of an individual (as in Fig. 5.4).

Autonomous Forces

Autonomous propulsion ~a is chosen to be [0 a] (where a is some positiveparameter); direction choice is arbitrary, having no effect on spatial distri-butions. Because we focus on the relative weight of the various interactionforces, we fix a = 0.5 as a reference parameter. Equilibrium speed in Eq.(5.2) can be solved as

v0 =|~a| + ωal

γ, (5.6)

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5.5. Parameters

Table 5.2: Summary of Parameters

fixed parametersparameter value sourcerrep 1.45 BL dataral 3 BL data (est.)ratt 5 BL chosenθ 60 dataa 0.5 chosen (reference)γ (ωal + a)/2 data (via Eq. 5.6)free parametersparameter range optimalωrep 1-20 10ωal 0-2 0.5ωatt 0.1-3 0.5ωfront 0-1 0.1ωξ 0.05-0.5 0.325

Table 5.3: Using Model V together with an optimization routine, an optimalparameter set is found that best matches simulated neighbor distributions toobserved neighbor distributions. Parameters are dimensionless unless unitsare given.

and so the drag coefficient γ is determined by ~a and ωal by matching equi-librium speed to mean speed calculated from empirical data (2.0 BL/s).

5.5.2 Free Parameters

In order to explore the effects of relative weighting of the various interactionforces, we leave ωrep, ωatt, ωal and ωfront as free parameters. We first explorechanges in distributions resulting from parameter variation, and then use anoptimization algorithm to find the set of parameters which provide the bestfit to the observed data. We also explore variations in the relative magnitudeof noise in the system, and so let ωξ vary.

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5.6. An Optimal Parameter Set

5.5.3 Parameter Effects on Radial Distributions

Matching simulated distributions to observed distributions involves accu-rately representing the radial distribution of neighbors in the data, as well asthe angular distribution. To first investigate radial distribution dependenceon parameters, we set ωfront = 0, and set the remaining free parameters totypical values (see caption of Fig. 5.14). Then, free parameters are variedone by one, keeping the remaining parameters fixed at the typical values. InFigure 5.14, radial neighbor density is plotted over ranges of the free param-eters ωrep, ωatt, ωal and ωξ. Radial neighbor density from the data is shownfor reference in Fig. 5.13.

In Fig. 5.14.a, increasing ωrep from 0 lowers density within R, and en-hances periodicity in density due to well-spaced group structure. In Fig.5.14.b, varying ωatt has little effect because at (quasi-)steady state, attractionforces occur in all directions, and tend to cancel. Self-propulsion is governedby the sum of ~a and ~fi,al; increasing ωal (Fig. 5.14.c) weights propulsionvia alignment more strongly than autonomous forcing from ~a, which reducesgroup cohesion imparted by ~a (individuals sharing a common preferred di-rection will naturally move cohesively). Less cohesiveness in turn increasesneighbor density at distances beyond preferred spacing rrep. Though it mayseem counterintuitive that increasing alignment reduces cohesion, it is due tothe tradeoff experienced by reducing the relative strength of ~a. It is only atconsiderably larger alignment radii ral (≈ 5 BL) that alignment forcing alone(i.e., with ~a = 0) leads to cohesive groups. In Fig. 5.14.d, increasing noisereduces well-spaced group structure, leading to decreased periodic structurein neighbor density.

5.6 An Optimal Parameter Set

5.6.1 Goodness-of-fit Measure

The modeling goal is to recover both the observed spatial distribution of near-est neighbors, and observed angular bias in neighbor location. To do this, wefirst develop a simple goodness-of-fit measure. Ideally, if the model was ableto completely recover Fig. 5.3, the error between the model-generated anddata-generated spatial distribution of neighbors would be sufficient. How-ever, real groups are much more complex than those generated by a model,and so Fig. 5.3 contains influences not included in the model developed

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Figure 5.13: Average radial neighbor density, from data.

here. As a result, to emphasize matching the angular bias in neighbors, thegoodness-of-fit measure used is a combination of overall spatial distributionfit, and matching angular variation in neighbor density in the radial regionof preferred distance. The measure is defined as

E = 〈|fdata(x, y) − fsim(x, y)|〉 + 〈|hdata(θ) − hsim(θ)|〉, (5.7)

where fdata (Fig. 5.3) and fsim are the observed and simulated two-dimensionalspatial distributions of neighbors, respectively, and hdata (Fig. 5.4) and hsim

are the observed and simulated angular variation in neighbors. 〈·〉 denotes anaverage (two-dimensional and one-dimensional, respectively, in Eq. (5.7)).

5.6.2 Optimization Process

Parameter space was first extensively investigated manually. From visualinspection, a reasonable set of parameters was chosen to initialize the freeparameters listed in Table 5.3 in the optimization routine. Lower and upperbounds are placed on each parameter, given in Table 5.3, obtained manuallyfrom inspection of model output, requiring that simulations appear realistic.Then, a series of 30 simulations to t = 200 are performed (randomized initialconditions), with data from the last half of the simulation used to constructfsim and hsim. These distributions are used to calculate E via Eq. (5.7). Apattern search algorithm (suited for optimization of a stochastic function)

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ωξ = 0.4

radius

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ity

(d) Effect of varyingωξ = 0, 0.1, 0.2, 0.3, 0.4.

Figure 5.14: Effect of parameter variation on radial neighbor distributionin basic model simulations. Unless varied as indicated, parameters areωrep = 5, ωatt = 1, ωal = 1, ωξ = 0.15. 50 Simulations of 100 individualswere performed to t = 100, and average densities over all simulations (afterquasi-equilibrium was reached, t = 50 to t = 100) were calculated.

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is then used to update the parameter set that, after another series of sim-ulations, attempts to reduce E. This process is continued iteratively, untilthe parameter set that leads to the minimum value of E is found, giving theoptimal set. The entire optimization process was run a number of times fromdifferent initial conditions to verify the computed minimum.

5.6.3 Optimization Results

Following the optimization, a set of parameters that minimized the error Ewas found. The optimal parameter set is listed in Table 5.3. Of all influences,repulsion is weighted most strongly ωrep = 10, an order of magnitude largerthan attraction (ωatt = 0.5). Interestingly, an optimal value of ωfront =0.1 (two orders of magnitude less than repulsion) suggests that the frontalinteraction tendency need not be very strong to have a clear effect on theangular preference of neighbors.

Alignment strength is optimal at ωal = 0.5, such that individuals arepropelled via a = 0.5 equally as much as the propulsion component dueto aligning with neighbors. Spatial distribution of neighbors for the opti-mal set is shown in Fig. 5.15.a together with the corresponding distributionfrom data (repeated from Fig. 5.3) in Fig. 5.15.b. The essential featuresof the spatial distribution are captured in the simulated data, although thesecondary shell of neighbors at radial distance 3 BL is more evident in sim-ulated data, likely due to irregularities in the shape and structure of realgroups that cannot be accounted for by white noise in simulations. In Fig.5.16, angular distribution of neighbors at the preferred distance is plotted forboth the data and simulations with the optimal parameter set. For simula-tions upon which Figs. 5.15.a and 5.16 are based, the mean absolute errorin spatial distributions is 〈|fdata(x, y) − fsim(x, y)|〉 = 0.0388, and the meanabsolute error in angular distribution is 〈|hdata(θ) − hsim(θ)|〉 = 0.201, for atotal error E = 0.0589.

5.6.4 Detailed Parameter Exploration Near Optimal

Set

To verify that the optimal parameter set minimizes error E (locally, at least),we vary each parameter in turn in a local region about the optimal value,keeping all others fixed at the respective optimal value. For each variation, 30

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(a) Simulations

(b) Data

Figure 5.15: A comparison of spatial neighbor density for [a] the best-fitsimulations of the most appropriate model, Model V , and [b] data (repeatingFig. 5.3). Essential features of the data are observed in simulated data,including the spatial extent of neighbors, and the frontal bias.

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1angular neighbor density at pref. dist.


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Figure 5.16: A comparison of average angular neighbor density at the pre-ferred distance for data (green), and simulation of Model V with optimalparameters (blue).

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5.7. Discussion

0.05

0.06

0.07

0.08

0.09

0.1opt. par.

erro

r

Parameter variation around optimal set

ωξω

att

ωrep

ωfront

ωal

Figure 5.17: Error as each parameter is varied about the optimal value. Eachparameter is scaled in the plot so that optimal values coincide (blue dottedline). Parameter values explored were as follows: ωrep = 5, 7.5, 10, 12.5, 15,ωatt = 0.5, 1, 1.5, 2, ωal = 0.25, 0.5, 0.75, 1, ωfront = 0.05, 0.1, 0.15, 0.2, andωξ = 0.275, 0.3, 0.325, 0.35, 0.375. In each case, the optimal value is a mini-mum.

simulations are run to generate an error measure E, three times. The averageof the three error values gives the error measure for a particular variation.Fig. 5.17 shows the results for varying each of the 5 free parameters about theoptimal set (where parameter values are all scaled to have coinciding optimalvalues). It is clear that for each parameter, the optimal value minimizeserror. We note that in Fig. 5.17, error values at the optimal parameterline do not coincide for each exploration due to the stochastic nature of thesystem (though parameters are equal at this line).

5.7 Discussion

Recent advances in camera technology have made empirical studies on largeanimal groups much more feasible, although many challenges still exist.Much of the previous work has focused on flocks of birds flying [2, 4, 14, 18],

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5.7. Discussion

or fish schooling [7, 8, 16, 17, 22], both of which occur naturally in threedimensions. In this study a group of surf scoters foraging collectively wasanalyzed because their motion of interest was limited to two dimensions (thewater surface). Such groups eliminate many of the challenges associatedwith empirical data collection of collective motion, yet still exhibit relativelycomplex responses and interactions.

Local interactions of individuals were investigated via spatial distribu-tions of alignment and relative positioning, indicating a frontal bias in neigh-bor position, and a distinct avoidance manoeuvre in the frontal direction.The frontal bias of neighbors suggests that scoters have a follow-the-leadertendency, within a two-dimensional group structure.

Interestingly, the scoters studied were observed on numerous occasions toform distinct follow-the-leader formations while approaching the dock, (seeFig. 5.18). It thus seems likely that scoters are able to modify group shapeby changing interactions, although frontal tendencies that entirely governone-dimensional formations are still present to a degree in two-dimensionalformations. Surf scoters observed in this study were foraging on musselsgrowing on dock pilings, and so one possible explanation for the variousgroup formations is to optimize foraging depending on the spatial propertiesof the food source (whether it is widely dispersed, or concentrated).

The spatial structure of deviation reinforces that interactions to the frontare the dominant influence. Should an individual scoter move too close toa neighbor in front, a strong deviation is taken to avoid the neighbor. Incontrast, individuals who are too close to neighbors at the side (though in-frequent, as shown in Fig. 5.3) do not show as strong an avoidance tendencyvia deviation. This mode of interaction is likely influenced by both theenvironment (swimming in water), and physiology of scoters. Turning ma-noeuvres are likely easier to perform than to modify speed while swimming,while avoiding many velocity changes would confer energetic benefits dur-ing approach to the dock for foraging. It is of future interest to develop amodel that simultaneously matches both neighbor distribution and deviationdistributions.

Another feature of deviation distribution is (at distances beyond preferredspacing) an increase with distance from a focal individual. This fact is strongevidence that alignment forces exist. Consider the alternative explanation:that heading is determined by the common influence of the goal (in ourstudy, approaching the dock). If this were the case, no clear correlationwould be observed between deviation and distance, as each individual would

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5.7. Discussion

Figure 5.18: An example of a follow-the-leader formation observed in surfscoters approaching a dock to forage on mussels. Here, interactions to thefront are dominant.

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5.7. Discussion

be governed by the same global influence, and deviation differences would becaused by noise.

The model proposed here, built step-by-step from general collective mo-tion principles, is an attempt to simulate the joint tendencies of existing ina well-spaced cohesive group, and having particular interaction directly infront (as per a follow-the-leader formation). By modifying a dynamic, zonalattraction-repulsion-alignment model to include a topological frontal inter-action, groups were formed having similar spatial distribution of neighbors,with a distinct frontal bias near the preferred spacing. Although the modelconvincingly accounts for the observed neighbor distribution, it is possiblethat other models are able to do so as well (or better). However, it is unlikelythat such a model is distinctly simpler than that proposed here, because thenecessity of each component included was shown via the series of more com-plex models.

Typical simulation studies of collective motion differ from this work in afundamental way. Here, empirical data motivates both the implementationof the model, and the choice of parameters. In studies based on simulation,modeling choices are hypothetical, and parameters often arbitrarily chosen.As more empirical data becomes available, more focus should be placed onevaluating competing hypotheses for collective motion.

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Bibliography

[1] I. Aoki, A simulation study on the schooling mechanism in fish, Bull.Jpn. Soc. Sci. Fish. 48 (1982), 1081–1088.


[3] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giar-dina, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, and V. Zdravkovic,An empirical study of large, naturally occurring starling flocks: a bench-mark in collective animal behaviour, arXiv:0802.1667 (2008).


[5] A. Cavagna, I. Giardina, A. Orlandi, G. Parisi, A. Procaccini, M. Viale,and V. Zdravkovic, The STARFLAG handbook on collective animal be-haviour: Part 1, empirical methods, to appear: Anim. Behav. (2008).




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[9] S. Gueron, S. A. Levin, and D. I. Rubenstein, The dynamics of herds:From individual to aggregations, J theor Biol 182 (1996), 85–98.



[12] Y.-X. Li, R. Lukeman, and L. Edelstein-Keshet, Minimal mechanismsfor school formation in self-propelled particles, Phys. D. 237(5), 699–720.

[13] R. Lukeman, Y.-X. Li, and L. Edelstein-Keshet, A mathematical modelfor milling formations in biological aggregates, Bull. Math. Biol. 71(2),352–382.







[20] J. Tien, S. Levin, and D. Rubenstein, Dynamics of fish shoals: identify-ing key decision rules, Evolutionary Ecology Research 6 (2004), 555–565.


155



[23] K. Warburton and J. Lazarus, Tendency-distance models of social cohe-sion in animal groups, J. theor. Biol. 150 (1991), 473–488.

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Chapter 6

Conclusion

6.1 Introduction

In this thesis, I explored collective motion in animal groups with two ap-proaches. First, perfect schools were studied with a minimal model, andexistence and stability conditions for particular formations were derived. Sec-ond, real data on motions of surf scoters swimming in polarized groups weregathered and used to test a series model hypotheses, leading to a specificmodel that matched observations of real scoter flocks.

6.2 Analysis of Perfect School Solutions

6.2.1 Summary

In Chapters 2 and 3, a minimal model for school formation was presented.Though the model exhibited a wide array of possible solutions, I focused ona number of solutions that featured regular spacing and group geometry, anda predictable interaction structure, so-called ‘perfect schools.’ The regular-ity of these perfect school solutions was exploited to obtain conditions onmodel parameters that guarantee existence of solutions, and conditions thatdetermine linear stability. Where a number of equilibria were possible, sta-bility conditions could determine the stability of each solution. For realisticinteractions, a unique stable solution was shown to exist.

The geometric regularity of these schools appeared in the model’s equa-tion system as a structured coefficient matrix. This structure was then usedto obtain a set eigenvalues for arbitrary dimension (i.e., n individuals). Inchapter 2, the matrix structure was block- and block-diagonal, and in Chap-ter 3, block-circulant. Sets of eigenvalues led to conditions on model inputsthat determine stability.

In Chapter 2, the stability condition was expressed in terms of the attraction-repulsion function g(x) governing interactions between individuals. I showed

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6.2. Analysis of Perfect School Solutions

that if the derivative of g(x) was positive at a given equilibrium school solu-tion, then that solution was stable. This was true for perfect schools in onedimension, and for ‘soldier formations’ in two dimensions.

In Chapter 3, these techniques were used to explore another perfectschool: individuals in a milling formation, following one another in a cir-cular path. Importantly, the change in group geometry changed conditionson g(x) leading to stability. Whereas explicit solutions for eigenvalues werefound for perfect schools in Chapter 2, the milling formation stability analysisled to a characteristic equation in the form of a complex-coefficient quarticpolynomial. This equation was studied numerically which led to an intervalof values of the derivative of g(x) at steady-state which lead to stability.In contrast with results in Chapter 2, an upper limit existed on g′(x) toguarantee stability, while also, negative values of this derivative led to stableformations under certain parameters.

Although the solutions studied in Chapters 2 and 3 were idealized, theyled to clear relationships between model parameters and observed group be-haviour. Previously, models were typically investigated by observing modeloutput in simulations while varying parameters, and then developing a hy-pothesis to explain the observed output. In contrast, the conclusions inChapters 2 and 3 state explicitly the properties of the group in terms ofparameters, eliminating the need for detailed parameter exploration (in thecase of the solutions we study).

6.2.2 Perfect School Analysis in Context

The types of solutions studied in this thesis (soldier formations and mill for-mations) are representative of two solutions that occur across many differentmodels for collective motion [2, 3, 4, 6, 8, 13, 14]: polarized motion, and cir-cular (milling) motion. Although the solutions obtained in these models aremore complex, and are truly two-dimensional, our results show that besidesmodel parameters, the group geometry itself is an important component indetermining stable patterns observed in a model.

There have been a number of other attempts to obtain analytical pre-dictions of model behavior. Analogy to statistical mechanics [2, 4] led toconditions on whether groups remained well-spaced (H-stable), or becameever more dense (catastrophic), as number of individuals increased. Relatedresults on existence of well-spaced groups were obtained using Lyapunov sta-bility [7] and assuming particular interaction forces. The analytical work in

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6.3. Empirical Data and Modeling

Chapters 2 and 3 is more general in that no particular form of interactionfunction is assumed, and stability results are to general perturbations, as op-posed to the particular case of group collapse (i.e. H-stable vs. catastrophicbehavior).

However, there are a number of limitations to this work apart from therequirement of regular geometry. In order to obtain results on stability forgroups of arbitrary size, a well-defined structure of interindividual interac-tion must exist, as interactions lead to coupling of model equations, which inturn leads to structure in the coefficient matrix. Without a clear connectionstructure, these methods become unwieldy. A particular example is in ex-tending the analysis to regular arrays of individuals, as in a lattice structure.It is a significant challenge to label individuals, and determine a connectionstructure, that leads to a structured matrix which can be solved. In the for-mations we studied, they shared the property of being organized relative toone another in a quasi-one-dimensional manner, thus avoiding the connectionand labeling challenges. Furthermore, these methods do not always general-ize to other Lagrangian models. The models of [4, 6] for example, containa non-linear drag term which complicates stability analysis of mill forma-tions substantially. However, in spite of the challenges, there remains thepotential to develop these techniques further to overcome these limitations.Additionally, analytical results obtained in this thesis have direct implicationfor designing stable multi-agent systems, especially in the case where motionis not subject to any constraints (such as underwater vehicles).

6.3 Empirical Data and Modeling

6.3.1 Summary

In Chapters 4 and 5, I described research collecting and analyzing data ongroups of surf scoters. Chapter 4 described the empirical methods used toobtain and process data, and also reported on basic statistics measured fromthe data. The experimental subject, flocks of scoters swimming collectively,did not possess the typical difficulties associated with gathering data on otheranimal groups. This was mostly due to their motion in two dimensions, butalso to their predictable location and relatively slow speeds of movement. Anarray of techniques were developed to process data, including image detec-tion software and routines for correcting camera angle, tracking, and current

159


effects.The main result of Chapter 4 was in obtaining the final data set. This data

set represents a significant step forward in data collection on animal groups,being the first empirical study (to my knowledge) to capture trajectories ofeach individual within large groups (a few hundred individuals). Previouswork has been limited to small groups (an order of magnitude less thanthose considered in this thesis) [5, 9, 10, 11, 12, 15], or studies that abandontracking altogether, instead gathering static images of groups without anydescription of movement [1].

In Chapter 5, the data set from Chapter 4 was used to build a modelto match properties of real groups. Spatial neighbor distributions were con-structed to deduce the spatial preference for neighbors in local space aroundan individual, and to relate deviation response to relative location. It wasshown that surf scoters diligently maintain empty space around themselves,and also have a bias to positioning themselves directly behind a neighbor,in a type of follow-the-leader formation. Surf scoters also exhibited a strongtendency to deviate sideways, should they approach a frontal neighbor tooclosely, whereas the avoidance tendency was much weaker with neighborsat the side. A positive correlation between distance from a neighbor andheading deviation gave direct evidence for alignment forces existing locallybetween individuals.

The spatial distribution of neighbors was then used to develop and testa model for collective motion. A model was built from generic collectivemotion principles of attraction, repulsion, and alignment. Showing that theseforces were not sufficient to account for observations of relative neighborlocations, an additional topological interaction to the front was proposed.By combining this frontal interaction force with the other typical interactionforces, essential elements of the neighbor distribution were captured, namelythe frontal preference. An error measure was devised to measure goodness-of-fit, and this measure was used to obtain an optimal set of parameters(representing relative weights of the constituent forces) that best describedthe empirical data.

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6.3.2 Empirical Data and Modeling in Context

Empirical Data: Strengths and Limitations

We have emphasized throughout Chapters 4 and 5 that the empirical datagathered is, in a sense, the first of its kind: large scale trajectory data of indi-viduals within groups of a few hundred individuals. There have been studiesof much larger groups (over 1000 starlings [1]), but the inherent complex-ity and three-dimensional structure prevented reconstructing all positions,which is a crucial requirement for generating trajectories. However, thereare a number of limitations, both in our data set, and in applying theseexperimental methods to other groups.

Foremost on the list of limitations is the temporal resolution of the data,gathered at 3 frames per second. At this resolution, tracking is much more ofa challenge than at higher resolutions, especially in analyzing non-polarizedgroups (for example, comparison to properties of polarized groups). Alsomovement at scales finer than our resolution is, of course, lost. This leadsto rather noisy descriptions of sensitive measurements such as acceleration.Another current limitation on the methods is in the need for manual inter-vention during image processing. Because there are situations that cannotcurrently be automatically detected (such as the separation of two individu-als who are very close together, appearing as one), one has to inspect eachimage and fix any detection errors. Continued refining of the software canalleviate much of this manual effort, however.

The methods here relied on an overhead location from which to takepictures, and also calibrate the experimental setup (i.e., using the dock widthand height above water level). To extend these methods to other groupsnot near convenient overhead locations, the camera could for example bemounted on a retractable pole that could be placed in a location of interest.I am currently working with Dr. J. Heath on extending these techniques toeider duck groups in the Canadian Arctic.

Modeling of Real Groups: Strengths and Limitations

In Chapter 5, the distribution of neighbors and deviation in a local neigh-borhood of an individual was used to uncover interaction structure betweenindividuals. These statistics did indeed show clear spatial patterns fromwhich interactions were inferred. In particular, the spatial distribution ofdeviation has never been shown for any group of animals, owing to the joint

161

6.4. Summary: Main Contributions

need for large spatial extent and trajectory reconstruction.Empirical data was used to refute or accept a series of models to describe

spatial structure in scoter flocks. The final model was successful in matchingthe particular quantity chosen for comparing simulation results to data, andit was clearly shown that the preceding sequence of null models were notsufficient to describe the data.

However, other quantities, such as the distribution of deviation, were notaccurately captured in the model. Additional complexity is likely neededto capture more aspects of the data. The presentation of a feasible modelfor frontal preference also does not preclude the possibility that anothercompeting hypothesis could account for the data as well, or better thanthe model proposed. However, it is a strong first step in linking data withdynamic models of collective motion.

Empirical Data and Modeling: Future Work

There remains much work to be done in interpreting the data set of Chapter4. In particular, attraction and repulsion responses can be measured, sothat attraction-repulsion curves can be empirically derived, as opposed tobeing formulated in terms of hypothetical functional forms (exponential [4, 6],inverse-power [7] etc.). Also, heading correlations to particular neighborscan be calculated to attempt to deduce with which neighbors an individualinteracts.

Another interesting aspect of the raw data is the sequences in whichindividuals do not move in polarized groups, but instead, move in disordereddirections, maintaining a stationary group location (such groups were notanalyzed in this thesis). As these transitions from disorder to order have beenstudied extensively in models (e.g., [3, 8, 14]), analyzing real occurrences ofsuch transitions would be very informative.

6.4 Summary: Main Contributions

My main contributions in this thesis were:

1. Obtaining existence and stability conditions on one-dimensional and‘soldier formation’ solutions in terms of an interaction force and itsderivative;

162

6.4. Summary: Main Contributions

2. Obtaining existence and stability conditions on ‘milling formation’ so-lutions in terms of an interaction force and its derivative;

3. Development of field methods to obtain a data set of hundreds of indi-vidual surf scoters moving collectively, where each individual’s trajec-tory has been reconstructed;

4. Analysis of the field data set via spatial distributions;

5. Development of a model, built from a series of null models of increasingcomplexity, that matches empirical spatial distributions well.

163

Bibliography










164








165

Appendix A

A.1 The Stability Analysis in 1D

We write (2.44)-(2.47) in matrix form as

d

dt

δ1δ2...δnω1

ω2...ωn

=

|0 | I

|−− −− −− | −− −− −−

|V | U

|

δ1δ2...δnω1

ω2...ωn

, (A.1)

where all the sub-matrices are n× n. In particular, 0 is the zero matrix andI is the identity matrix. V and U are given by

V =

−g′−(d) g′−(d) 0 0 · · · 0 0g′+(d) −σ g′−(d) 0 · · · 0 0

0 g′+(d) −σ g′−(d) · · · 0 0· · · · · · · · · · · · · · · · · · · · ·0 0 0 0 · · · −σ g′−(d)0 0 0 0 · · · g′+(d) −g′+(d)

,

U =

−h′−(0) − γ h′−(0) 0 0 · · · 0 0h′+(0) −s h′−(0) 0 · · · 0 0

0 h′+(0) −s h′−(0) · · · 0 0· · · · · · · · · · · · · · · · · · · · ·0 0 0 0 · · · −s h′−(0)0 0 0 0 · · · h′+(0) −h′+(0) − γ

,

where s = h′+(0) + h′−(0) + γ and σ = g′+(d)+ g′−(d). Note that U and V aretridiagonal matrices.

166

A.1. The Stability Analysis in 1D

The eigenvalues of J are obtained by solving

Λ(λ) ≡ det[λI − J ] = det

[

λI −I−V λI − U

]

= 0. (A.2)

We can simplify this eigenvalue equation via the result stated in Eq.(3.30). In this case, because λI and −I commute, then

det[λI − J ] = det[λI − UλI − IV ] = det[λ2I − λU − V ]. (A.3)

Since U and V are tridiagonal matrices, so is the matrix in (A.3). We cannow write (A.2) explicitly as

det

χ+ η α 0 · · · 0 0η χ α · · · 0 0· · · · · · · · · · · · · · · · · ·0 0 0 · · · χ α0 0 0 · · · η χ+ α

= 0, (A.4)

where χ = λ2 + sλ+ σ, η = −h′+(0)λ− g′+(d), and α = −h′−(0)λ− g′−(d).Thus, the eigenvalue equation is a polynomial of λ that is defined by the

determinant of a tridiagonal matrix. Although the determinant can be cal-culated recursively, the complexity of the resulting formula prevents us fromsolving the eigenvalues easily. However, the eigenvalues are easily solvable inthe case when either g+ or g− is absent, i.e., when individuals interact onlywith neighbours in front or behind them.

Let us consider the case that particles see only what is in front of them.This is the model for schools formed by following one immediate predecessor.In this case, h′−(0) = g−(d) = 0. Thus, α = 0, χ = λ2 +(h′+(0)+γ)λ+g′+(d),and χ+ η = λ2 + γλ = λ(λ+ γ). The eigenvalue equation now requires onlythe determinant of a triangular matrix which is given by the simple formulaΛ(λ) = (χ + η)χn−1. Thus, the eigenvalues are determined by

Λ(λ) = (χ+ η)χn−1 = λ(λ+ γ)[λ2 + (h′+(0) + γ)λ+ g′+(d)]n−1 = 0. (A.5)

Therefore, the eigenvalues are λ1 = 0, λ2 = −γ, and λ± = −(h′+(0) + γ)/2±√

[(h′+(0) + γ)/2]2 − g′+(d). Both λ+ and λ− have multiplicity n− 1.Similarly, if each particle can only detect the a particle behind but not a

particle in front (a rather peculiar situation), η = 0 and χ = λ2 + (h′−(0) +γ)λ+ g′−(d).

Λ(λ) = (χ + α)χn−1 = λ(λ+ γ)[λ2 + (h′−(0) + γ)λ+ g′−(d)]n−1 = 0. (A.6)

167

A.2. Stability Analysis for the Soldier Formation in 2D

The eigenvalues are λ1 = 0, λ2 = −γ, and λ± = −(h′−(0)+γ)/2±√

[(h′−(0) + γ)/2]2 − g′−(d).

Both λ+ and λ− have n− 1 multiplicity. We find that the roles of h′−(0) andg′−(d) in determining the stability are quite similar to the roles of h′+(0) andg′+(d) in the case of forward sensing.

A.2 Stability Analysis for the Soldier

Formation in 2D

We can explicitly write the Jacobian matrix ∂ ~fi

∂~ziin Eq. (2.78) as

J =

∂fix

∂zix

∂fix

∂ziy

∂fiy

∂zix

∂fiy

∂ziy

=

g+(|~zi|)|~zi|

+g′+

(|~zi|)z2ix

|~zi|2− z2

ixg+(|~zi|)

|~zi|3g′+

(|~zi|)zixziy

|~zi|2− g+(|~zi|)zixziy

|~zi|3

g′+

(|~zi|)zixziy

|~zi|2− g+(|~zi|)zixziy

|~zi|3g+(|~zi|)

|~zi|+

g′+

(|~zi|)z2iy

|~zi|2− z2

iyg+(|~zi|)

|~zi|3

.

(A.7)

Evaluating J at steady state ~zi = ~zsi simplifies the matrix considerably, due to

the judicious choice of coordinate system aligned with the axis of the school.Specifically, since

~zsi = (zix , ziy)

s = (d , 0), (A.8)

substituting zix = |~zi| = d, ziy = 0 into (A.7) yields

Df ≡

∂ ~fi

∂~zi

|~zi=~zsi

=

[

g′+(d) 00 g+(d)/d

]

.

Next, we write Eqs. (2.81)–(2.84) in matrix form as

168


d

dt

~δ1~δ2...~δn~ω1

~ω2...~ωn

=

|0 | I

|−− −− −− | −− −− −−

|D | −γI

|

~δ1~δ2...~δn~ω1

~ω2...~ωn

, (A.9)

where

D =

0 0 0 . . . 0 0Df −Df 0 . . . 0 00 Df −Df 0 . . . 0

...0 0 . . . 0 Df −Df

, (A.10)

where each 0 in the matrix represents a 2×2 matrix of zeros. D is a bidiagonalblock matrix. We obtain stability conditions by evaluating the eigenvalues ofthe coefficient matrix in Eq. (A.9) and determining under what conditionsthese eigenvalues have negative real part.

To find the eigenvalues of the coefficient matrix in Eq. (A.9), we formthe determinant equation given by this coefficient matrix:

det

[

−λI ID −(λ+ γ)I

]

= 0.

Next, we use the block matrix property in Eq. (3.30). Noting that −λI andI commute, we have

det

[

−λI ID −(λ+ γ)I

]

= det (λ(γ + λ)I −D) .

Recalling the definition of D in (A.10), where

Df =

[

g′+(d) 00 g+(d)/d

]

,

169


it is clear that D is a lower triangular matrix, and thus (λ(γ + λ)I −D) isalso a lower triangular matrix. λ(γ + λ) appears in the first two diagonalentries (since each D entry is 2 × 2), while the remaining 2(n− 1) diagonalentries contain n− 1 pairs

λ(γ + λ) − g′+(d),

λ(γ + λ) − g+(d)/d.

The lower triangular structure allows us to immediately infer the eigenvalueequation, given by the product of diagonal terms, i.e.,

(

λ(γ + λ)2) (

λ(γ + λ) − g′+(d))n−1

(λ(γ + λ) − g+(d)/d)n−1 = 0.

We thus get solutionsλ = 0

with multiplicity 2,λ = −γ

with multiplicity 2,

λ = −γ2±√

γ2 − 4g′+(d)

2,

each with multiplicity n− 1, and

λ = −γ2±√

γ2 − 4g+(d)/d

2

each with multiplicity n− 1.

170

Appendix B

B.1 Transformation to Relative Coordinates

Expanding (3.18)–(3.19) gives

xi = cosφi (xi+1 − xi) + sinφi (yi+1 − yi) , (B.1)

yi = − sinφi (xi+1 − xi) + cosφi (yi+1 − yi) , (B.2)

ui = cosφi (ui+1 − ui) + sinφi (vi+1 − vi) , (B.3)

vi = − sinφi (ui+1 − ui) + cosφi (vi+1 − vi) , (B.4)

where ~xi = (xi, yi), and ~vi = (ui, vi). We can obtain the first model equationthrough time-differentiation of xi and yi, which gives

d

dtxi = −ω0 sinφi (xi+1 − xi) + cosφi (xi+1 − xi)

+ω0 cosφi (yi+1 − yi) + sinφi (yi+1 − yi) ,

d

dtyi = −ω0 cosφi (xi+1 − xi) − sinφi (xi+1 − xi)

−ω0 sinφi (yi+1 − yi) + cosφi (yi+1 − yi) .

Using (B.1)–(B.4), we rewrite the above in the more compact vector no-tation

d

dt~xi = ~vi + ω0k × ~xi, (B.5)

where

k × ~xi = det

i j kxi yi 00 0 1

= (yi,−xi).

Similarly, we time-differentiate (B.3)–(B.4), giving

˙ui = −ω0 sin φi (ui+1 − ui) + cosφi (ui+1 − ui)

+ω0 cosφi (vi+1 − vi) + sin φi (vi+1 − vi) , (B.6)˙vi = −ω0 cos φi (ui+1 − ui) − sin φi (ui+1 − ui)

−ω0 sin φi (vi+1 − vi) + cosφi (vi+1 − vi) . (B.7)

171

B.1. Transformation to Relative Coordinates

This can be rewritten more compactly using (B.3)–(B.4) as

˙~vi = ω0k × ~vi +R(φi)

(

~vi+1 − ~vi

)

= ω0k × ~vi +R(φi)(

~f (~xi+2 − ~xi+1) − ~f (~xi+1 − ~xi))

−R(φi)γ (~vi+1 − ~vi) . (B.8)

Following (3.3), we write the interaction force terms in (B.8) in terms of unitdirection vectors and magnitudes:

R(φi)(

~f (~xi+2 − ~xi+1) − ~f (~xi+1 − ~xi))

= R(φi)(~xi+2 − ~xi+1)

|~xi+2 − ~xi+1|g (|~xi+2 − ~xi+1|)

−R(φi)(~xi+1 − ~xi)

|~xi+1 − ~xi|g (|~xi+1 − ~xi|) . (B.9)

We make a number of observations. First, the rotational matrix operator Ris distance-preserving, and so

|~xi+1 − ~xi| = |~xi|.

Second,R(a)R(b) = R(a + b).

We use this property to recast (B.9), using (3.18), as

R(φi)(~xi+2 − ~xi+1)

|~xi+2 − ~xi+1|g (|~xi+2 − ~xi+1|) −R(φi)

(~xi+1 − ~xi)

|~xi+1 − ~xi|g (|~xi+1 − ~xi|)

= R(−2π

n

)

~xi+1

|~xi+1|g(

|~xi+1|)

− ~xi

|~xi|g(

|~xi|)

,

where we have used R−1(a) = R(−a). We now write the equation system inrelative coordinates:

˙~xi = ω0k × ~xi + ~vi, (B.10)

˙~vi = ω0k × ~vi +R

(−2π

n

)

~xi+1

|~xi+1|g(

|~xi+1|)

− ~xi

|~xi|g(

|~xi|)

− γ~vi. (B.11)

172

B.1. Transformation to Relative Coordinates

Next, we calculate the steady-state quantities. In equations (B.1)–(B.2),we replace the steady-state expressions for xi, xi+1, yi and yi+1 to obtain

xsi = − cosφi

(

r0 cos(

θi +2π

n

)

− r0 cos θi

)

− sin φi

(

r0 sin(

θi +2π

n

)

− r0 sin θi

)

, (B.12)

ysi = sin φi

(

r0 cos(

θi +2π

n

)

− r0 cos θi

)

−

cosφi

(

r0 sin(

θi +2π

n

)

− r0 sin θi

)

, (B.13)

where θi = 2πi/n + ω0t. These expressions can be simplified using trigono-metric difference formulas to

xsi = −r0 cos

(

φi − θi −2π

n

)

+ r0 cos(

φi − θi

)

,

ysi = r0 sin

(

φi − θi −2π

n

)

+ r0 sin(

θi − φi

)

,

which we can further simplify using the definition of φi in (3.21) to

xsi = −r0 cos

(

−π2− π

n

)

+ r0 cos(

π

n− π

2

)

,

ysi = r0 sin

(

−π2− π

n

)

+ r0 sin(

−πn

+π

2

)

.

With some simple trigonometric manipulation, we write

xsi = −r0 sin

(

−πn

)

+ r0 sin(

π

n

)

,

ysi = −r0 cos

(

−πn

)

+ r0 cos(

π

n

)

,

i.e.,

xsi = 2r0 sin

(

π

n

)

= d, (B.14)

ysi = 0. (B.15)

The steady-state positions in this coordinate frame are independent of theindex i, and each position is a vector parallel to the x-axis with magnitude

173

B.2. Derivation of the Linearized Perturbed System

d, the interindividual distance at steady state in the original coordinates.Instead of calculating us

i and vsi directly as we did with position, we can

evaluate (B.10) at steady state, which implies that

~vs

i = −ω0k × ~xs

i ,

i.e.,

usi = 0,

vsi = ω0d,

both of which are independent of index.

B.2 Derivation of the Linearized Perturbed

System

Substituting (3.25)–(3.26) into (3.22)–(3.23) gives

d

dt

(

~xs

i + ~δi(t))

= ω0k ×(

~xs

i + ~δi(t))

+ ~vs

i + ~ξi(t), (B.16)

d

dt

(

~vs

i + ~ξi(t))

= ω0k ×(

~vs

i + ~ξi(t))

−R−

(−2π

n

)

~f(

~xs

i+1 + ~δi+1(t))

−~f(

~xs

i + ~δi(t))

− γ(

~vs

i + ~ξi(t))

, (B.17)

where i = 1, . . . , n, and n+ 1 is identified with 1 (henceforth assumed to be

the case). Because k× (~a+~b) = k×~a+ k×~b, (B.16) can be separated intosteady-state and perturbed components. Using

ω0k × ~xs

i + ~vs

i = 0,

(B.16) can be simplified to

d

dt~δi = ω0k × ~δi + ~ξi, (B.18)

174

B.2. Derivation of the Linearized Perturbed System

where we have dropped arguments of ~δi and ~ξi. We expand the nonlinearinteraction terms in (B.17) via a linear approximation:

R(−2π

n

)

~f(

~xs

i+1 + ~δi+1(t))

− ~f(

~xs

i + ~δi(t))

≈ R(−2π

n

)

f(

~xs

i+1

)

+∂ ~f

∂~xi

|(d,0)~δi+1

−~f(

~xs

i

)

− ∂ ~f

∂~xi

|(d,0)~δi. (B.19)

Using this expansion in (B.17) and noting that

ω0k ×(

~vs

i

)

+R(−2π

n

)

~f(

~xs

i+1

)

− ~f(

~xs

i

)

− γ(

~vs

i

)

= 0,

we simplify (B.17) as

d

dt~ξi = ω0k × ~ξi +R

(−2π

n

)

∂ ~f

∂~xi

|(d,0)~δi+1

− ∂ ~f

∂~xi

|(d,0)~δi − γ~ξi.

Combining with (B.18), we now write the set of equations for the perturbedsystem:

d

dt~δi = ω0k × ~δi + ~ξi, (B.20)

d

dt~ξi = ω0k × ~ξi +R

(−2π

n

)

∂ ~f

∂~xi

|(d,0)~δi+1

− ∂ ~f

∂~xi

|(d,0)~δi − γ~ξi. (B.21)

We explicitly calculate the Jacobian matrix ∂ ~f

∂~xiby expanding ~f as in (B.11),

i.e.,

~f(~xi) =xi

|~xi|g(

|~xi|)

i +yi

|~xi|g(

|~xi|)

j ≡ f1ii + f2ij,

where|~xi| =

√

x2i + y2

i .

175

B.3. Eigenvalue Equation

We can now calculate the Jacobian matrix

∂f1i

∂xi

∂f1i

∂yi∂f2i

∂xi

∂f2i

∂yi

,

via derivatives:

∂f1i

∂xi

=g(|~xi|)|~xi|

+g′(|~xi|)x2

i

|~xi|2− x2

i g(|~xi|)|~xi|3

,

∂f1i

∂yi

=g′(|~xi|)xiyi

|~xi|2− g(|~xi|)xiyi

|~xi|3,

∂f2i

∂xi

=g′(|~xi|)xiyi

|~xi|2− g(|~xi|)xiyi

|~xi|3,

∂f2i

∂yi

=g(|~xi|)|~xi|

+g′(|~xi|)y2

i

|~xi|2− y2

i g(|~xi|)|~xi|3

.

Evaluating these matrix entries at the steady state values (xi, yi) = (d, 0)gives

∂f1i

∂xi

|(d,0) =g(d)

d+g′(d)d2

d2− d2g(d)

d3= g′(d),

∂f1i

∂yi

|(d,0) = 0,

∂f2i

∂xi

|(d,0) = 0,

∂f2i

∂yi

|(d,0) =g(d)

d.

Then we can write the Jacobian matrix as

D =∂ ~f

∂~xi

|(d,0) =

[

g′(d) 00 g(d)/d

]

.

With definitions of Ω and RD from (3.27), we now can write the system in(3.28).

B.3 Eigenvalue Equation

The set of diagonal blocks is given by Di = Λ + D +i−1RD, i = 1, . . . , n.We equivalently consider the set Di+1, i = 0, . . . , n − 1, for notational ease.

176

B.4. Derivation of Inequality (3.37)

Figure B.1: A schematic diagram of particles at the threshold of mill break-ing. The angle between particles is π/2, beyond which particle i no longersenses particle i+ 1. ~vm

i indicates the velocity of particle i in the absence ofautonomous self-propulsion.

Expanding detDi+1 = 0 gives

p(λ) =(

λ(λ+ γ) − ω20 + g′(d) − exp

(

2πij

n

)

cos−2π

ng′(d)

)

·(

λ(λ+ γ) − ω20 + g(d)/d− exp

(

2πij

n

)

cos−2π

n

g(d)

d

)

+

(

γω0 + 2λω0 + exp(

2πij

n

)

sin−2π

n

g(d)

d

)

·(

γω0 + 2λω0 + exp(

2πij

n

)

sin−2π

ng′(d)

)

= 0. (B.22)

Using the expression for ω0 in (3.16), and for g(d)/d in (3.29), we reduce theparameters to γ, n, and g′(d), while the index i = 0, . . . , n−1 (correspondingto D1, . . . , Dn).

B.4 Derivation of Inequality (3.37)

At the steady state of the moving mill, we have

~vsi = ω0r0

− sin(

θi(t))

cos(

θi(t))

+~a

γ, ~ui =

sin(

θi(t) + πn

)

cos(

θi(t) + πn

)

,

177

B.4. Derivation of Inequality (3.37)

where θi(t) = 2πi/n+ω0t. The threshold of ~a occurs when these two vectorsare perpendicular, that is, ~vs

i · ~ui = 0 (see Figure B.1). Expanding thiscondition gives

− ω0r0 sin(

θi(t))

sin(

θi(t) +π

n

)

− a1

γsin

(

θi(t) +π

n

)

+

ω0r0 cos(

θi(t))

cos(

θi(t) +π

n

)

+a2

γcos

(

θi(t) +π

n

)

= 0, (B.23)

where ~a = [a1 a2]T . This equation simplifies using trigonometric identities

toa1

γsin

(

θi(t) +π

n

)

− a2

γcos

(

θi(t) +π

n

)

= ω0r0 cos(

π

n

)

.

Using the expressions for ω0 and r0 from the existence conditions, we furthersimplify the threshold condition to

a1 sin(

θi(t) +π

n

)

− a2 cos(

θi(t) +π

n

)

(B.24)

= γ

(

γ sin(πn)

cos(πn)

)(

g(d) cos2(πn)

γ2 sin(πn)

)

cos(

π

n

)

,

= g(d) cos2(

π

n

)

. (B.25)

Mill breaking actually occurs whenever ~vsi · ~ui ≤ 0, i.e.,

a1 sin(

θi(t) +π

n

)

− a2 cos(

θi(t) +π

n

)

≥ g(d) cos2(

π

n

)

,

so we must maximize the left-hand side over all possible angles to find theminimal values of a1 and a2 that lead to mill breaking. It can easily beshown using calculus that the left-hand side attains a global maximum value

of |~a| =√

a21 + a2

2. This gives our mill breaking condition in (3.37).

178

Appendix C

C.1 Data Sequences: Details

14 sequences were gathered over 4 days in March, 2008. From all raw datagathered, sequences to be analyzed were chosen based on quality of data(including image quality, lack of external disturbance, having large groupswithin the frame). Due to limitations of the tracking software used, sequenceslonger than 50 frames are split and subcategorized by letters. Details ofanalyzed sequences are listed in Table C.1. We note that all groups analyzedshowed a high degree of polarization, and all groups were two-dimensionalin shape, except sequence S-14, wherein individuals approached the dock ina one-dimensional, follow-the-leader organization. These data were excludedfrom statistical measures.

C.2 Data Analysis: Details on Currents

For each data sequence, average alignment is manually calculated via a graph-ical user interface in MATLAB and compared with actual velocity (as de-termined by particle-tracking). Then, tracer velocity is calculated for eachsequence, and together this information is used to recover individual align-ment, so as to account for effects of currents. Details are shown in TableC.2.

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C.2. Data Analysis: Details on Currents

Table C.1: Data Sequences: Details

Event Date Frames Total (w/o edge)S-1 mar 1 25 4206 (2980)S-2 mar 6 42 2294 (903)S-3 mar 6 40 3920 (2739)

S-4(a,b) mar 7 62 9257 (6277)S-5 mar 7 47 2833 (1533)

S-6(a,b,c) mar 7 137 12423 (7665)S-7(a,b) mar 7 91 7285 (2571)

S-8(a,b,c) mar 7 101 11023 (6667)S-9(a,b) mar 7 70 6765 (2995)S-10(a,b) mar 7 61 6628 (4057)

S-11 mar 12 31 1250 (500)S-12 mar 12 22 2540 (1536)

S-13(a,b) mar 12 55 3886 (2176)S-14 mar 12 44 959 (n/a)

total: 828 75269 (42599)

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C.2.

Data

Analy

sis:D

etailson

Curren

ts

Table C.2: Current statistics: by sequence

date event frame b v cx (angle diff.) tracer (BL/s)mar 1 S-1 img. 10 [-0.88 -0.46] [-0.88 -0.46] -0.0058 (0.15o) 0mar 6 S-2 img. 38 [0.54 -0.83] [0.53 -0.84] 0.0176 (0.84o) 0

S-3 img. 3 [-0.27 0.96] [-0.28 0.96] 0.0147 (0.81o) 0mar 7 S-4 img. 25 [-0.47 -0.88 ] [-0.03 -0.99] -0.5020 (26.34o) 0.75

S-5 img. 39 [-0.25 -0.96 ] [ 0.10 -0.99] -0.3680 (20.85o) 0.74S-6 img. 76 [0.24 0.97] [0.47 0.88] -0.2487 (13.95o) 0.69,0.67,0.62S-6 img. 111 [-0.26 0.96] [0.07 0.99] -0.3410 (19.23o) -S-7 img. 16 [-0.17 0.98] [0.07 0.99] -0.2477 (14.13o) 0.67S-8 img. 38 [ -0.76 -0.64] [-0.41 -0.91 ] -0.6858 (26.11o) 0.61S-8 img. 82 [-0.08 -0.99] [0.34 -0.94] -0.4276 (25.21o) -S-9 img. 59 [-0.17 -0.98] [0.04 -0.99] -0.2190 (12.45o) 0.40S-10 img. 2 [-0.52 -0.85] [-0.24 -0.97] -0.3588 (17.77o) 0.50

mar 12 E11 img. 21 [-0.29 -0.95] [-0.18 -0.97] -0.1213 (6.70o) 0.35S-12 img. 18 [-0.01 0.99] [0.02 0.99] -0.0363 (2.08o) 0.15S-13 img. 11 [0.12 -0.99] [0.19 -0.98] -0.0752 (4.28o) 0.14S-14 img. 27 [0.53 -0.82] [0.60 -0.79] -0.0825 (3.97o) 0.44

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