ESPM/IB/ERG 205Instructors: Wayne Getz & Zack Powell

Course Outline: HandoutCourse Website: The site is on BSPACE with a back up at http://www.cnr.berkeley.edu/~getz/205Fall07Required Text: Otto and Day, 2007. A Biologist’s Guide to Mathematical Modeling in Ecology and Evolution, Princeton UP.

Programming Languages/Software: Your choice: MATLAB, R+, other. Word processing with equation editor untility, spread sheet with graphics, drawing (could use power point).

Computers: computers available in lab, but best solution to use your own laptop with installed software.

Lab Assignments: Must be submitted through BSPACE site on time to get graded--missing assignments get 0 added to cumulative score

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Models of dynamic processes

Temporal and spatial processes

Discrete vs continuous

Deterministic versus stochastic

Individual (Lagrangian) versus population (Eulerian) levels of description

Mechanistic versus phenomenological

Theoretical versus empirical

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Components of modelsvariables:

independent (time t, space (x, y))

dependent on time (x1(t), x2(t), ... ), on space & time (z(x,y,t)...)

parameters (a, b, c, ...)

equations

principle of parsimony

principle of hierarchical epistemology

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Steps to modeling (Box 2.1 modified!)Formulate the question

what do you want to know?state question as precisely as possible

Determine basic ingredientsstart with simplest sufficient biological description:

define variablesidentify flows (draw flow diagram)list variables influencing flows describe flow rate forms phenomenologically (e.g. growth rate is increasing but saturating function of resource intake)

Translate into equations write down mathematical equations with general flow designators fi (i=1,2,3...)develop expressions for fi that conform to phenomenological descriptions

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E.g. Growth of a populationFormulate the question

what do you want to know?the size of a population at various times in the future?state question as precisely as possiblewhat are the fundamental process leading to population change and how do they combine to determine the trajectory of a spatially homogeneous population over time?

Determine basic ingredientsstart with simplest sufficient biological description:

define variablesx(t): density of a closed population at time t

(assumption: spatially homogeneous, class-less population)identify flows (draw flow diagram)

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Population growth: flow diagram

x(t)

influences on flows

b d

f1=bx f2=dx

x(t)flow in flow out

f1 f2

variables and flows (a la Stella or Berkeley Madonna)

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Formulate equation

x(t)

influences on flows

b d

f1=bx f2=dx

x(t +1) = x(t)+ bx(t) − dx(t)

Discrete: Pop(t+1)=Pop(t)+births-deaths

dxdt

= bx − dx

Continuous: Pop rate of change=birth rate - death rate

dxdt

= bx − dx

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Elaborate: density-dependent death rate

x(t)

influences on flows

b d

f1=bx f2=dx

dxdt

= bx − dx

Continuous: Pop rate of change=births rate - death rate

d = d0 + d1x⇒ f2 = d0x + d1x

2

dxdt

= bx − d0x − d1x2 = (b − d0 )x 1−

d1xb − d0

⎛⎝⎜

⎞⎠⎟

Logistic: r=b-d0 and K= d1/(b-d0)9

Deriving the logistic from a “metaphysiological” point of view

dxdt

= bx − dx

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Resource extraction functions, density dependence, andcarrying capacity: a metaphysiological viewpoint.

•Getz, W. M. 1991. A unified approach to multispecies modeling. Natural ResourceModeling 5:393-421.•Getz, W. M. 1993. Metaphysiological and evolutionary dynamics of populations exploitingconstant and interactive resources: r-K selection revisited. Evol. Ecol. 7:287-305•Getz, W. M. 1994. A metaphysiological approach to modeling ecological populations andcommunities. In: S. A. Levin (ed), Frontiers in Mathematical Biology (Lecture Notes inBiomathematics, Vol. 100), Springer-Verlag, New York, pp 411-442•Getz, W. M., 1996. A hypothesis regarding the abruptness of density dependence and thegrowth rate of populations. Ecology 77:2014-2026•Getz, W. M. and N. Owen-Smith, 1999. A metaphysiological population model of storage invariable environments. Natural Resource Modeling 12:197-230•Getz, W. M., 1999. Population and Evolutionary Dynamics of Consumer-ResourceSystems. In: J. McGlade (ed.), Theoretical Ecology: Advances in Principles andApplications, Blackwell Science Limited, Oxford, England, pp 194-231

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Rfextraction

process

f(R,x)

x g(f)

growthprocess

R: resource; x population density f(R,x): per capita resource extraction rateg(f): per capita growth rate

Population growth rate = per capita growth rate X population density

dxdt

= xg( f ) = xg( f (R,x)) = xG(R, x) ; G = g f

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R: resourcex population density

dxdt

= xg( f (R, x))

dxdt

= xg( f (R, x)) = xr 1−m

f (R,x)⎛⎝⎜

⎞⎠⎟= xr 1−

m b + cR + x( )aR

⎛⎝⎜

⎞⎠⎟= hx 1−

xK

⎛⎝⎜

⎞⎠⎟

h == r 1− ma

⎛⎝⎜

⎞⎠⎟

, K =(a − m)cm

R −ba

f(R,x): per capita resource extraction rate

f (R, x) = aRb + cR + x

0

1

0 25 50 75 100R

f

decreasing x

-20

-15

-10

-5

0

5

0 25 50 75 100

g

fm

g(f): per capita growth rate

g( f ) = r 1−mf

⎛⎝⎜

⎞⎠⎟

m: metabolic breakeven point

0

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We have derived the logistic in 2 ways each with an entirely different interpretation!

dxdt

= bx − dx

Which is correct?What does this mean?

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Steps to modeling cont’d (Box 2.1)Verify your equations

check units of parameterscheck dimensions of expressionscheck dimensions balance in the equations

Analyze the equationsanalytical results where possible (equilibria, stability)graphical analyses where possible (phase plane; null isoclines)carry out simulations and make various plotscarry out sensitivity studies

Checks and balances verify your outputhave you addressed the questions you formulated

Relate results back to question have you answered your questiondo the answers make sense? have you learned something new?should you elaborate the model to explore more deeply?do results suggest experiments? identify gaps in understanding?

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The logistic model can be extended in many ways:e.g. Lotka-Volterra prey-

predator systems and beyond to trophic cascades

dxdt

= bx − dx

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Prey-Predator model (L-V)Formulate the question

what do you want to know?what is the dynamic nature of prey-predator interactions?state question as compactly as possibledo prey-predator interactions support sustained oscillations?

Determine basic ingredientsstart with simplest sufficient biological description:

define variablesx(t): density of predator populations at time ty(t): density of prey populations at time t

(implicit assumption: spatially homogeneous, all individuals identical)identify flows (draw flow diagram)

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x

y

* functions to be constructed

Fox extraction rate

b

*Rabbit

growth ratea

*

Fox growth in terms of rabbit

extractionc

*

d

*

Fox death rate

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dycxydtdy

bxyaxdtdx

−=

−=The simplest possible prey predator model

Simulation output using Stella (or Berkeley Madonna)

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2

Trophic Cascade

dxidt

= xigi fi( ) − xi+1 fi+1, i = 1,2, 3,... Getz 1991

growth(conversion)

feeding(extraction)

�

x0

�

x1

�

x2

�

x3

�

fi(xi−1, xi)

�

gi fi( )

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3

Feeding functions f

fi (xi−1,xi) = ai xi−1

Lotka-Volterragrowth

(conversion)

feeding(extraction)

�

x0

�

x1

�

x2

�

x3

�

fi(xi−1, xi)

�

gi fi( )

fi (xi−1,xi) =aixi−1

bi + xi−1

Holling Type II

Resource per-capita type II

fi (xi−1,xi) =ai xi−1

cixi + xi−1

Beddington (DeAngelis et al.)

fi (xi−1,xi) =aixi−1

bi + cixi + xi−1

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4

Growth functions g

growth(conversion)

feeding(extraction)

�

x0

�

x1

�

x2

�

x3

�

fi(xi−1, xi)

�

gi fi( )gi( fi) = ri fi −miLinear:

Lotka-Volterra, Rosenzweig-MacArthur

Hyperbolic: gi( fi) = ri 1− mi

fi

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Getz (1991)

Metaphysiological: gi( fi) = ri fi −mi −qifi

Getz and Owen-Smith (1999)

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Logistic growth

Hyperbolicgrowth: (not linear) g( f ) = r 1− m

f

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Beddington feeding: (not Holling II)

f (R,x) =aR

b+ cx + R

dxdt

= g f (R, x)( ) = hx 1 − xK

⎛ ⎝ ⎜

⎞ ⎠ ⎟

h = r 1 − ma

⎛ ⎝ ⎜

⎞ ⎠ ⎟ −

rmbaR

and K =(a −m)cm

R−bc

Logistic model:

where

growth(conversion)

feeding(extraction)

R

x

buffered resource

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6

Two species cascade on a constant resource

x0

�

x2

�

dx1

dt= r1φ1 −m1( )x1 − φ2x2

dx2

dt= r2φ2 −m2( )x2

x1

linear growth

�

dx1

dt= r1 1− κ1

φ1

⎛

⎝ ⎜

⎞

⎠ ⎟ x1 −φ2x2

dx2

dt= r2 1− κ2

φ2

⎛

⎝ ⎜

⎞

⎠ ⎟ x2

hyperbolic growth

metaphysiologicalgrowth

�

dx1

dt= r1φ1 −m1 −

q1

φ1

⎛

⎝ ⎜

⎞

⎠ ⎟ x1 −φ2x2

dx2

dt= r2φ2 −m2 −

q2

φ2

⎛

⎝ ⎜

⎞

⎠ ⎟ x2

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Two species cascades

�

dx1

dt= r1φ1 −m1( )x1 − φ2x2

dx2

dt= r2φ2 −m2( )x2

Linear growth

�

φ2(x1,xi) = a2x1

b2 + x1

Holling Type II extraction

+

=

�

dx1

dt= r1φ1 −m1( )x1 −

a2x1x2

b2 + x1

dx2

dt= r2a2x1x2

b2 + x1

−m2x2

This isRosenzweig-MacArthur

provided φ1 is

Beddingtonand underlying resources are constant

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8

Metaphysiological model

with storage

Metaphysiological with storagesee Getz and

Owen-Smith, 1999

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