CEE262C Lecture 3: Predator-prey models 1

CEE262C Lecture 3: The predator-prey problem

Overview

• Lotka-Volterra predator-prey model– Phase-plane analysis– Analytical solutions– Numerical solutions

References: Mooney & Swift, Ch 5.2-5.3;

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Compartmental Analysis

• Tool to graphically set up an ODE-based model– Example: Population

Immigration: ix

Births: bx

Deaths: dx

Emigration: ex

Population: x

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Logistic equation

Population: x

Can flow both directions but thedirection shown is defined aspositive

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Income class model

Lowerx

Middley

Upperz

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• For a system

the fixed points are given by the Null

space of the matrix A. For the income class

model:

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Classical Predator-Prey Model

Predator y Prey x

Die-off in absenceof prey

dy

Growth in absence ofpredators

ax

bxycxy

Lotka-Volterra predator-preyequations

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Assumptions about the interaction term xy

• xy = interaction; bxy: b = likelihood that it results in a prey death; cxy: c = likelihood that it leads to predator success. An "interaction" results when prey moves into predator territory.

• Animals reside in a fixed region (an infinite region would not affect number of interactions).

• Predators never become satiated.

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Phase-plane analysis

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Analytical solution

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Solution with Matlab

% Initial condition is a low predator population with

% a fixed-point prey population.

X0 = [x0,.25*y0]';

% Decrease the relative tolerance

opts = odeset('reltol',1e-4);

[t,X]=ode23(@pprey,[0 tmax],X0,opts);

lvdemo.m

function Xdot = pprey(t,X)

% Constants are set in lvdemo.m (the calling function)

global a b c d

% Must return a column vector

Xdot = zeros(2,1);

% dx/dt=Xdot(1), dy/dt=Xdot(2)

Xdot(1) = a*X(1)-b*X(1)*X(2);

Xdot(2) = c*X(1)*X(2)-d*X(2);

pprey.m

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0 10 20 30 40 50 600

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time

x or

yx

y

at t=0,x=20y=19.25

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x (Prey)

y (P

reda

tors

)NonlinearLinear