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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time
x or
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at t=0,x=20y=19.25
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x (Prey)
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reda
tors
)NonlinearLinear