LynxHareDemo.m

Contents

Overview

This script illustrates the solution of a system of first-order Lotka-Volterra (predator-prey) IVPs:

$$ \begin{array}{rcl} H'(t) & = & a H(t) - b H(t) L(t),\quad H(0) = 75 \\ L'(t) & = & -c L(t) + d H(t) L(t),\quad L(0) = 30 \end{array} $$

for finding the hare and lynx population (and to compare with the data recorded by the Hudson Bay Company).

Here $t$ stands for time, $H$ and $L$ stand for hare and lynx populations, and

The right-hand side function is coded in LynxHare.m

Initialization

clear all
clc
close all
f = @LynxHare;              % function handle for right-hand side function
t0 = 1855; y0 = [75; 30];   % initial time and initial population in thousands
tend = 1930;                % end time
N = 10000;                  % number of time steps
h = (tend-t0)/N;            % stepsize

Call Euler

We will discuss this function later

[t,y] = Euler(t0,y0,f,h,N);

We could also use a MATLAB ODE solver such as

[t,y] = ode23(f,[t0 tend],y0);

Plot

of the populations vs. time

figure
hold on
set(gca,'Fontsize',14)
xlabel('t','FontSize',14);
ylabel('Populations','FontSize',14);
plot(t,y(:,1),'r','LineWidth',2);    % hares
plot(t,y(:,2),'b','LineWidth',2);    % lynx
legend('Hares','Lynx');
hold off

Plot of the populations in phase space

figure
hold on
title('Approximate solution in phase plane', 'Fontsize',16)
set(gca,'Fontsize',14)
xlabel('Hares','Fontsize',14)
ylabel('Lynx','Fontsize',14)
plot(y(:,1),y(:,2),'g','LineWidth',1);   % lynx vs. hares (phase plane)
hold off


© Greg Fasshauer, IIT. Published with MATLAB® 7.11