Troubleshooting - A Guide to MATLAB for Beginners and Experienced Users - page 289

Thegraphicalevidencesuggeststhat:Thesolutionthatstartsatzerostays
there; all the others tend toward the constant solution 1.

(c)

clear all; close all; hold on
f = inline('[x(1) - x(1)ˆ2 - 0.5*x(1)*x(2); x(2) - x(2)ˆ2 -
0.5*x(1)*x(2)]', 't', 'x');
for a = 0:1/12:13/12
for b = 0:1/12:13/12
[t, xa] = ode45(f, [0 3], [a,b]);
plot(xa(:, 1), xa(:, 2))
echo off
end
end
axis([0 13/12 0 13/12])

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(d)

The endpoints on the curves are the start points. So clearly any curve
that starts out inside the first quadrant, that is, one that corresponds to
a situation in whichbothpopulations are present at the outset, tends
toward a unique point — which from the graph appears to be about
(2/3,2/3). In fact if x = y = 2 / 3, then the right sides of both equations in
(4) vanish, so the derivatives are zero and the values of x ( t ) and y ( t )
remain constant — they don't depend on t . If only one species is present
at the outset, that is, you start out on one of the axes, then the solution