Troubleshooting - A Guide to MATLAB for Beginners and Experienced Users - page 290

tends toward either (1,0) or (0,1) depending on whether x or y is the
species present. That is precisely the behavior we saw in part (b).

(e)

close all; hold on
f = inline('[x(1) - x(1)ˆ2 - 2*x(1)*x(2); x(2) - x(2)ˆ2 -
2*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

This time most of the curves seem to be tending toward one of the points
(1,0) or (0,1) — in particular, any solution curve that starts on one of the
axes (corresponding to no initial poulation for the other species) does so. It
seems that whichever species has a greater population at the outset will
eventually take over all the population — the other will die out. But there
is a delicate balance in the middle — it appears that if the two populations
are about equal at the outset, then they tend to the unique population
distribution at which, if you start there, nothing happens. That value
looks like (1 / 3,1 / 3). In fact this is the value that renders both sides of (5)
zero and its role is analogous to that played by (2 / 3,2 / 3) in part (d).