ORDINARY DIFFERENTIAL EQUATIONS

COMPETING SPECIES

The following differential equation describes the populations of two competing species

x' (t) = 3x(1-x - y)

y 3

(y'(t) = 2y (1-2-2x)

The question I am going to ask you requires some context. To help understand the behavior of these

equations, I'd like you first to make the following plots:

First show x(t) and y(t) on the same axes under the following initial conditions. You should have one

figure window per set of initial conditions

x(0) = 0, y(0) = 1

x(0) = 0.5, y(0) = 0

What do these 2 plots tell you about the population of the species under these initial conditions?

Show x(t), y(t) on the same axes under the following initial conditions. You should have one figure

window per set of initial conditions.

•

x(0) = 1, y(0) = 1

• x(0) = 5, y(0) = 1

What do these 2 plots tell you about the population of the species under these initial conditions?

For y(0) = 1 (same as above), use an iterative method to determine the smallest value of x(0) for

which x eventually comes to dominate y. (Something like bisection works well)

Hint: If you plot it, it's easy to tell which is dominating, but you need MATLAB to do it automatically.

Think about what you look for, and think about how you could get MATLAB to check that automatically.