Chapter 12.3, Problem 25E

When one of the populations in a competing species model is being harvested, the phase plane diagram is completely changed. For example, consider the competition model

$\frac{dx}{dt}=x\left(1-4x-y\right)-h$,

$\frac{dy}{dt}=y\left(1-2y-5x\right)$,

where $h$ is the constant harvest rate. For $h=0,1/32,\text{and }5/32$, determine the critical points, discuss the type and stability of the critical points, and sketch the phase plane diagrams. In each case discuss the implications of the completion model.