Chapter 7.4, Problem 13P

In the Lotka-Volterra equations, the interaction between the two species is modeled by terms proportional to the product $xy$ of the respective populations. If the prey population ismuch larger than the predator population, this may overstate the interaction; for example, a predator may hunt only whenit is hungry, and ignore the prey at other times. In this problem, we consider an alternative model of a type proposed byRosenzweig and MacArthur.

a) Consider the system

$\begin{array}{l}x\prime =x\left(1-0.2x-\frac{2y}{x+6}\right),\\ y\prime =y\left(-0.25+\frac{x}{x+6}\right)\end{array}$

Find all of the critical points of this system.

b) Determine the type and stability characteristics of eachcritical point.

c) Draw a direction field and phase portrait for this system