Chapter 7.5, Problem 18P

Problems 18 and 19 extend the consideration of the Rosenzweig-MacArthr predator-prey model introduced in Problem 13 of section 7.4.

Consider the system

$\begin{array}{l}x\text{'}=x\left(2.4-0.2x-\frac{2y}{x+6}\right),\\ y\text{'}=y\left(-0.25+\frac{x}{x+6}\right).\end{array}$

Observe that this system differs from that in Problem 13 of section 7.4 only in the growth rate for the prey.

(a) Find all the critical points.

(b) Determine the type and stability of each critical point.

(c) Draw a phase portrait in the first quadrant and conclude that there is an asymptotically stable limit cycle. Thus this model predicts a stable long-term oscillation of the prey and predator populations.