Chapter 7.5, Problem 19P

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

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

Where $a$ is a positive parameter. Observe that this system includes the one in Problem 18 above and also the one in Problem 13 of section 7.4.

(a) Find all the critical points.

(b) Consider the critical point in the interior of the first quadrant. Find the eigenvalues of the approximate linear system. Determine the value $a_0$ where this critical point changes from asymptotically stable to unstable.

(c) Draw a phase portrait for a value of $a$ slightly greater than $a_0$. Observe that a limit cycle has appeared. How does the limit cycle changes as $a$ increases further?