Chapter 7.4, Problem 9P

Consider the system

$\begin{array}{cc}\frac{dx}{dt}=ax\left(1-\frac{y}{2}\right),& \frac{dy}{dt}=bx\left(-1+\frac{x}{3}\right),\end{array}$

Where $a$ and $b$ are positive constants. Observe that this system is the same as in the example in the text if $a=1$ and $b=0.75$. Suppose the initial conations are $x(0)=5$ and $y(0)=2$.

a) Let $a=1$ and $b=1$. Plot the trajectory in the phase plane and determine (or estimate) the period of the oscillation.

b) Repeat part (a) for $a=3$ and $a=\frac{1}{3}$, with $b=1$.

c) Repeat part (a) for $b=3$ and $b=\frac{1}{3}$, with $a=1$.

d) Describe how the period and the shape of the trajectory depend on $a$ and $b$.