Chapter 7.4, Problem 11P

In Problems 11 and 12, we consider the effect of modifying the equation for the prey $x$ by including a term $-\sigma x^2$ so that this equation reduces to a logistic equation in the absence of the predator $y$. Problem 11 deals with a specific system of this kind and Problem 12 takes up this modification to the general Lotka-Volterra system. The systems in Problems 3 and 4 are other examples of this type.

Consider the system

$\begin{array}{cc}x\text{'}=x(1-\sigma x-0.5y),& y\text{'}=y(-0.75+0.25x),\end{array}$

where $\sigma >0$. Observe that this system is a modification of the system $(2)$ in Example 1.

a) Find all of the critical points. How does their location change as $\sigma$ increases from zero? Observe that there is a critical point in the interior of the first quadrant only if $\sigma <\frac{1}{3}$.

b) Determine the type and stability property of each criticalpoint. Find the value ${\sigma }_1<\frac{1}{3}$ where the nature of the criticalpoint in the interior of the first quadrant changes. Describe the change that takes place in this critical point as $\sigma$ passesthrough ${\sigma }_1$.

c) Draw a direction field and phase portrait for a value of $\sigma$ between zero and ${\sigma }_1$; for a value of $\sigma$ between ${\sigma }_1$ and $\frac{1}{3}$

d) Describe the effect on the two populations as $\sigma$ increasesfrom zero to $\frac{1}{3}$.