Chapter 7.4, Problem 12P

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

$\begin{array}{cc}\frac{dx}{dt}=x(a-\sigma x-\alpha y),& \frac{dy}{dt}=y(-x+\gamma x),\end{array}$

Where $a,\sigma ,\alpha ,c,\text{and}\gamma$ are positive constants

a) Find all critical points of the given system. How does their location change as $\sigma$ increases from zero ? Assume that $a/\sigma >c/\gamma$ or that $\sigma <a\gamma /c$. Why is this assumption necessary?

b) Determine the nature and stability characteristics of each critical point.

c) Show that there is a value of $\sigma$ between zero and $a\gamma /c$ where the critical point in the interior of the first quadrant change from a spiral point to a node.

d) Describe the effect on the two population as $\sigma$ increases from zero to $a\gamma /c$.