Chapter 7.3, Problem 16P

In each of Problem 15 and 16:

a) Find the critical points.

b) Determine the value of $\alpha$, denoted by ${\alpha }_0$, where two critical points coincide.

c) By finding the approximating linear systems and their eigenvalues, determine how the stability properties of these two critical points change as $\alpha$ passes through the bifurcation point ${\alpha }_0$.

d) Draw phase portraits for values of $\alpha$ near ${\alpha }_0$, to show how the transition through the bifurcation point occurs.

$\begin{array}{cc}x\text{'}=y(\alpha -2x+3y),& y\text{'}=(4-x)(3+y)\end{array}$