Chapter 7.3, Problem 15P

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.

$\begin{array}{l}x\text{'}=(3+x)(1-x+y),\\ \begin{array}{cc}y\text{'}=(y-1)(1+x+\alpha y);& \alpha >0\end{array}\end{array}$