Chapter 7.3, Problem 17P

Suppose that a certain pair of competing species are described by the system

$\begin{array}{l}\frac{dx}{dt}=x(4-x-y),\\ \frac{dy}{dt}=y(2+2\alpha -y-\alpha x),\end{array}$

Where $\alpha >0$ is a parameter.

a) Find the critical points. Note that $(2,2)$ is a critical points for all values of $\alpha$.

b) Determine the nature of the critical points $(2,2)$ for $\alpha =0.75$ and for $\alpha =1.25$. There is a value of $\alpha$ between $0.75$ and $1.25$ where the nature of the critical point changes abruptly. Denote this value by ${\alpha }_0$; it is also called a bifurcation points.

c) Find the approximate linear system near the point $(2,2)$ in term of $\alpha$.

d) Find the eigenvalues of the linear system in part (c) as functions of $\alpha$. Then determine ${\alpha }_0$.

e) Draw phase portraits near $(2,2)$ for $\alpha ={\alpha }_0$ and for values of $\alpha$ slightly less than, and slightly greater than, ${\alpha }_0$. Explain how the transition in the phase portrait takes place as $\alpha$ passes through ${\alpha }_0$.