Chapter 7.3, Problem 9P

Consider the system (3) in Example 1 of the text. Recall that this system has an asymptotically stable critical points at $(0.5,0.5),$ corresponding to the stable coexistence of the two population species. Now suppose that immigration or emigration occurs at the constant rates of $\delta a$ and $\delta b$ for the species $x$ and $y$, respectively. In this case, Eqs. (3) are replaced by

$\begin{array}{l}\frac{dx}{dt}=x(1-x-y)+\delta a,\\ \frac{dy}{dt}=y(0.75-y-0.5x)+\delta b.\end{array}$ (i)

The question is what effect this has on the location of the stable equilibrium point.

a) To find the new critical points, we must solve the equations

$\begin{array}{r}x(1-x-y)+\delta a=0,\\ y(0.75-y-0.5x)+\delta b=0.\end{array}$ (ii)

One way to proceed is to assume that $x$ and $y$ are given by power series in the parameter $\delta ;$ thus

$\begin{array}{cc}x=x_0+x_1\delta +\mathrm{...},& y=y_0+y_1\delta +\mathrm{...}\end{array}$ (iii)

Substitute Eqs. (iii) into Eqs. (ii) and collect terms according to powers of $\delta$.

b) From the constant terms (the terms not involving $\delta$), show that $x_0=0.5$ and $y_0=0.5$, thus confirming that in the absence of immigration or emigration, the critical points is $(0.5,0.5),$

c) From the terms that are linear in $\delta$, show that

$\begin{array}{cc}{x}_1=4a-4b& y_1=-2a+4b.\end{array}$ (iv)

d) Suppose that $a>0$ and $b>0$ so that immigration occurs for both species. Show that the resulting equilibrium solution may represent an increase in both populations, or an increase in one but a decrease in the other. Explain intuitively why this is a reasonable result.