Chapter 7.P2, Problem 1P

Consider again the system

$\begin{array}{c}\frac{dx}{dt}=x(1-x-y),\\ \frac{dy}{dt}=y(0.75-y-0.5x),\end{array}$ (i)

Which appeared in Example 1 of Section7.3. A constant effort model, applied to the species $x$ alone, assumes that the rate of growth of $x$ is altered by including the term $-Ex$, where $E$ is a positive constant measuring the effort investedin harvesting members of species $x$. This assumption means that for a given effort $E$, the rate of catch is proportional to the population $x$, and that for a given population $x$, the rate ofcatch is proportional to the effort $E$. Based on this assumption, Eqs. (i) are replaced by

$\begin{array}{c}\frac{dx}{dt}=x(1-x-y)-Ex=x(1-E-x-y),\\ \frac{dy}{dt}=y(0.75-y-0.5x),\end{array}$ (ii)

(a) For $E=0$, the critical points of Eqs. (ii) are as in Example 1 of Section 7.3. As $E$ increases, some critical points move, while others remain fixed. Which ones moveand how?

(b) For a certain value of $E$, denoted by $E_0$, the asymptotically stable node originally at $(0.5,0.5)$ coincides with thesaddle point $(0,0.75)$. Find the value of $E_0$.

(c) Draw a direction field and/or a phase portrait for $E=E_0$, and for values of $E$ slightly less than and slightly greater than $E_0$.

(d) How does the nature of the critical point $(0,0.75)$ change as $E$ passes through $E_0$?

(e) What happens to the species $x$ for $E>E_0$?