Chapter 7.P2, Problem 2P

Consider the system

$\begin{array}{c}\frac{dx}{dt}=x(1-x-y),\\ \frac{dy}{dt}=y(0.8-0.6y-x),\end{array}$ (iii)

Which appeared in Example 2of Section 7.3.If constant-effort harvesting is applied to species $x$, then the modified equations are

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

(a) For $E=0$, the critical points of Eqs. (iv) are as in Example 2 of Section 7.3. As $E$ increases, some critical points move, while others remain fixed. Which ones move and how?

(b) For a certain value of $E$, denoted by $E_0$, the asymptotically stable node originally at $(0.5,0.5)$ coincides with the saddle point $(1,0)$. 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 $E_0$. Estimate the basin of attraction for each asymptotically stable critical point.

(d) Consider the asymptotically stable node originally at $(1,0)$. How does the nature of the critical point change as $E$ passes through $E_0$?

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