Chapter 7.P2, Problem 3P

Consider the system (i) in Problem 1, and assume now that both $x$ and $y$ are harvested, with efforts $E_1$ and $E_2$, respectively. Then the modified equations are

$\begin{array}{c}\frac{dx}{dt}=x(1-E_1-x-y),\\ \frac{dy}{dt}=y(0.75-E_2-y-0.5x).\end{array}$ (v)

(a) When $E_1=E_2=0$, there is an asymptotically stable node at $(0.5,0.5)$. Find conditions on $E_1$ and $E_2$ that permit thecontinued long-term survival of both species.

(b) Use the conditions found in part (a) to sketch the region in the $E_1{E}_2$-plane that corresponds to the long-term survival of both species. Also identify regions where one species survives but not the other, and a region where both decline toextinction.

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