Chapter 2.P1, Problem 1P

Constant Effort Harvesting. At a given level of effort, it is reasonable to assume that the rate at which fish are caught depends on the population $y$: the more fish there are, the easier it is to catch them. Thus we assume that the rate at which fish are caught is given by $H\left(y,t\right)=Ey$, where $E$ is a positive constant, with units of $\text{1/time}$, that measures the total effortmade to harvest the given species of fish. With this choice for $H\left(y,t\right)$, Eq. $\left(1\right)$ becomes

$dy/dt=r\left(1-y/K\right)y-Ey$. $\left(\text{i}\right)$

This equation is known as the Schaefer model after the biologist M. B. Schaefer, who applied it to fish populations.

(a) Show that if $E<r$, then there are two equilibrium points, $y_1=0$ and $y_2=K\left(1-E/r\right)>0$.

(b) Show that $y=y_1$ is unstable and $y=y_2$ is asymptotically stable.

(c) A sustainable yield $Y$ of the fishery is a rate at which fish can be caught indefinitely. It is the product of the effort $E$ and the asymptotically stable population $y_2$. Find $Y$ as a function of the effort $E$. The graph of this function is known as the yield–effort curve.

(d) Determine $E$ so as to $Y$ and thereby find the maximum sustainable yield $Y_m$.