Chapter 2.P1, Problem 2P

Constant Yield Harvesting. In this problem, we assume that fish are caught at a constant rate $h$ independent of the size of the fish population, that is, the harvesting rate $H\left(y,t\right)=h$. Then $y$ satisfies

$dy/dt=r\left(1-y/K\right)y-h=f\left(y\right)$. $\left(\text{ii}\right)$

The assumption of a constant catch rate $h$ may be reasonable when $y$ is large but becomes less so when $y$ is small.

(a) If $h<rK/4$, show that Eq. $\left(\text{ii}\right)$ has two equilibrium points $y_1$ and $y_2$ with $y_1<y_2$; determine these points.

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

(c) From a plot of $f\left(y\right)$ versus $y$, show that if the initial population $y_0>y_1$, then $y\to y_2$ as $t\to \infty$, but if $y_0<y_1$, then $y$ decreases as $t$ increases. Note that $y=0$ is not an equilibrium point, so if $y_0<y_1$, then extinction will be reached in a finite time.

(d) If $h>rK/4$, show that $y$ decreases to zero as $t$ increases regardless of the value $y_0$.

(e) If $h=rK/4$, show that there is a single equilibrium point $y=K/2$ and that this point is semistable. Thus the maximum sustainable yield is $h_m=rK/4$, corresponding to the equilibrium value $y=K/2$. Observe that $h_m$ has the same value as $Y_m$ in Problem $1\left(\text{d}\right)$. The fishery is considered to be overexploited if $y$ is reduced to a level below $K/2$.

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 effort made 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 $Y_m$.