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 Ey, where E is a positive constant, with units of 1/time, that measures the total effort made to harvest the given species of fish. To include this effect, the logistic equation is replaced by
dy/dt = r(1 − y/K)y − Ey.
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 1 = 0 and y2 = K(1 − E/r) > 0. (b) Show that y = y1 is unstable and y = y2 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 y2. 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 maximize Y and thereby find the maximum sustainable yield Ym.
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