At a given level of effort, it is reasonable to assume

Chapter 2, Problem 20

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At a given level of effort, it is reasonable to assume that the rate at which fish are caughtdepends on the population y: the more fish there are, the easier it is to catch them. Thus weassume 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 offish. To include this effect, the logistic equation is replaced bydy/dt = r(1 y/K)y Ey. (i)This equation is known as the Schaefer model after the biologist M. B. Schaefer, whoapplied it to fish populations.(a) Show that if E < r, then there are two equilibrium points, y1 = 0 andy2 = 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 afunction of the effort E; the graph of this function is known as the yieldeffort curve.(d) Determine E so as to maximize Y and thereby find the maximum sustainable yield Ym.