At a given level of effort, it is reasonable to assume that the rate at which fi

Question

At a given level of effort, it is reasonable to assume that the rate at which fish are caught depends on the population the more fish there arc, 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. Show that if E 0. Show that y = y1 is unstable and y = y2 is asymptotically stable. 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 Determine E so as to maximize Y and thereby find the maximum sustainable yield Ym

Explanation / Answer

The equilibrium points of an autonomous differential equation are the values of the dependent variable, y, for which dy/dt = 0 In this case, we have that dy/dt = r(1 - y/K)y - Ey Set dy/dt equal to zero and solve for y to determine the equilibrium points: 0 = ry - r*K*y^2 - Ey 0 = (r - E)*y - r*y^2/K 0 = y*[(r-E) - (r/K)*y] y_eq1 = 0 and y_eq2 = (r-E)/(r/K) = K(1 - E/r) Note that these are always the equilibrium points of this equation; however, the second point is only physically reasonable if E

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