Suppose that the population y of a certain species of fish (e.g., tuna or halibut) in a given area of the ocean is described by the logistic equation dy=r (1-). dt 1 If the population is subjected to harvesting at a rate H(y, t) members per unit time, then the harvested population is modeled by the differential equation (1) dy = 1 r (1 - x) y y-H(y,t). K Although it is desirable to utilize the fish as a food source, it is intuitively clear that if too many fish are caught, then the fish population may be reduced below a useful level and possibly even driven to extinction. The following problems explore some of the questions involved in formulating a rational strategy for managing the fishery. 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(y,t) = 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. With this choice for H(y, t), Equation (1) becomes dy dt =1 (1- ) y - Ey. This equation is know as the Schaefer model after the biologist M. B. Schaefer, who applied it to fish populations. a) Show that if E 0. b) Show that y = y is unstable and 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 y2. Find Y as a function of the effort E. d) Determine E so as to maximize Y and thereby find the maximum sustainable yield Ym.