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Project 1 Harvesting a Renewable Resource 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/dt =r(1 - y/K)y If the population is subjected to harvesting at a rate H(v.) members per unit time, then the harvested population is modeled by the differential equation dy/di=r(1 - y/Kly H(y, 1). (1) 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. 1. 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 pos- itive 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), Eq. (1) becomes dy/dt = r(1 y/K)y Ey. (i) 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 0. (b) Show that y = yi 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