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 β y/k)y β 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 < r, then there are two equilibrium points, y1 = 0 and y2 = k(1 β e/r) > 0. (b) show that y = y1 is unstable and y = y2 is asymptotically stable.
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Suppose that f : rn β rm and that a β k, where k is a connected subset of rn . suppose further that for each x β k there exists a Ξ΄x > 0 such that f(x) = f(y) for all y β bΞ΄x (x). prove that f is constant on k; that is, f(x) = f(a) for all x β k
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At a given level of effort, it is reasonable to assume that the rate at which fish are caught depend...
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