A model that describes the population of a fishery in which harvesting takes place at a constant rate is given by dP dt = kP − h
, where k and h are positive constants. (a) Solve the DE subject to P(0) = P0. P(t) = (b) Describe the behavior of the population P(t) for increasing time in the three cases P0 > h/k, P0 = h/k, and 0 < P0 < h/k. P0 > h/k P(t) decreases as t increases P(t) increases as t increases P(t) = 0 for every t P(t) = P0 for every t P0 = h/k P(t) decreases as t increases P(t) increases as t increases P(t) = 0 for every t P(t) = P0 for every t 0 < P0 < h/k P(t) decreases as t increases P(t) increases as t increases P(t) = 0 for every t P(t) = P0 for every t (c) Use the results from part (b) to determine whether the fish population will ever go extinct in finite time, that is, whether there exists a time T > 0 such that P(T) = 0. If the population goes extinct, then find T. (If the population does not go extinct, enter DNE.)
