A pond in a fish hatchery starts out with 1000 fish, which is also the capacity...

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A pond in a fish hatchery starts out with 1000 fish, which is also the capacity of the pond. The fish population is known to grow at the rate of r = 0.01/day. They are harvested at the rate of h fish/day. (a) A simple exponential growth model for the fish population is given by dP = rP - h dt With this model, how many fish hmax can be caught at most per day without causing the population to go extinct in the long run? What will the fish population P(t) be in this case? Explain your answer. (Hint: draw a phase line showing the equilibrium solution Pequil of the ODE, and explain what happens if P(0) > Pequil, P(0) = Pequil, P(0) < Pequil.) (b) Another model for the fish hatchery is given the logistic growth law dP = rP(1 ) - P - h dt K With this model, how many fish hmax can be caught at most per day without causing the population to go extinct in the long run? What will the fish population P(t) be in this case? Explain your answer. (Hint: sketch P' vs P for various values of h, as well as the corresponding phase line, then find the critical value hmax that still gives you a nonzero equilibrium population.) (c) Which model do you think is more realistic? A pond in a fish hatchery starts out with 1000 fish, which is also the capacity of the pond. The fish population is known to grow at the rate of r = 0.01/day. They are harvested at the rate of h fish/day. (a) A simple exponential growth model for the fish population is given by dP = rP - h dt With this model, how many fish hmax can be caught at most per day without causing the population to go extinct in the long run? What will the fish population P(t) be in this case? Explain your answer. (Hint: draw a phase line showing the equilibrium solution Pequil of the ODE, and explain what happens if P(0) > Pequil, P(0) = Pequil, P(0) < Pequil.) (b) Another model for the fish hatchery is given the logistic growth law dP = rP(1 ) - P - h dt K With this model, how many fish hmax can be caught at most per day without causing the population to go extinct in the long run? What will the fish population P(t) be in this case? Explain your answer. (Hint: sketch P' vs P for various values of h, as well as the corresponding phase line, then find the critical value hmax that still gives you a nonzero equilibrium population.) (c) Which model do you think is more realistic?