Harvesting Fish from a Pond: A new pond is built at a recreation park and is sto

Question

Harvesting Fish from a Pond: A new pond is built at a recreation park and is stocked for fishing. Assume that the fish population if left alone would grow exponentially with a growth constant of r = 0.6 (the units are 1/months). Park visitors are allowed limited fishing, harvesting exactly 270 fish each month.

(a) Write down the differential equation that models the system. (Assume time is measured in months.)

(b) Find the equilibrium population (by hand).

(c) Use Maple to plot the slope field, and plot solution curves for various different initial population sizes (say 100, 300, 500, 700 fish). (Print and hand in Maple plots. Maple plots are HUGE when printed. Please use "print preview" and reduce the plots to a reasonable size.)

(d) Describe in a sentence or two the "big picture" -- What seems to happen to the fish population for various initial values? Do they approach the equilibrium solution?

(e) Based on this model, is it possible that the fish population stays at the equilibrium level? Is it likely? (Explain briefly.)

(f) Solve the differential equation by direct integration.

(g) Solve the initial value problem, assuming that 350 fish are initially placed in the pond. What happens to the population in this case as t increases? (Find find the limit and interpret what this means for the fish population. Also compare your solution to the slope field.)

(h) Now suppose the manager of the pond decides to change the number of fish removed every month to 180.

(i) Write down the new d.e.

(ii) Find the new equilibrium population.

(iii) Explain the change in the equilibrium population - why does it make sense in terms of the fish population? [Respond with a few clear sentences.]

(iv) What will happen now if the initial population is 350 fish? (You don't have to solve again - just think about the slope field.)

Explanation / Answer

a) r=0.6

harvesting exactly 270 fish

F = 270*(1-e^-0.6/t)

b)

Differentiating we get

F=850

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