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1. Consider a lake that is stocked with walleye pike and the population of pike is governed by the logistic equation:

P' = 0.1P(1-P/10)

where time is measured in days and P in thousands of fish. Supposed that fishing is started in this lake and that 100 fish are removed each day.

a) Modify the logistic model to account for the fishing.

b) Find and classify the equilibrium points for your model.

c) Use qualitative analysis to completely discuss the fate of the fish population with this model. In particular, if the initial fish population is 1000, what happens to the fish as time passes? What will happen to an initial population of 2000 fish?

2. The attempt at a solution

a) I got: P' = 0.1P(1-P/10) - .1

b) I set P' equal to 0 and solved, I then got 1.12702 and 8.87298.

c) Drawing a phase line, I see that any number less than 1.12702 results in a negative rate in population, therefore an initial population of 1000 would die off and eventually reach 0. As for the initial population of 2000, the phase line shows a negative growth until the equilibrium point, therefore the initial 2000 fish will die off until they reach that equilibrium point of 1.12702.

Is what I did correct? I'm not sure, and was hoping to get insight.