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## Homework Statement

Consider the population model

dP/dt = -P^2/50 + 2P

for a species of fish in a lake. Suppose it is decided that fishing will be allowed, but

it is unclear how many fishing licenses should be issued. Suppose the average catch

of a fisherman with a license is 3 fish per year (these are hard fish to catch).

(a) What is the largest number of licenses that can be issued if the fish are to have

a chance to survive in the lake?

(b) Suppose the number of fishing licenses in part (a) is issued. What will happen

to the fish population-that is, how does the behavior of the population depend

on the initial population?

(c) The simple population model above can be thought of as a model of an ideal

fish population that is not subject to many of the environmental problems of an

actual lake. For the actual fish population, there will be occasional changes in

the population that were not considered when this model was constructed. For

example, if the water level increases due to a heavy rainstorm, a few extra fish

might be able to swim down a usually dry stream bed to reach the lake, or the

extra water might wash toxic waste into the lake, killing a few fish. Given

the possibility of unexpected perturbations of the population not included in

the model, what do you think will happen to the actual fish population if we

allow fishing at the level determined in part (b)?

## The Attempt at a Solution

I am not sure how to start part a. Should I model the equation first like so :

dp/dt = -P^2/50 + 2P - 3C

Where C is license?