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?