Chapter 1.7, Problem 19E

Consider the population model
   $\frac{dP}{dt}=2P-\frac{P^2}{50}$
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 arc hard fish to catch).
(a) What is the largest number of licenses that can be issued lithe 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 no included in the model, what do you think will happen to the actual fish population if we allow fishing at the level determined in pan (b)?