Consider the population model

$\frac{dP}{dt}=2P-\frac{P^2}{50}$

for a species of fish in a lake $($notice this is a Logistic model$).$ 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?

$($Hint: think about a bifurcation $--$ a qualitative change in the behavior of solutions such as population stability at a non$-$zero value vs die$-$off to population of zero.$)$