Homework: Modeling Applications

Consider a population $P(t)$ of fish living in a fish farm with limited food resources. Assume that his population is modeled by the logistic equation with a harvesting term, where the growth rate coefficient is $3$ per year, the food resources can sustain $900$ individuals, and suppose that every year we harvest $H$ fish.

$($a$)$ Write the differential equation $P^{\text{'}}=f(P)$ satisfied by the deer population.

$f(P)=$

Note: Type P for $P(t)$ and H for $H.$

$($b$)$ Find the extinction zone for a given harvesting rate $H.$

Extinction Zone:

$($c$)$ Suppose we want to harvest $60$ fish every year. What has to be the minimum initial population of fish in the farm, $P_{\text{m}\text{i}\text{n}},$ so the fish do not go extinct for any initial population larger than $P_{\text{m}\text{i}\text{n}}?$

$P_{\text{m}\text{i}\text{n}}=$

Note: Enter the exact value $($decimal or formula$)$ not an integer approximation.