Textbook Problem

In a fish farm, a population of fish is introduced into a pond and harvested regularly. A model for the rate of change of the fish population is given by the equation

$\frac{dP}{dt}=r^0\left(1-\frac{P\left(t\right)}{P_c}\right)P\left(t\right)-\text{β+}P\left(t\right)$

where r₀ is the birth rate of the fish, $P_c$, is the maximum population that the pond can sustain (called the carrying capacity), and $\text{β}$ is the percentage of the population that is harvested.

(a) What value of dP/dt corresponds to a stable population?

(b) If the pond can sustain 10, 000 fish, the birth rate is 5%, and the harvesting rate is 4%, find the stable population level.

(c) What happens if $\text{β}$ is raised to 5%?