Problem 36E

36. 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

$$ \frac{d P}{d t}=r_{0}\left(1-\frac{P(t)}{P_{c}}\right) P(t)-\beta P(t) $$

where $r_{0}$ is the birth rate of the fish, $P_{c}$ is the maximum population that the pond can sustain (called the carrying capacity), and $\beta$ is the percentage of the population that is harvested.

(a) What value of $d P / d t$ 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 $\beta$ is raised to $5 \%$ ?

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