In a fish farm a population of fish is introduced into a pon

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 where tau 0 is the birth rate of the fish, Pc is the maximum population that the pond can sustain (called the carrying capacity), and beta is the percentage of the population that is harvested. What value of dP/dt corresponds to a stable population? If the pond can sustain 10,000 fish, the birth rate is 5%, and the harvesting rate is 4%, find the stable population level. What happens if ,beta is raised to 5%?

Solution

(a) d2P/dt2 = r0 - 2P(t)/Pc - = 0

Pc (r0 - ) /2 = P(t)

dP/dt = P(t) [ r0 (1 - P(t)/Pc) - ] = P(t) [ r0 (2 - r0 + )/2 - ] = Pc (-r0)(ro^2 - 2r0 + (1-r0))/4

(b) P(t) = Pc(R0-)/2 = 10000 * (5 - 4) / 2 = 5000

(c) if = 5%,

P(t) = 0