In a fish farm, a population of fish is introduced into a pond and is harvested

Question

In a fish farm, a population of fish is introduced into a pond and is harvested regularly. A model for the rate of change of the fish population is given by the equation dP/dt = r_0 (1 - P(t)/P_c) P(t) = beta P(t), where r_0 is the birth rate of the fish, Pc is the maximum population that the pond can sustain, and beta is the percentage of the population that can be harvested. What value of dP/dt corresponds to a stable maximum population? If the pond can sustain 8,000 fish, the birth rate is 13 percent and the harvesting rate is 8 percent, find the stable population level and round down to the number of whole fish.

Explanation / Answer

a) If dP/dt = 0; then the population will remain stable.

Under such conditions,the number of new fish born will be harvested and number of fish in the original pond (which was stable) will remain constant.

b) dP/dt =0

13 (1- P(t) / 8000) P(t) = 8 * P(t)

(104000- 13Pt) = 8000 *8

104000 - 64000 = 13Pt

40000 = 13P(t) ; P(t) ~ 3077 = This much population should be present after harvesting , to attain a stable population...

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