Biologists stocked a lake with 400 fish and estimated the carrying capacity (the

Question

Biologists stocked a lake with 400 fish and estimated
the carrying capacity (the maximal population for the
fish of that species in that lake) to be 7300. The number of fish
tripled in the first year.
(a) Assuming that the size of the fish population satisfies the
logistic equation
dP/dt = kP 1? (P/K)
determine the constant k, and then solve the equation to find an
expression for the size of the population after t years.
k = ?
P(t) = ?
(b) How long will it take for the population to increase to 3650
(half of the carrying capacity)?
It will take ? years

Explanation / Answer

The logistic equation is: P(t) = [KPoe^rt] / [K + Po(e^rt - 1)] The data given are: K = 7300, Po = 400, P(1) = 1200, and t =1 Solving the equation for r, r = (1/t) ln[(KP(t) - PoP(t)) / (KPo - PoP(t))] Solving the equation for t, t = (1/r) ln[(KP(t) - PoP(t)) / (KPo - PoP(t))] Use the given data and the 2nd equation to get r; it is 1.185 /yr. Use this r, P(t) = 3650, and the 3rd equation, to get t; is is 2.68 yr, the answer.

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