Biologists stocked a lake with \\(400\\) fish and estimated the carrying capacit

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 (7500). The number of fish doubled in the first year.

(a) Assuming that the size of the fish population satisfies the logistic equation [rac{dP}{dt} = kP left( 1 - rac{P}{K} ight) ,] 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 (3750) (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 = 7500, Po = 400, P(1) = 800, 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 0.857 /yr.
b)

Use this r, P(t) = 3750, and the 3rd equation, to get t; is is3.337 yr,

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