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## Homework Statement

Biologists stocked a lake with 143 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 8000. The number of fish doubled in the first year.

(a) Assuming that the size of the fish population satisfies the logistic equation

dP/dt = kP(1-P/N)

determine the constant , and then solve the equation to find an expression for the size of the population after years.

## Homework Equations

none in particular

## The Attempt at a Solution

I didn't think that this problem would be difficult but the answer I am getting doesn't make any sense. Basically I try to solve for P to start with. So dP/(P-P^2/N) = kdt and integrate

ln(P/(P-N)) = kt+c

It would make the most sense to solve for k right away so

(ln(P/(P-N)) - c)/t = k

The problem though is that N = 8000 while P will always be less than this. This will cause there to be a negative number inside the natural log and you can't take the log of a negative number. Where am I going wrong?

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