# Population change differential equations

Lake has 400 fish, estimated carrying capacity is 10,000 for that year. Fish tripled in first year. a) Assuming size of the fish population satisfies the logisitic equation find an expression for the size of population after t years. b) How long will it take for population for increase to 5000?

I do not know which logisitical equation to use or how to put the numbers in it, this is not like the practice problems.

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5
Am away from your book. The DE will have shape $\frac{dy}{dt}=ky(10000-y)$. – André Nicolas Jun 20 '12 at 0:18

## 1 Answer

As Andre commented above, the logistic equation has the form $$\frac{dy}{dt}=ry(10000-y),$$ and your initial condition $y(0)=400$.

This is a separable equation, whose solution is given by $$y(t)=\frac{1000 e^{10000rt}}{24+e^{10000 rt}}.$$ You also have the condition that $y(1)=3\cdot y(0)=1200$, from which it is possible to estimate $r$: $$r\approx 0.0001185.$$ Now you can find the moment $t$ when $y(t)=5000$ (it is $2.6805$).

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