A logistic differential equation can be written as follows:
[tex] \frac{dP}{dt} = rP[1- \frac{P}{K}] [/tex]
where r = growth parameter and K = carrying parameter.
In order to write you equation in this form, you have to regroup 2:
[tex] \frac{dP}{dt} = 2P[1- \frac{P}{10000}] [/tex]
Therefore, in you case r = 2 and K = 10000
To solve the logistic differential equation you need to solve:
[tex] \int { \frac{1}{[P(1- \frac{P}{K})] } } \, dP = \int {r} \, dt [/tex]
The soution will be:
P(t) = [tex] \frac{P(0)K}{P(0)+(K-P(0)) e^{-rt} } [/tex]
where P(0) is the initial population.
In your case, you'll have:
P(t) = [tex] \frac{3E7}{3E3+7E3 e^{-2t} } [/tex]
Now you have to calculate the limit of P(t).
We know that
[tex] \lim_{t \to \infty} e^{-2t} -\ \textgreater \ 0
[/tex]
hence,
[tex] \lim_{t \to \infty} P(t) = \lim_{t \to \infty} \frac{3E7}{3E3+0} = 10^{4} [/tex]
