Question

The population P(t) of a species satisfies the logistic differential equation dP/dt= P(2-(P/5000)) where the initial population 3,000 and t is the time in years. What is the lim t>infinity P(t)?

Answer 1

A logistic differential equation can be written as follows:
$\frac{dP}{dt} = rP[1- \frac{P}{K}]$

where r = growth parameter and K = carrying parameter.

In order to write you equation in this form, you have to regroup 2:
$\frac{dP}{dt} = 2P[1- \frac{P}{10000}]$

Therefore, in you case r = 2 and K = 10000

To solve the logistic differential equation you need to solve:

$\int { \frac{1}{[P(1- \frac{P}{K})] } } \, dP = \int {r} \, dt$

The soution will be:

P(t) = $\frac{P(0)K}{P(0)+(K-P(0)) e^{-rt} }$

where P(0) is the initial population.

In your case, you'll have:

P(t) = $\frac{3E7}{3E3+7E3 e^{-2t} }$

Now you have to calculate the limit of P(t).
We know that
$\lim_{t \to \infty} e^{-2t} -\ \textgreater \ 0$

hence,

$\lim_{t \to \infty} P(t) = \lim_{t \to \infty} \frac{3E7}{3E3+0} = 10^{4}$