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:

`(dP)/(dt) = rP[1- (P)/(K)]`

where r = growth parameter and K = carrying parameter.

In order to write you equation in this form, you have to regroup 2:

`(dP)/(dt) = 2P[1- (P)/(10000)]`

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

To solve the logistic differential equation you need to solve:


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

The soution will be:

P(t) =
`(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) =
`(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) (3E7)/(3E3+0) = 10^(4)`