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)?

A logistic differential equation can be written as follows:


where r = growth parameter and K = carrying parameter.

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


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

To solve the logistic differential equation you need to solve:



The soution will be:

P(t) = 

where P(0) is the initial population.

In your case, you'll have:

P(t) = 

Now you have to calculate the limit of P(t).
We know that


hence,