Question

The population of Cook Island was always been modeled by a logistic equation with growth rate r=19 and carrying capacity N=8000, with time t measured in years. However, beginning in 2000, 9 citizens of Cook Island have left every year to become a mathematician, never to return. Find the new differential equation modeling the population of the island P(t) after 2000. Use P for P(t) and P' for P′(t)

The answer is P' = P/9(1-P/8000)-9

Answer 1

`P'(t) = 19P(1 - (P)/(8000)) - 9`

Explanation:

The logistic equation is given by Equation 1):

1)
`(dP)/(dt) = rP(1 - (P)/(N))`

In which P represents the population,
`(dP)/(dt) = P'(t)` is the variation of the population in function of time, r is the growth rate of the population and N is the carrying capacity of the population.

Now for your system:

The problem states that the population has growth rate r=19.

The problem also states that the population has carrying capacity N=8000.

We can replace these values in Equation 1), so:

`P'(t) = 19P(1 - (P)/(8000))`

However, beginning in 2000, 9 citizens of Cook Island have left every year to become a mathematician, never to return. So, we have to subtract these 9 citizens in the P'(t) equation. So:

`P'(t) = 19P(1 - (P)/(8000)) - 9`