Question

Assume that in the absence of immigration and emigration, the growth of a country's population P(t) satisfies dP/dt = kP for some constant k > 0.

a. Determine a differential equation governing the growing population P(t) of the country when individuals are allowed to immigrate into the country at a constant rate r > 0.
b. What is the differential equation for the population P(t) of the country when individuals are allowed to emigrate at a constant rate r > 0?

Answer 1

`(a)\ (dP)/(dt) = kP + r`

`(b)\ (dP)/(dt) = kP - r`

Explanation:

Given

`(dP)/(dt) = kP`

Solving (a): Differential equation for immigration where
`r > 0`

We have:

`(dP)/(dt) = kP`

Make dP the subject

`dP =kP \cdot dt`

From the question, we understand that:
`r > 0`. This means that

`dP =kP \cdot dt + r \cdot dt` --- i.e. the population will increase with time

Divide both sides by dt

`(dP)/(dt) = kP + r`

Solving (b): Differential equation for emigration where
`r > 0`

We have:

`(dP)/(dt) = kP`

Make dP the subject

`dP =kP \cdot dt`

From the question, we understand that:
`r > 0`. This means that

`dP =kP \cdot dt - r \cdot dt` --- i.e. the population will decrease with time

Divide both sides by dt

`(dP)/(dt) = kP - r`