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. 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. (Use P for P(t).)

dP/dt = _____

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?

dP/dt =______

Answer 1

The rate of change is proportional to the population itself If we ignore  immigration and emigration $\frac{ \text{ dp}}{ \text{ dt}}=Kp \Rightarrow \text{ dp}=Kp \text{ dt} \\[1 em]$

If the immigration is a constant rate, the population will increase by $r$ dt  due to immigration.

$\therefore \text{ dp}=Kp \text{ dt} +r \text{ dt},r>0 \\[1 em] \therefore \frac{dp}{dt}=Kp+r ,r>0 \\[1 em]$

If the emigration is a constant rate, the population will decrease by $r$ dt due to emigration.

we get.

$\therefore \frac{dp}{dt}=Kp-r ,r>0\\[1 em]$