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 (see Equation (1) of Section 1.3). 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

A) Differential equation for population growth in case of individual immigration is:

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

B) Differential equation for population growth in case of individual emigration is:

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

Explanation:

Population growth rate in the absence of immigration and emigration is given as:

`(dP)/(dt)=kP--(1)`

A) When individuals are allowed to immigrate:

Let r be the constant rate of individual immigration given that r >0.

Differential equation for population growth in this case is:

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

B) In case of individual emigration:

Let r be the constant rate of individual emigration given that r >0.

Differential equation for population growth in this case is:

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