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

Answer:

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

$\frac{dP}{dt}=kP+r$

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

$\frac{dP}{dt}=kP-r$

Step-by-step explanation:

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

$\frac{dP}{dt}=kP--(1)$

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:

$\frac{dP}{dt}=kP+r$

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:

$\frac{dP}{dt}=kP-r$