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 =

Answer 1

If $P(t)$ is the country's population at the instant $t$, its variation over a small time interval $\textrm{d}t$ is given by:

$\textrm{d}P = \dfrac{\textrm{d}P}{\textrm{d}t}\textrm{ d}t,$

where $\dfrac{\textrm{d}P}{\textrm{d}t}$ is the instantaneous rate of growth.

In the absence of both immigration and emigration, this rate of change is proportional to the population itself:

$\dfrac{\textrm{d}P}{\textrm{d}t} = kP, \quad\textrm{with } k>0.$

When we consider only immigration at a constant rate, we get:

$\dfrac{\textrm{d}P}{\textrm{d}t} = r, \quad\textrm{with } r>0.$

So over a small time interval $\textrm{d}t$, we have two contributions:

• On the one hand, the population will increase by $kP\textrm{ d}t$ on its own;
• On the other hand, the population will increase by $r\textrm{ d}t$ due to immigration.

All in all, we get:

$\textrm{d}P = kP\textrm{ d}t + r\textrm{ d}t  \iff \dfrac{\textrm{d}P}{\textrm{d}t} = kP + r,$

which is the differential equation we were looking for.