A population grows exponentially when its net rate of increase per individual remains constant.

Let $N(t)$ be population size and let $r$ be the per-capita growth rate: the net contribution to population growth per individual per unit time. Then

$$\frac{dN}{dt}=rN.$$

The equation says that total growth is proportional to how many individuals are already present. Its solution is

$$N(t)=N_0e^{rt},$$

where $N_0$ is the initial population size.

If $r>0$, population size increases exponentially; if $r=0$, it remains constant; if $r<0$, it declines exponentially.

Worked example

A microbial population begins with $N_0=100$ cells and has $r=0.20\ \text{h}^{-1}$. After 10 hours,

$$N(10)=100e^{(0.20)(10)}=100e^2\approx739.$$

The important feature is not simply that the curve is J-shaped. Because each individual contributes at the same average rate, adding individuals increases the absolute number of births that can occur next. Growth therefore compounds.

Exponential growth is an idealization. It can approximate a population when resources are abundant and density-dependent limits are weak, especially during early colonization or short experimental intervals. It cannot continue indefinitely in a finite environment.