Population growth

Consider the growth of a population — human, animal, or a bacteria colony — as a function of time $t$. Let $x(t)$ denote the population size at time $t$; although genuinely integer-valued, $x(t)$ is approximated as a differentiable function so the tools of differential equations can be applied.

Model. If the population grows at a rate directly proportional to the population already present,

Solution. This is separable: $\dfrac{dx}{x}=k\,dt$, integrating to $x(t)=Ce^{kt}$; writing $x_0$ for the population at $t=0$ gives $C=x_0$, so

The population therefore increases exponentially with time — this is the Malthusian law of population growth. …