Why Populations Explode: Exponential vs. Logistic Growth Models Explained

Exponential Growth Model

More than two centuries ago, British demographer Thomas Malthus analyzed over a hundred years of British population data and assumed a constant per‑capita growth rate, leading to the famous exponential growth model.

Let the population at time t be P(t) . Treating P(t) as a continuous differentiable function, with initial population P(0)=P_0 and a constant growth rate r , the change in population per unit time is dP/dt = rP . Solving this differential equation by separation of variables yields P(t) = P_0 e^{rt} , indicating unlimited exponential increase.

Logistic (Stagnation) Growth Model

When population size approaches environmental limits, the growth rate declines because resources and conditions impose a restraining effect that intensifies as the population grows. Modifying the exponential assumption gives the logistic model.

Assume the per‑capita growth rate decreases linearly with population: r(P) = r\left(1 - \frac{P}{K}\right) , where r is the intrinsic growth rate (growth when population is very small) and K is the environmental carrying capacity. Substituting into the differential equation yields dP/dt = rP\left(1 - \frac{P}{K}\right) .

Separating variables and integrating leads to the solution P(t) = \frac{K}{1 + \left(\frac{K - P_0}{P_0}\right) e^{-rt}} . This logistic equation, first proposed by the Dutch biologist Verhulst in the mid‑19th century, describes how populations (and many other quantities such as forest trees or fish stocks) grow rapidly at first and then level off as they approach the carrying capacity.

The logistic model also finds wide applications in economics and social sciences for phenomena that follow a similar S‑shaped curve.