Why Populations Explode: Understanding Exponential and Logistic Growth Models

1 Exponential Growth Model

Over two hundred years ago, British demographer Thomas Malthus analyzed more than a century of British population data and assumed a constant per‑capita growth rate. From this assumption he derived the classic exponential growth model.

Let P(t) denote the population at time t, with initial population P0 at t = 0. Assuming a constant growth rate r, the differential equation is

dP/dt = r * P

Separating variables and integrating gives

P(t) = P0 * e^{r t}

This solution shows that, without any limiting factors, the population grows without bound following an exponential law.

2 Logistic (Limited‑Growth) Model

When a population approaches the limits imposed by resources and environment, the growth rate declines. The logistic model incorporates this “crowding” effect by modifying the exponential assumption.

Let r be the intrinsic growth rate (the rate when the population is very small) and K the carrying capacity, the maximum population the environment can sustain. The differential equation becomes

dP/dt = r * P * (1 - P/K)

Here the factor (1‑P/K) reduces the growth rate as P approaches K.

Solving by separation of variables yields the logistic solution

P(t) = K / (1 + ((K - P0)/P0) * e^{-r t})

When P = K, the growth rate is zero, indicating the population has reached its carrying capacity.

The logistic model, first proposed by the Dutch biologist Verhulst in the mid‑19th century, describes not only human population dynamics but also the growth of many biological species and finds applications in economics and other social sciences.

References

ThomsonRen GitHub https://github.com/ThomsonRen/mathmodels