In 1838 the Belgian mathematician Verhulst suggested the differential equation

as a model for population growth. N(t) is the population at time t. For small N, we have dN/dt ≈ rN(t) and so approximately exponential growth with growth rate r. However, as N becomes comparable to a carrying capacity K, the effective growth rate reduces. The general solution is

where A is a positive constant. The graph of N(t) is an increasing S-shaped curve with N(t) tending to 0 as t→−∞ and N(t) tending to K as t→∞.