Population Growth: Exponential and Logistic Growth Curves

When resources such as food, space, and nutrients are effectively unlimited, a population is free to grow at the fastest rate its biology allows, a pattern called exponential growth. The instantaneous rate of change of population size is given by the equation dN/dt = rN, where N is the population size, t is time, and r is the intrinsic rate of natural increase -- a constant, characteristic of the species, that reflects the balance of that species' own birth and death rates under the most favourable conditions. Because the rate of increase is proportional to the population size already present, growth accelerates continuously as N grows larger, and when population size is plotted against time the resulting curve rises ever more steeply without ever levelling off, producing the characteristic J-shaped curve. In reality, truly unlimited resources are rare and usually only temporary -- exponential growth is typically seen only in a population colonising a brand-new, resource-rich habitat, or briefly, in laboratory cultures with a constantly replenished food supply.

In nature, however, resources are always finite: food, space, light, and nutrients are all ultimately limited, which places an upper ceiling on how large a population can grow. This more realistic pattern is called logistic growth, and it is described by the Verhulst-Pearl logistic growth equation, dN/dt = rN(K − N)/K, where K is the carrying capacity -- the maximum population size that a given environment can sustainably support given its available resources. This equation modifies the exponential growth equation with an additional factor, (K − N)/K, which grows smaller and smaller as N approaches K. When N is very small relative to K, this factor is close to 1 and growth proceeds at nearly the same explosive rate as exponential growth; but as N climbs closer to K, the factor shrinks toward zero, so the growth rate itself slows down, and once N actually reaches K, the factor becomes zero and population growth stops altogether, with the population size levelling off at K. Plotted against time, this produces the characteristic S-shaped (sigmoid) curve: a slow initial rise, a rapid middle phase of near-exponential growth, and a final flattening phase as the population approaches its environment's carrying capacity. …