Logistic Population Growth Calculator

Logistic growth describes how a population grows rapidly when small but slows as it approaches the carrying capacity of its environment. Unlike exponential growth, which continues without limit, the logistic model produces an S-shaped (sigmoid) curve that levels off at the carrying capacity. It captures the reality that resources such as food and space are finite.

Carrying capacity, written K, is the maximum population size that an environment can sustain indefinitely given its resources. As a population nears K, growth slows because competition for food, space and other resources intensifies. At exactly K, births balance deaths and the population stabilises. Carrying capacity is a property of the environment and the species, not a fixed universal number.

The intrinsic growth rate r is the per-capita rate at which a population would grow if resources were unlimited, expressed per unit time. A larger r means faster initial growth. In the logistic model, the realised growth rate is r times (1 minus N over K), so growth is fastest at low population and approaches zero as the population nears carrying capacity.

This calculator uses the closed-form solution of the logistic differential equation: N(t) = K divided by (1 + A times e to the power of minus r t), where A = (K minus N0) divided by N0 and N0 is the initial population. This gives the population at time t directly without step-by-step simulation. The result approaches K as t grows large.