Logistic growth

Logistic growth models a population that grows quickly at first and then slows as it approaches a maximum sustainable size. The classic model is

dP/dt = r · P · (1 − P/K)

where P is the population, r the growth rate, and K the carrying-capacity. Near zero the term (1 − P/K) is close to 1, so growth looks exponential; as P nears K the term shrinks and growth levels off.

The result is a stretched S-shape: fast rise, an inflection point, then approach to a plateau.