The S-shaped growth curve is also called a logistic growth curve. It describes a situation in which (in a new environmental condition) the population density of an organism increases slowly establishing itself then increasing rapidly, approaching an exponential growth rate. Many population of micro-organisms broadly follow this basic sigmoidal pattern. For example, when a fresh culture
medium is inoculated with bacteria, sigmoidal or S-shaped growth curve is observed. The S-shaped curve is generated when a population approaches the environmental's carrying capacity. Carrying capacity is the maximum number of individuals of a population that can be supported in a given time.


The S-shaped growth form is represented by the following equation -
$d N / d t=r N[K-N / K]$
Where, $r=$ intrinsic rate of natural increase
$N=$ population density at time $t$
$K=$ carrying capacity