Organisms and Populations

When resources are unlimited, a population grows exponentially: $\frac{dN}{dt} = rN$ (J-shaped curve).
In nature resources are limited, and a habitat can support only a maximum number of individuals, the carrying capacity K.
Such a population shows a lag phase, then acceleration, deceleration and finally an asymptote when N reaches K (sigmoid curve). This is Verhulst–Pearl logistic growth.
It is described by $\frac{dN}{dt} = rN\left(\frac{K - N}{K}\right)$, where N is the population density at time t and r is the intrinsic rate of natural increase.
The factor $\frac{K - N}{K}$ is the environmental resistance term; as N approaches K it becomes zero and growth stops.
Hence the logistic equation is $dN/dt = rN\left(\frac{K - N}{K}\right)$.