Populations grow in respect to their consumption. As long as there are ample resources, their growth can be unrestricted. Once resources get sparse however, growth is slowed down by the growing exploitation of resources. Modeled, this shows in a typical s-shaped (or sigmoid) growth curve that runs up to a limit \(K\) (see below) indicating a population size that corresponds to the amount of resources available for subsistence. This population is said to have reached the carrying capacity of its environment.

In the logistic growth model (aka Verhulst model, after the Belgian mathematician Pierre-Francois Verhulst), \(\gamma\) is proportional to \(N_t\) and to the difference of \(N_t\) and the limit \(K\). This is expressed as \(\gamma\) being proportional to the product of \(N_t\) and the difference of \(N_t\) and \(K\).

\[\frac{dN}{dt}=\frac{\gamma*N*(K-N)}K\]Dividing both sides by \(K\) and defining \(x=\frac N K\) gives the differential equation \(\frac{dx}{dt}=\gamma*x*(1-x)\)

Modeled with the modeling platform VENSIM (which is introduced here):