Logistic growth: The logistic function
Behaviour of the logistic function
We consider the differential equation
The derivative can be regarded as a function of , which is a parabola with zeros and . This function has a maximum midway between the zeros, i.e., in . The maximum growth rate is therefore .
The graph of the increasing logistic function
Phase I If is very small, then is approximately equal to .
So, in this stage .
In other words, in the early stages of growth there is growth that is very similar to exponential growth with growth rate .
Phase II In this phase, the growth rate reaches a maximum value at the moment when . We can calculate the time at which this happens: you just need to solve for . This can be done as follows:
From the derivation above it follows that the time at which the growth rate reaches a maximum is
Phase III When is very close to the value , then , and therefore
Thus, in the final stage we are dealing with growth that resembles limited exponential growth.