Logistic Regression

Regression is about determining the unknown parameters of a growth model or a given function for a series of measurement data in such a way that the final model adapts to the data as best as possible.
Frequently considered models are:

Linear Growth
With linear growth, the rate of change, i.e. the derivation of the growth function, is constant.
The corresponding diagram is a straight line.
Exponential Growth
With exponential growth, the rate of change is proportional to the population:  
Limited Growth
Whith limited growth, the rate of change is proportional to the saturation deficit, that is the difference between the saturation limit S  and the population:  
Logistic Growth
With logistic growth, it is assumed that the population grows essentially exponentially at the beginning, but that growth is slowed down more and more as the saturation limit is approached. It is therefore assumed that the rate of change is proportional to both the population and the saturation deficit. This results in the differential equation:
 
Which has the solution:

For a given saturation limit S , the program determines the initial value f(0)  and the proportionality factor k  for adapting the function f (t)  to the given value pairs.

The term logistic growth goes back to the Belgian mathematician Pierre François Verhulst (1804-1849), who in 1837 formulated the so-called logistic equation to describe demographic developments.

See also:

Wikipedia: Logistic regression | Logistic function | Pierre François Verhulst