The generalized logistic function or curve is an extension of the logistic or sigmoid functions. Originally developed for growth modelling, it allows for more flexible S-shaped curves. The function is sometimes named Richards's curve after F.J.Richards, who proposed the general form for the family of models in 1959.

Definition

Richards's curve has the following form:

= time. It has six parameters:

• : the left horizontal asymptote;

• is called the carrying capacity;

• : the growth rate;

• : affects near which asymptote maximum growth occurs.

The equation can also be written:

can be convenient:

at that time.

The logistic function.

Generalised logistic differential equation

A particular case of the generalised logistic function is:

which is the solution of the Richards's differential equation (RDE):

with initial condition

where

, whereas the Gompertz curve provided that:

it is

The RDE models many growth phenomena, arising in fields such as oncology and epidemiology.

Gradient of generalized logistic function

(see,

Special cases

The following functions are specific cases of Richards's curves:

• Logistic function

• Gompertz curve

• Von Bertalanffy function

• Monomolecular curve

Footnotes

References