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The logistic growth curve (sometimes called the Verhulst model as it was first proposed as a model of population growth Pierre Verhulst 1845, 1847) is one of the simplest of the S-shaped growth curves. A generalization of the logistic, which is no longer symmetrical about the point of inflection, was developed by Richard (1959) and is termed the Richards curve or generalized logistic


Assume that the rate of growth of an organism declines with size so that the rate of change in size may be described by:




where t is time, l is length (size), K is the growth rate and delta a term which expresses the rate at which growth declines with size.



After integration and some rearrangement we arrive at the 3 parameter logistic growth curve:





where  I is the age at the inflection point and L is the upper asymptote (maximum size reach after infinite growing time).


The 3 parameter logistic has a lower asymptote of 0. The point of inflection on the y-axis occurs at



This last formula states that the point of inflection is always at at 50 % of the asymptotic size (L). This does not hold true for all growth processes. You should consider using the Logistic growth curve to model sigmoid growth processes in which the point of inflection is  approximately 1/2 of the maximum possible size.


If a non-zero asymptote is required then the 4 parameter version of the equation is required this is expressed by the equation:





where a is the lower asymptote and d is a shape parameter that determines the steepness of rising curve.


The following graphs show example plots of the 3 and 4 parameter Logistic growth curve.