Gompertz growth curve
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The Gompertz curve was originally derived to estimate human mortality by Benjamin Gompertz (Gompertz, B. "On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies." Phil. Trans. Roy. Soc. London 123, 513-585, 1832). Charles Winsor (1932) presents an early description of the use of this equation to describe growth processes.
Assume that the rate of growth of an organism declines with size so that the rate of change in length, l, (or any other measure of size of weight) may be described by:
where K is the growth rate and L∞, termed 'L infinity', is the asymptotic length at which growth is zero. (This equation has the same form as the von Bertalanffy but with log length replacing length)
Integrating this becomes:
where t is age and I is the age at the inflection point.
The equation above is the 3 parameter version of the Gompertz growth curve (see below for an example plot). Growth II can also fit the 4 parameter version:
in which A is the lower asymptote (see below for an example plot) and B is the upper asymptote minus A.
The point of inflection on the y-axis occurs at
This last formula states that the point of inflection is always at at about 36.8 % of the asymptotic size (L∞). This does not hold true for all growth processes. You should consider using the Gompertz growth curve to model sigmoid growth processes in which the point of inflection is approximately 1/3 of the maximum possible size.
The following graphs show example plots of the 3 and 4 parameter Gompertz.