Richards curve

The Richards curve or generalized logistic is a widely used growth model that will fit a wide range of S-shaped growth curves. There are both 4 and 5 parameter versions in common use. The logistic curve is symmetrical about the point of inflection of the curve. To deal with situations where the growth curve is asymmetrical, Richards (1959)  added an additional parameter producing the equation:

where l = length, (or weight, height, size), δ ≠ 1 and t = time.

The four parameters are:

L∞, the upper asymptote;

k, the growth rate;

γ, the point of inflection on the x axis

δ, a parameter that in part determines the point of inflection on the y axis;

The y-ordinate of the point of inflection is determined from

The average normalized growth rate is:

Richards curve requires one more parameter than the logistic curve to generate asymmetry; you can avoid this additional parameter by using the Gompertz curve which generates an asymmetrical growth curve with only 3 parameters.

The Richards curve can be difficult to solve because of numerical difficulties. This particularly the case with 5 parameter model, where the choice of initial parameters is critical. The Janoschek model has much of the flexibility of the Richards model and is far easier to solve.

The 5 parameter version of the Richards curve is:

where

β, is the lower asymptote;
L∞, is the upper asymptote;
tm, is the time of maximum growth;
k, is the growth rate and
T, is a variable which fixes the point of inflection.

The following graphs show example plots of the 4 and 5 parameter Richards growth curve. The first 3 graphs show the effect of varying, η (nu), which is the parameter determining the degree of asymmetry. The final plot is the 5 parameter version with a lower asymptote greater than zero.