Journal Article

A New Generalized Logistic Sigmoid Growth Equation Compared with the Richards Growth Equation

BIRCH, COLIN P.D.

1999

Overview

A new sigmoid growth equation is presented for curve-fitting, analysis and simulation of growth curves. Like the logistic growth equation, it increases monotonically, with both upper and lower asymptotes. Like the Richards growth equation, it can have its maximum slope at any value between its minimum and maximum. The new sigmoid equation is unique because it always tends towards exponential growth at small sizes or low densities, unlike the Richards equation, which only has this characteristic in part of its range. The new sigmoid equation is therefore uniquely suitable for circumstances in which growth at small sizes or low densities is expected to be approximately exponential, and the maximum slope of the growth curve can be at any value. Eleven widely different sigmoid curves were constructed with an exponential form at low values, using an independent algorithm. Sets of 100 variations of sequences of 20 points along each curve were created by adding random errors. In general, the new sigmoid equation fitted the sequences of points as closely as the original curves that they were generated from. The new sigmoid equation always gave closer fits and more accurate estimates of the characteristics of the 11 original sigmoid curves than the Richards equation. The Richards equation could not estimate the maximum intrinsic rate of increase (relative growth rate) of several of the curves. Both equations tended to estimate that points of inflexion were closer to half the maximum size than was actually the case; the Richards equation underestimated asymmetry by more than the new sigmoid equation. When the two equations were compared by fitting to the example dataset that was used in the original presentation of the Richards growth equation, both equations gave good fits. The Richards equation is sometimes suitable for growth processes that may or may not be close to exponential during initial growth. The new sigmoid is more suitable when initial growth is believed to be generally close to exponential, when estimates of maximum relative growth rate are required, or for generic growth simulations.

• Publisher: Elsevier Science Ltd,Oxford University Press,Academic Press Limited,Oxford Publishing Limited (England)
• Subject: Asymptote / Asymptote,Cucumis melo,curve-fitting, exponential growth, intrinsic rate of increase, logistic equation, maximum growth rate, model, non-linear least-squares regression, numerical algorithm, point of inflexion, relative growth rate, Richards growth equation, sigmoid growth curve / Asymptotes / Asymptotic value / Average linear density / Botany / Cucumis melo / curve-fitting / Datasets / duration / equations / exponential growth / Geometric growth / growth / growth models / growth rate / height / intrinsic rate of increase / logistic equation / Logistic equations / Logistics / mathematical models / maximum growth rate / Modeling / non-linear least-squares regression / numerical algorithm / Plant growth / plants / point of inflexion / population density / relative growth rate / Richards growth equation / sigmoid growth curve / simulation models / size / Zero