A proposed family of Unified models for sigmoidal growth

Abstract

The four-parameter Unified Richards model has been applied to the growth of different animal taxa. Traditionally researchers of animal growth have favoured three-parameter models such as the logistic, Gompertz, and von Bertalanffy models. However, the growth-rate parameters of the traditional versions of these models are incomparable, and model forms returning starting points for the curves are not available. Therefore, we have reviewed and developed the family of Unified growth models (U-models), including U-versions of the logistic, Gompertz, and von Bertalanffy models, which each have an inflection placement at a fixed percentage of the upper asymptote. Consequently, in order to accommodate for those who prefer a three-parameter model, we also deduce (show) how to derive from the U-Richards a new, generic, three-parameter U-models with any predetermined inflection placement. This means an indefinite number of U-models, which will cover inflection placements at any percentage of the upper asymptote. All U-models have been re-parameterized to exhibit a unified set of parameters, which measure the same thing cross all family members, hence these models are termed the Unified family (or simply U-family). We also discuss how to interpret parameter values and whether to restrain the asymptote to a fixed value. All U-family models can be fitted to data in either of two forms: the first where one of the parameters represents the time of inflection, and the second with a parameter representing the starting value (intersection with the x-axis). Each parameter in these models only affects a single curve-shape characteristic. We show, also by fitting the models to bird growth data, how only a complete U-Richards family of models and accompanying parameter-translation equations will guarantee that we will be able to choose a model that returns realistic values and provides a consistent interpretation of growth data. There should be no enticement to choose other tools for analysing sigmoidal growth. Traditional versions of the Gompertz, logistic, von Bertalanffy or Richard’s models found in the literature have various shortcomings.