A mathematical framework for the statistical interpretation of biological growth models

Abstract

Biological entities are inherently dynamic. As such, various ecological disciplines use mathematical models to describe temporal evolution. Typically, growth curves are modelled as sigmoids, with the evolution modelled by ordinary differential equations. Among the various sigmoid models, the logistic, Gompertz and Richards equations are well-established and widely used for the purpose of fitting growth data in the fields of biology and ecology. The present paper puts forth a mathematical framework for the statistical analysis of population growth models. The analysis is based on a mathematical model of the population–environment relationship, the theoretical foundations of which are discussed in detail. By applying this theory, stochastic evolutionary equations are obtained, for which the logistic, Gompertz, Richards and Birch equations represent a limiting case. To substantiate the models of population growth dynamics, the results of numerical simulations are presented. It is demonstrated that a variety of population growth models can be addressed in a comparable manner. It is suggested that the discussed mathematical framework for statistical interpretation of the joint population–environment evolution represents a promising avenue for further research.