A statistical interpretation of biologically inspired growth models

TL;DR

This paper provides a statistical framework for interpreting biological growth models like logistic and Gompertz equations, linking deterministic models to stochastic processes and enabling modifications based on population-environment relationships.

Contribution

It introduces a novel statistical interpretation of growth models, connecting them to stochastic evolutionary equations and expanding their applicability.

Findings

Logistic equation as a limiting case of stochastic models

Gompertz and related models can be similarly interpreted

Numerical simulations support the statistical framework

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 and Gompertz equations are well established and widely used in fitting growth data in the fields of biology and ecology. This paper suggests a statistical interpretation of the logistic equation within the general framework. This interpretation is based on modelling the population environment relationship, the mathematical theory of which we discuss in detail. By applying this theory, we obtain stochastic evolutionary equations, for which the logistic equation is a limiting case. The prospect of modifying logistic population growth is discussed. We support our statistical interpretation of population growth dynamics with test numerical simulations. We show that the Gompertz equation and other related models can be treated in a similar way.