On discretely structured growth models and their moments

TL;DR

This paper generalizes the logistic growth model to discretely structured populations, deriving conditions for exact low-dimensional moment equations and demonstrating their use in understanding structured biological dynamics.

Contribution

It introduces a broad class of discretely structured growth models and provides a framework for deriving exact moment equations based on polynomial kinetics.

Findings

Necessary and sufficient conditions for low-dimensional moment closure.

Exact moment equations for structured population dynamics.

Potential applications in model selection and hypothesis testing.

Abstract

The logistic equation is ubiquitous in applied mathematics as a minimal model of saturating growth. Here, we examine a broad generalisation of the logistic growth model to discretely structured populations, motivated by examples that range from the ageing of individuals in a species to immune cell exhaustion by cancerous tissue. Through exploration of a range of concrete examples and a general analysis of polynomial kinetics, we derive necessary and sufficient conditions for the dependence of the kinetics on structure to result in closed, low-dimensional moment equations that are exact. Further, we showcase how coarse-grained moment information can be used to elucidate the details of structured dynamics, with immediate potential for model selection and hypothesis testing. This paper belongs to the special collection: Problems, Progress and Perspectives in Mathematical and Computational Biology.