# On Discretely Structured Growth Models and Their Moments

## TL;DR

This paper generalizes the logistic growth model to structured populations and derives conditions for exact low-dimensional moment equations.

## Contribution

The paper introduces a new generalization of logistic growth models to discrete structures and derives exact moment equations.

## Key findings

- Necessary and sufficient conditions for exact low-dimensional moment equations are derived.
- Coarse-grained moment information can reveal structured dynamics and aid in model selection.
- Examples include population ageing and immune cell exhaustion by cancerous tissue.

## 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.