Random walk models in the life sciences: including births, deaths and local interactions

TL;DR

This paper reviews the development of random walk models in the life sciences, emphasizing models with interactions like adhesion and competition, and discusses their continuum limits, patterns, and future research directions.

Contribution

It provides a comprehensive overview of models including interactions, derivation of continuum limits, and simplified deterministic descriptions, with practical examples and future challenges.

Findings

Models with interactions capture crucial population behaviors.

Continuum-limit descriptions facilitate analysis and fitting.

Interacting models reveal complex spatial patterns.

Abstract

Random walks and related spatial stochastic models have been used in a range of application areas including animal and plant ecology, infectious disease epidemiology, developmental biology, wound healing, and oncology. Classical random walk models assume that all individuals in a population behave independently, ignoring local physical and biological interactions. This assumption simplifies the mathematical description of the population considerably, enabling continuum-limit descriptions to be derived and used in model analysis and fitting. However, interactions between individuals can have a crucial impact on population-level behaviour. In recent decades, research has increasingly been directed towards models that include interactions, including physical crowding effects and local biological processes such as adhesion, competition, dispersal, predation and adaptive directional bias. In this article, we review the progress that has been made with models of interacting individuals. We aim to provide an overview that is accessible to researchers in application areas, as well as to specialist modellers. We focus particularly on derivation of asymptotically exact or approximate continuum-limit descriptions and simplified deterministic models of mean-field behaviour and resulting spatial patterns. We provide worked examples and illustrative results of selected models. We conclude with a discussion of current areas of focus and future challenges.