Mathematical models for chemotaxis and their applications in self-organisation phenomena

TL;DR

This paper reviews the mathematical modeling of chemotaxis using the Patlak-Keller-Segel system, highlighting its applications across biology and social phenomena, and discusses its patterning properties and historical development.

Contribution

It provides a comprehensive review of PKS models in explaining self-organization, including their foundation, patterning capabilities, and diverse applications in science and social contexts.

Findings

PKS models effectively describe chemotactic pattern formation.

Applications span microbiology, development, immunology, cancer, ecology, and social sciences.

A humorous model illustrates academic clique formation through research clustering.

Abstract

Chemotaxis is a fundamental guidance mechanism of cells and organisms, responsible for attracting microbes to food, embryonic cells into developing tissues, immune cells to infection sites, animals towards potential mates, and mathematicians into biology. The Patlak-Keller-Segel (PKS) system forms part of the bedrock of mathematical biology, a go-to-choice for modellers and analysts alike. For the former it is simple yet recapitulates numerous phenomena; the latter are attracted to these rich dynamics. Here I review the adoption of PKS systems when explaining self-organisation processes. I consider their foundation, returning to the initial efforts of Patlak and Keller and Segel, and briefly describe their patterning properties. Applications of PKS systems are considered in their diverse areas, including microbiology, development, immunology, cancer, ecology and crime. In each case a historical perspective is provided on the evidence for chemotactic behaviour, followed by a review of modelling efforts; a compendium of the models is included as an Appendix. Finally, a half-serious/half-tongue-in-cheek model is developed to explain how cliques form in academia. Assumptions in which scholars alter their research line according to available problems leads to clustering of academics and the formation of "hot" research topics.