Non-Local Cell Adhesion Models: Steady States and Bifurcations

TL;DR

This paper analyzes the mathematical properties of non-local cell adhesion models, focusing on steady states and bifurcations, which are crucial for understanding pattern formation in biological tissues.

Contribution

It provides the first rigorous analysis of steady states and bifurcation structures in non-local adhesion models, extending mathematical understanding of pattern formation in tissue modeling.

Findings

Global bifurcation results for non-trivial solutions

Identification of steady-state patterns in tissue models

Application of equivariant bifurcation theory to biological models

Abstract

In this manuscript, we consider the modelling of cellular adhesions, which is a key interaction between biological cells. Continuum models of the diffusion-advection-reaction type have long been used in tissue modelling. In 2006, Armstrong, Painter, and Sherratt proposed an extension to take adhesion effects into account. The resulting equation is a non-local advection-diffusion equation. While immensely successful in applications, the development of mathematical theory pertaining to steady states and pattern formation is lacking. The mathematical analysis of the non-local adhesion model is challenging. In this monograph, we contribute to the analysis of steady states and their bifurcation structure. The importance of steady-states is that these are the patterns observed in nature and tissues (e.g. cell-sorting experiments). In the case of periodic boundary conditions, we combine global bifurcation results pioneered by Rabinowitz, equivariant bifurcation theory, and the mathematical properties (maximum principle) of the non-local term to obtain a global bifurcation result for the branches of non-trivial solutions.