Study of a degenerate reaction-diffusion system arising in particle dynamics with aggregation effects

TL;DR

This paper analyzes a degenerate reaction-diffusion system modeling cell adhesion at the protein level, establishing existence, uniqueness, positivity, and stability of solutions, and examining conditions for pattern formation through analytical and numerical methods.

Contribution

It provides the first rigorous analysis of a degenerate reaction-diffusion system in biological cell adhesion, including solution existence, stability, and conditions preventing pattern formation.

Findings

Existence, positivity, and boundedness of solutions are proven.

Uniqueness and stability of equilibrium solutions are established.

Simple affine aggregation functions do not produce patterns as expected.

Abstract

In this paper we are interested in a degenerate parabolic system of reaction-diffusion equations arising in biology when studying cell adhesion at the protein level. In this modeling the unknown is the couple of the distribution laws of the freely diffusing proteins and of the fixed ones. Under sufficient conditions on the aggregation and unbinding probabilities, we prove the existence of solution of the considered system, as well as their positivity, boundedness and uniqueness. Moreover, we discuss the stability of the equilibrium solution. Finally, we show that the simpler and particular choice of an affine aggregation function and of a constant unbinding probability do not lead to pattern formation as expected in the application. These analytical results are also supported by some numerical simulation.