Long-time behaviour and bifurcation analysis of a two-species aggregation-diffusion system on the torus

TL;DR

This paper analyzes the long-term behavior and bifurcation phenomena in a two-species aggregation-diffusion system on a torus, revealing stability exchanges and segregation patterns relevant to cell sorting.

Contribution

It extends bifurcation analysis to a two-species nonlocal system with minimal regularity assumptions, classifying all solution branches and their stability.

Findings

Existence and stability of stationary states established.

Bifurcation structure classified for both scalar and two-species systems.

Stable segregation patterns emerge at the onset of cell sorting.

Abstract

We investigate stationary states, including their existence and stability, in a class of nonlocal aggregation-diffusion equations with linear diffusion and symmetric nonlocal interactions. For the scalar case, we extend previous results by showing that key model features, such as existence, regularity, bifurcation structure, and stability exchange, continue to hold under a mere bounded variation hypothesis. For the corresponding two-species system, we carry out a fully rigorous bifurcation analysis using the bifurcation theory of Crandall & Rabinowitz. This framework allows us to classify all solution branches from homogeneous states, with particular attention given to those arising from the self-interaction strength and the cross-interaction strength, as well as the stability of the branch at a point of critical stability. The analysis relies on an equivalent classification of solutions through fixed points of a nonlinear map, followed by a careful derivation of Fr\'echet derivatives up to third order. An interesting application to cell-cell adhesion arises from our analysis, yielding stable segregation patterns that appear at the onset of cell sorting in a modelling regime where all interactions are purely attractive.