Journal

Calculus of Variations and Partial Differential Equations

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 Fréchet 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.

Overview

This paper is really about pushing the scalar aggregation-diffusion picture into the two-species setting in a fully rigorous way. In the scalar case, a lot of the main ideas were already known under fairly nice assumptions, and one part of the paper is showing that much of that structure still survives under the weaker assumption that the kernel is only of bounded variation. The more interesting part is the two-species problem, where the bifurcation picture becomes much richer because there are now self-interaction and cross-interaction parameters to vary.

A big goal of the paper is to classify the branches of nontrivial stationary states that emerge from the homogeneous state, and to understand which of those branches are stable. To do that, we set the problem up in a way that lets us apply Crandall–Rabinowitz bifurcation theory, and then compute enough derivatives of the associated nonlinear map to determine the local branch structure. In the end, this gives a fairly complete picture of how patterned two-species states appear and persist on the torus.

One nice application is to cell-cell adhesion. In that setting, the analysis shows that stable segregation patterns can already appear right at the onset of bifurcation, even in a regime where all of the underlying interactions are purely “attractive” (in the sense that the kernel is roughly bowl-shaped). That gives a mathematical mechanism for the onset of cell sorting in a model where the pattern formation is not being driven by explicit repulsion.

Keywords

aggregation-diffusion systems; bifurcation from simple eigenvalue; long-time behaviour; supercritical bifurcations; subcritical bifurcations; exchange of stability

MSC 2020

35B32; 35Q92; 35R09; 35R05; 35P05

Links

• DOI / Journal page
• Preprint (arXiv)