Turing instability analysis of a singular cross-diffusion problem

TL;DR

This paper investigates the stability and pattern formation in a regularized population model with singular cross-diffusion, showing that certain instabilities vanish in the limit, supported by theoretical analysis and finite element simulations.

Contribution

It provides a weakly nonlinear stability analysis of regularized problems and demonstrates the disappearance of high wave number patterns in the limit, advancing understanding of singular cross-diffusion systems.

Findings

Pattern formation with unbounded wave numbers occurs in regularized problems.

In the limit, oscillation amplitudes decay, preventing pattern formation.

Finite element simulations confirm the stability analysis results.

Abstract

The population model of Busenberg and Travis is a paradigmatic model in ecology and tumour modelling due to its ability to capture interesting phenomena like the segregation of populations. Its singular mathematical structure enforces the consideration of regularized problems to deduce properties as fundamental as the existence of solutions. In this article we perform a weakly nonlinear stability analisys of a general class of regularized problems to study the convergence of the instability modes in the limit of the regularization parameter. We demonstrate with some specific examples that the pattern formation observed in the regularized problems, with unbounded wave numbers, is not present in the limit problem due to the amplitude decay of the oscillations. We also check the results of the stability analysis with direct finite element simulations of the problem.