Segregation and Gap Formation in Cross-Diffusion Models

TL;DR

This paper investigates how small changes in cross-diffusion systems for two species can cause significant shifts between segregation and mixing behaviors, revealing critical sensitivities in these models.

Contribution

It demonstrates the critical nature of population pressure functions in cross-diffusion systems and how symmetry-breaking perturbations lead to different population distributions.

Findings

Adding cross-diffusion terms can cause segregation or mixing.

Systems are critical with respect to perturbations in population pressure.

Numerical simulations illustrate the transition dynamics.

Abstract

In this paper we analyse a class of nonlinear cross-diffusion systems for two species with local repulsive interactions that exhibit a formal gradient flow structure with respect to the Wasserstein metric. We show that systems where the population pressure is given by a function of the total population are critical with respect to cross-diffusion perturbations. This criticality is showcased by proving that adding an extra cross-diffusion term that breaks the symmetry of the population pressure in the system leads to completely different behaviours, namely segregation or mixing, depending on the sign of the perturbation. We show these results at the level of the minimisers of the associated free energy functionals. We also analyse certain implications of these results for the gradient flow systems of PDEs associated to these functionals and we present a numerical exploration of the time evolution of these phenomena.