On the influence of cross-diffusion in pattern formation

TL;DR

This paper investigates how cross-diffusion influences pattern formation in a two-species competition model, revealing conditions for stable inhomogeneous states and demonstrating their existence beyond traditional parameter ranges.

Contribution

It provides a detailed analysis of the conditions for non-homogeneous steady states in the SKT model, combining linearized analysis with numerical bifurcation methods, and explores the effects of cross-diffusion and self-diffusion.

Findings

Stable non-homogeneous steady states can exist outside the parameter range of homogeneous coexistence.

Cross-diffusion significantly affects pattern formation and stability.

Numerical experiments support the theoretical analysis of pattern emergence.

Abstract

In this paper we consider the Shigesada-Kawasaki-Teramoto (SKT) model to account for stable inhomogeneous steady states exhibiting spatial segregation, which describe a situation of coexistence of two competing species. We provide a deeper understanding on the conditions required on both the cross-diffusion and the reaction coefficients for non-homogeneous steady states to exist, by combining a detailed linearized analysis with advanced numerical bifurcation methods via the continuation software pde2path. We report some numerical experiments suggesting that, when cross-diffusion is taken into account, there exist positive and stable non-homogeneous steady states outside of the range of parameters for which the coexistence homogeneous steady state is positive. Furthermore, we also analyze the case in which self-diffusion terms are considered.