Cross-diffusion and fast-reaction in pattern formation: a structural analysis

TL;DR

This paper analyzes how biologically derived cross-diffusion influences pattern formation and Turing instability in generalized SKT models, revealing conditions that promote or inhibit spatial self-organization.

Contribution

It characterizes instability conditions for cross-diffusion in a generalized SKT framework and explores how different fast-reaction formulations affect pattern formation.

Findings

Cross-diffusion can induce or prevent Turing patterns depending on the model structure.

Different fast-reaction limits lead to distinct diffusion effects and pattern formation outcomes.

The sign of the reaction Jacobian interacts with cross-diffusion to determine pattern onset.

Abstract

Cross-diffusion systems play a central role in mathematical modelling, in which density-dependent dispersal and multiscale mechanisms can lead to spatial segregation and diffusion-driven instabilities. In several relevant examples, including generalised SKT-type competition models, cross-diffusion terms can be rigorously derived as fast-reaction limits, thereby providing a clear biological interpretation while posing significant analytical challenges. In this work, we investigate the impact of biologically derived cross-diffusion on Turing instability. For a generalised SKT framework, we characterise instability conditions for a broad class of cross-diffusion functions arising from fast-reaction mechanisms. We then propose an alternative fast-reaction formulation leading to a different diffusion structure and show that, in this case, diffusion-driven pattern formation is prevented. We further discuss an example motivated by dietary diversity and starvation dynamics, and analyse how the sign structure of the reaction Jacobian interacts with cross-diffusion in determining the onset of patterns. Our results clarify structural features that promote or inhibit spatial self-organisation in competitive systems.