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arXiv · 2603.22177

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

Brocchieri Elisabetta·Soresina Cinzia
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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.

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Brocchieri Elisabetta, Soresina Cinzia. 2026-03-23. Cross-diffusion and fast-reaction in pattern formation: a structural analysis. https://arxiv.org/abs/2603.22177

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