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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2603.22177 |
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| _version_ | 1866915883753930752 |
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| author | Elisabetta, Brocchieri Cinzia, Soresina |
| author_facet | Elisabetta, Brocchieri Cinzia, Soresina |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22177 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cross-diffusion and fast-reaction in pattern formation: a structural analysis Elisabetta, Brocchieri Cinzia, Soresina Analysis of PDEs 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. |
| title | Cross-diffusion and fast-reaction in pattern formation: a structural analysis |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.22177 |