Will a large complex system form Turing patterns?

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Will a large complex system form Turing patterns?

Authors

Piskovsky, V.; Maini, P. K.

Abstract

From atomic spins in magnets to galaxies, and from embryonic development to patchy vegetation in arid environments, the physical world is filled with complex systems that spontaneously form spatial patterns. In the context of reaction-diffusion systems, pattern formation can be explained by the Turing mechanism, which postulates that diffusion can destabilise spatially uniform steady states. However, the resulting Turing patterns have only been analyzed in systems with a small number of diffusing variables, even though the physical systems they aim to describe are highly complex. In this study, we utilize ideas from random matrix theory to explore how spontaneously formed Turing patterns emerge in random complex systems with many variables. Crucially, we find that diffusion-driven instability is not sufficient to explain pattern formation. Nevertheless, modifications of the Turing mechanism that include various forms of local and non-local attraction and repulsion are shown to reliably promote pattern formation in large complex systems. Altogether, this study assesses the robustness of multiple pattern-forming mechanisms in large complex systems, providing a guide for recognising which mechanisms might be important for pattern formation in empirical studies.

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