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Discreteness-induced spatial chaos versus fluctuation-induced spatial order in stochastic Turing pattern formation

Yusuke Yanagisawa* and Shin-ichi Sasa

  • *Contact author: yanagisawa.yusuke.l42@kyoto-u.jp
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Phys. Rev. Research 8, 023052 – Published 16 April, 2026

DOI: https://doi.org/10.1103/7sw9-zjfs

Abstract

We investigate Turing pattern formation in a stochastic reaction-diffusion model defined on N lattice sites, where each lattice site is associated with a reaction vessel of volume Ω. We focus on a regime where spatial discreteness plays a crucial role, namely, when the characteristic length of patterns is comparable to the lattice spacing. This setting is motivated by mesoscopic and microscopic reaction-diffusion systems, in which continuum descriptions can break down. In this regime, we compare two different limiting procedures and show that they lead to qualitatively different outcomes. If we first take the deterministic limit Ω→∞ and then the long-time limit t→∞, the stationary solutions of the corresponding spatially discrete deterministic equations become spatially chaotic in the limit N→∞. In contrast, if we first take the limit t→∞ and then take an appropriate limit of Ω→∞ and N→∞, the resulting patterns are spatially periodic.

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Phys. Rev. Research 8, 023052– Published 16 April, 2026
  • Received 11 December 2025
  • Accepted 11 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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