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

TL;DR

This paper compares two limiting procedures in stochastic Turing pattern formation, revealing that the order of limits determines whether spatial chaos or order emerges in the patterns.

Contribution

It demonstrates how different limiting procedures in stochastic reaction-diffusion models lead to qualitatively different spatial patterns, highlighting the importance of discreteness effects.

Findings

Deterministic limit followed by long-time limit yields spatial chaos.

Long-time limit followed by large volume and lattice size yields periodic patterns.

Order of limits critically influences pattern type in stochastic Turing systems.

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 $\Omega$. 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. In this setting, we compare two different limiting procedures and show that they lead to qualitatively different outcomes. If we first take the deterministic limit $\Omega \to \infty$ and then the long-time limit $t \to \infty$, the stationary solutions of the corresponding spatially discrete deterministic equations become spatially chaotic in the limit $N\to\infty$. In contrast, if we first take the limit $t \to \infty$ and then take an appropriate limit of $\Omega \to \infty$ and $N\to\infty$, the resulting patterns are spatially periodic.