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

TL;DR

This work addresses Turing pattern formation in a stochastic, lattice-based reaction-diffusion system where the pattern length is comparable to the lattice spacing ($\Xi$ near 1). By comparing two limit orders—first taking $\Omega\to\infty$ (discrete deterministic limit) and then $t\to\infty$, $N\to\infty$ versus first taking $t\to\infty$ with finite $\Omega$ and then $\Omega\to\infty$, $N\to\infty$—the authors uncover a dichotomy: discreteness induces spatial chaos in the stationary discrete deterministic equations, while fluctuation-driven effects yield spatially periodic order in the stochastic steady state. Spectral and finite-size scaling analyses reveal that the steady-state spectrum peak scales linearly with system size $N$ in the fluctuation-driven limit, signaling periodic patterns, whereas the discrete deterministic limit lacks a characteristic wavelength. The results underscore a novel limit-order sensitivity arising from spatial discreteness, with potential relevance for mesoscopic and nanoscale pattern-forming systems and informing theories beyond macroscopic reaction-diffusion models.

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. 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 $Ω\to \infty$ 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 $Ω\to \infty$ and $N\to\infty$, the resulting patterns are spatially periodic.

Figures (11)

• Figure 1: Graphical summary of this paper.
• Figure 2: Schematic figure of a stochastic reaction-diffusion model.
• Figure 3: Examples of stationary solutions of Eqs. \ref{['ODE_u']} and \ref{['ODE_v']} with $N = 180$. The initial conditions in (a) and (b) are different.
• Figure 4: Spectrum $\chi_n^{\mathrm{(det)}}$ of the stationary patterns for several system sizes $N$. For each $N$, the spectrum is calculated by averaging $|\tilde{u}_n|^2$ over $1000$ stationary solutions obtained from different initial conditions.
• Figure 5: $N$-dependence of the distribution of maximum Lyapunov exponent. For each system size $N$, the average over $1000$ orbits is plotted. The error bars represent the standard deviation.