Spatio-temporal patterns driven by autocatalytic internal reaction noise

TL;DR

This paper investigates how intrinsic reaction noise influences spatial pattern formation in reaction-diffusion systems, revealing that strong internal fluctuations can both destroy and induce complex patterns in the Gray-Scott model.

Contribution

It introduces a Langevin-based stochastic framework for the Gray-Scott model that captures the effects of strong internal reaction noise on pattern formation.

Findings

Strong internal noise can obliterate existing patterns.

Reaction noise can induce novel spatial patterns.

Patterns are significantly affected in parameter regions with homogeneous solutions.

Abstract

The influence that intrinsic local density fluctuations can have on solutions of mean-field reaction-diffusion models is investigated numerically by means of the spatial patterns arising from two species that react and diffuse in the presence of strong internal reaction noise. The dynamics of the Gray-Scott (GS) model with constant external source is first cast in terms of a continuum field theory representing the corresponding master equation. We then derive a Langevin description of the field theory and use these stochastic differential equations in our simulations. The nature of the multiplicative noise is specified exactly without recourse to assumptions and turns out to be of the same order as the reaction itself, and thus cannot be treated as a small perturbation. Many of the complex patterns obtained in the absence of noise for the GS model are completely obliterated by these strong internal fluctuations, but we find novel spatial patterns induced by this reaction noise in regions of parameter space that otherwise correspond to homogeneous solutions when fluctuations are not included.