Consequences of imperfect mixing in the Gray-Scott model

TL;DR

This paper investigates how imperfect mixing affects the Gray-Scott reaction-diffusion model by deriving stochastic PDEs that include fluctuations, revealing significant deviations from mean-field predictions.

Contribution

It introduces a novel numerical method to solve complex stochastic PDEs for reaction-diffusion systems with imperfect mixing, extending analysis beyond mean-field models.

Findings

Fluctuations significantly alter stationary states compared to mean-field results.

The method accurately captures the interplay between reaction noise and mixing regimes.

Results demonstrate the importance of stochastic effects in reaction-diffusion dynamics.

Abstract

We study an autocatalytic reaction-diffusion scheme, the Gray-Scott model, when the mixing processes do not homogenize the reactants. Starting from the master equation, we derive the resulting coupled, nonlinear, stochastic partial differential equations that rigorously include the spatio-temporal fluctuations resulting from the interplay between the reaction and mixing processes. The fields are complex and depend on correlated complex noise terms. We implement a novel method to solve for these complex fields numerically and extract accurate information about the system evolution and stationary states under different mixing regimes. Through this example, we show how the reaction induced fluctuations interact with the temporal nonlinearities leading to results that differ significantly from the mean-field (perfectly mixed) approach. This procedure can be applied to an arbitrary non-linear reaction diffusion scheme.