Fluctuation Induced Instabilities in Front Propagation up a Co-Moving Reaction Gradient in Two Dimensions

TL;DR

This paper investigates how fluctuations affect the stability and velocity of reaction-diffusion fronts in two-dimensional channels, revealing that stochastic effects induce instabilities and alter front velocities beyond deterministic predictions.

Contribution

It demonstrates that stochastic fluctuations destabilize planar fronts in 2D channels when the width exceeds a critical value, and introduces a cutoff mean-field approach that aligns with stochastic results.

Findings

Planar solutions are stable in deterministic systems but unstable in stochastic systems beyond a critical channel width.

Stochastic front velocity depends on channel width in a simple, predictable manner.

A density cutoff in the reaction rate captures stochastic effects, aligning mean-field predictions with stochastic simulations.

Abstract

We study 2D fronts propagating up a co-moving reaction rate gradient in finite number reaction-diffusion systems. We show that in a 2D rectangular channel, planar solutions to the deterministic mean-field equation are stable with respect to deviations from planarity. We argue that planar fronts in the corresponding stochastic system, on the other hand, are unstable if the channel width exceeds a critical value. Furthermore, the velocity of the stochastic fronts is shown to depend on the channel width in a simple and interesting way, in contrast to fronts in the deterministic MFE. Thus, fluctuations alter the behavior of these fronts in an essential way. These affects are shown to be partially captured by introducing a density cutoff in the reaction rate. Some of the predictions of the cutoff mean-field approach are shown to be in quantitative accord with the stochastic results.