Stochastic approach to Fisher and Kolmogorov, Petrovskii, and Piskunov wave fronts for species with different diffusivities in dilute and concentrated solutions

TL;DR

This paper investigates wave front propagation in a two-species autocatalytic reaction using a stochastic master equation approach, revealing diffusion-dependent front speeds and the mitigating effects of cross-diffusion in concentrated solutions.

Contribution

It introduces a stochastic framework for Fisher-KPP wave fronts with different diffusivities, highlighting the influence of stochastic effects and cross-diffusion on front dynamics.

Findings

Front speed depends on species B's diffusion coefficient in dilute solutions.

Deterministic cutoff models cannot explain the observed speed decrease.

Cross-diffusion in concentrated solutions mitigates the impact of differing diffusivities.

Abstract

A wave front of Fisher and Kolmogorov, Petrovskii, and Piskunov type involving two species A and B with different diffusion coefficients $D_A$ and $D_B$ is studied using a master equation approach in dilute and concentrated solutions. Species A and B are supposed to be engaged in the autocatalytic reaction A+B -> 2A. Contrary to the results of a deterministic description, the front speed deduced from the master equation in the dilute case sensitively depends on the diffusion coefficient of species B. A linear analysis of the deterministic equations with a cutoff in the reactive term cannot explain the decrease of the front speed observed for $D_B > D_A$. In the case of a concentrated solution, the transition rates associated with cross-diffusion are derived from the corresponding diffusion fluxes. The properties of the wave front obtained in the dilute case remain valid but are mitigated by cross-diffusion which reduces the impact of different diffusion coefficients.