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

Highlights

• Stochastic description of FKPP wave front for 2 species with different diffusivities.

• Unexpected decrease of propagation speed for larger diffusivity of consumed species.

• Non-standard master equation for reaction and cross-diffusion processes.

• Cross-diffusion mitigates the impact of different 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 $\to$ 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.

Introduction

Wave fronts propagating into an unstable state according to the model of Fisher and Kolmogorov, Petrovskii, and Piskunov (FKPP) [1], [2] are encountered in many fields [3], in particular biology [4] and ecology [5]. Phenotype selection through the propagation of the fittest trait [6] and cultural transmission in neolithic transitions [7] are a few examples of applications of FKPP fronts. The model introduces a partial differential equation with a logistic growth term and a diffusion term.

The effect of non standard diffusion on the speed of FKPP front is currently investigated [8], [9], [10], [11] and we recently considered the propagation of a wave front in a concentrated solution in which cross-diffusion cannot be neglected [12]. Experimental evidence of cross-diffusion has been given in systems involving ions, micelles, surface, or polymer reactions and its implication in hydrodynamic instabilities has been demonstrated [13], [14], [15], [16], [17], [18]. In parallel, cross-diffusion is becoming an active field of research in applied mathematics [19], [20], [21], [22], [23], [24].

The sensitivity of FKPP fronts to fluctuations has been first numerically observed [25], [26]. An interpretation has been then proposed in the framework of a deterministic approach introducing a cutoff in the logistic term [27]. In mesoscopic or microscopic descriptions of the invasion front of A particles engaged in the reaction $\mathrm{A}+\mathrm{B}\to 2\mathrm{A}$, the discontinuity induced by the rightmost particle in the leading edge of species A profile amounts to a cutoff in the reactive term. The inverse of the number of particles in the reactive interface gives an estimate of the cutoff [28]. The study of the effect of fluctuations on FKPP fronts remains topical [29], [30]. In this paper we perform a stochastic analysis of a reaction–diffusion front of FKPP type in the case of two species A and B with different diffusion coefficients [31], giving rise to cross-diffusion phenomena in concentrated solutions.

The paper is organized as follows. Section 2 is devoted to a dilute system without cross-diffusion. The effects of the discrete number of particles on the front speed, the shift between the profiles of the two species and the width of species A profile are deduced from a master equation approach. In Section 3, we derive the expression of the master equation associated with a concentrated system inducing cross-diffusion and compare the properties of the FKPP wave front in the dilute and the concentrated cases. Conclusions are given in Section 4.