Stochastic approach to Fisher and Kolmogorov, Petrovskii, and Piskunov wave fronts for species with different diffusivities in dilute and concentrated solutions
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 , 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.
Section snippets
Dilute system
We consider two chemical species A and B engaged in the reaction where is the rate constant. The diffusion coefficient, , of species A may differ from the diffusion coefficient, , of species B.
In a deterministic approach, the reaction–diffusion equations are where the concentrations of species A and B are denoted by and . The system admits wave front solutions propagating without deformation at constant speed. For sufficiently steep initial conditions and
Concentrated system
In a dilute system, the solvent S is in great excess with respect to the reactive species A and B. The concentration of the solvent is then supposed to remain homogeneous regardless of the variation of concentrations and . In a concentrated solution, the variation of the concentration of the solvent cannot be ignored. In the linear domain of irreversible thermodynamics, the diffusion fluxes are linear combinations of the concentration gradients of the different species. The flux of
Conclusion
We have performed kinetic Monte Carlo simulations of the master equation associated with a chemical system involving two species A and B. The two species have two different diffusion coefficients, and , and are engaged in the autocatalytic reaction . The effects of fluctuations on the FKPP wave front have been studied in the cases of a dilute solution and a concentrated solution in which cross-diffusion cannot be neglected.
In the case of a dilute system, the linearization of the
CRediT authorship contribution statement
Gabriel Morgado: Methodology, Software, Investigation, Writing - original draft, Visualization. Bogdan Nowakowski: Conceptualization, Validation, Writing - review & editing, Supervision. Annie Lemarchand: Definition, Conceptualization, Validation, Resources, Writing - original draft, Review & editing, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This publication is part of a project that has received funding from the European Union’s Horizon 2020 (H2020-EU.1.3.4.) research and innovation program under the Marie Sklodowska-Curie Actions (MSCA-COFUND ID 711859) and from the Ministry of Science and Higher Education, Poland, for the implementation of an international cofinanced project.
References (39)
Phys. Rep.
(2003)- et al.
C. R. Acad. Sci., Paris I
(2012) - et al.
Physica D
(2003) - et al.
C. R. Acad. Sci., Paris I
(2009) - et al.
J. Math. Anal. Appl.
(2015) - et al.
J. Differ. Equ.
(2019) - et al.
Physica D
(1994) Phys. Rep.
(2004)- et al.
Physica A
(2015) Ann. Eugen.
(1937)