Fluctuations and Stability of Fisher Waves

Abstract

We have performed direct Monte Carlo simulations of the reversible diffusion-limited process A + A~A to study the effect of fluctuations on a propagating interface between stable and unstable phases. The mean-field description of this process, Fisher's reaction-diffusion equation, admits stable nonlinear wave fronts. We find that this mean-field description breaks down in spatial dimensions 1 and 2, while it appears to be qualitatively and quantitatively accurate at and above 4 dimensions. In

Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka–Volterra Models

“…2), such 'pursuit and evasion' waves in the predator and prey density fields arise naturally in the SLLVM in a certain region of parameter space, even in the absence of explicit diffusion of either species. We also mention that the problem of velocity selection for reaction fronts starting from a microscopic description, e.g., from the associated master equation, for the underlying stochastic processes is a rather subtle issue [33,34]. For some two-state models, a field-theoretic representation has been useful to derive a stochastic differential equation that properly represents the underlying stochastic process [34].…”

External Fluctuations in Front Propagation

“…Here we will focus on the simplest case in which a globally stable state invades an unstable or metastable state. This problem has been extensively studied in the recent literature [2][3][4][5][6][7][8] particularly concerning the issue of velocity selection.On the other hand, in the last few years there has been a growing interest in the theoretical study of the role of fluctuations in front propagation [7,[9][10][11][12][13][14], and in particular there have been some experiments on the effects of stochastic turbulence in front propagation in the context of chemical fronts [15]. These studies have been basically concerned with the modification of the front velocity and the spreading of the front due to fluctuations.…”

“…On the other hand, in the last few years there has been a growing interest in the theoretical study of the role of fluctuations in front propagation [7,[9][10][11][12][13][14], and in particular there have been some experiments on the effects of stochastic turbulence in front propagation in the context of chemical fronts [15]. These studies have been basically concerned with the modification of the front velocity and the spreading of the front due to fluctuations.…”

“…In these circumstances the ensemble average ͗c͑x, t͒͘ exhibits a width which grows in time as t 1͞2 (in the frame moving with the average velocity) [12], but which is not the actual front width. In fact, the front has a well defined mean shape which is different from the deterministic kink solution and which is obtained when the roughness of the front profile produced by noise is appropriately averaged out.…”

“…Internal [9][10][11][12] and external [13,14] fluctuations have been introduced in particular models using both Langevin [9,11,13,14] and master equation formalisms [10,12], but no systematic studies have been carried out concerning the modification of the well established selection criteria of the deterministic case. For internal fluctuations mostly numerical studies of different situations have obtained distinct effects on the front propagation.…”