Front propagation into unstable states

Abstract

This paper is an introductory review of the problem of front propagation into unstable states. Our presentation is centered around the concept of the asymptotic linear spreading velocity v*, the asymptotic rate with which initially localized perturbations spread into an unstable state according to the linear dynamical equations obtained by linearizing the fully nonlinear equations about the unstable state. This allows us to give a precise definition of pulled fronts, nonlinear fronts whose asymptotic propagation speed equals v*, and pushed fronts, nonlinear fronts whose asymptotic speed v^dagger is larger than v*. In addition, this approach allows us to clarify many aspects of the front selection problem, the question whether for a given dynamical equation the front is pulled or pushed. It also is the basis for the universal expressions for the power law rate of approach of the transient velocity v(t) of a pulled front as it converges toward its asymptotic value v*. Almost half of the paper is devoted to reviewing many experimental and theoretical examples of front propagation into unstable states from this unified perspective. The paper also includes short sections on the derivation of the universal power law relaxation behavior of v(t), on the absence of a moving boundary approximation for pulled fronts, on the relation between so-called global modes and front propagation, and on stochastic fronts.Comment: final version with some added references; a single pdf file of the published version is available at http://www.lorentz.leidenuniv.nl/~saarloo

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

“…The effects of cut-offs are also investigated in section 7 of [31], as one aspect of a broad study of fronts propagating into unstable states. It is shown there that the results of Brunet and Derrida on the shift in propagation speed hold more generally for a large class of so-called fluctuating pulled fronts in the large-N limit.…”

“…Also, we note that for ε = 1/N > 0, these fronts are actually pushed fronts. We refer the reader to [31, equation (246)], and more generally to section 7.1 of [31], for further analysis.…”

Genetic hitchhiking in spatially extended populations

“…However, the speed of the Fisher wave is determined by its behaviour in regions where the frequency of the favoured allele is extremely low and, consequently, it is very sensitive to stochastic fluctuations. In recent years there has been considerable progress in understanding the coupling between such fluctuations and the progress of the 'bulk' of the wave Derrida (1997, 2001);van Saarloos (2003); Brunet et al (2006); Hallatschek and Nelson (2008); Mueller et al (2011);Berestycki et al (2012)). Much of this work is concerned with the spread of a new species into an empty habitat, but the mathematical models apply equally to the spread of a selectively favoured allele through a stable population.…”

Solitary Waves of Nonlinear Nonintegrable Equations

“…Computer simulations as well as real experiments (for a recent review, see [34]) sometimes display regular patterns in the (x, t) plane, and some of them are indeed described by some analytic solution. For the remaining patterns, the guess is that there should exist analytic expressions, to be found, corresponding to these patterns.…”

“…[5]), together with the assumption that a 2 obeys a first order second degree elliptic equation. This allows one to retrieve (34) in the particular case r 0 = 0, c = 0.…”