Effect of noise on front propagation in reaction-diffusion equations of KPP type

Abstract

We consider reaction-diffusion equations of KPP type in one spatial dimension, perturbed by a Fisher-Wright white noise, under the assumption of uniqueness in distribution. Examples include the randomly perturbed Fisher-KPP equationswhereẆ =Ẇ (t, x) is a space-time white noise.

Genetic hitchhiking in spatially extended populations

“…This argument is not rigorous, but for (3), writing p cη (z) for the allele frequency in the stationary wavefront, at least where allele frequencies are small we have (Brunet et al, 2006;Mueller et al, 2011;Berestycki et al, 2012) p cη (z) ∝ W π sin πz W e −c∞z/σ 2 , (A.4) where in these coordinates the 'front' of the wave is at z = W = (log η + 3 log log η)ℓ/2.…”

“…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.…”

“…Three ramifications of the presence of small amounts of genetic drift have been explored: first, drift slows down the rate of advance of the wave by a factor proportional to the dimensionless quantity 1/(log(ρσ s/2)) 2 where ρ is population density, σ 2 is the rate of diffusion and s is the selection coefficient (Brunet et al, 2006;Mueller et al, 2011); second, the resulting cline shape (that is the shape of the wavefront) is well-approximated by a truncated Fisher wave (described in more detail in §2.1), Brunet and Derrida (1997); Mueller et al (2011); and third, the genealogy of a sample from the wavefront will be dominated by 'founder effects' (resulting from fluctuations in the wavefront) and in an appropriate timescale is approximated not by a Kingman coalescent, but instead by the so-called Bolthausen-Sznitman coalescent in which, in particular, multiple lineages merge in a single event (Brunet et al, 2006;Berestycki et al, 2012).…”

“…In the case of (3) it has been proved rigorously that A = π 2 /2 (Mueller et al, 2011). The heuristic arguments of Brunet and Derrida, and indeed the rigorous proof of Brunet-Derrida behaviour of (3), rest on comparing the solution to the travelling wave solution of the deterministic Fisher-KPP equation (1) corresponding to the wavespeed c η .…”

Effect of stochastic perturbations for front propagation in Kolmogorov Petrovskii Piscunov equations

“…The behaviour of SPDEs of this type, whose initial conditions have compact support is markedly different; it is well known that the solutions themselves have compact support for all time. Indeed, the same carries over when there is a right limit for the support of a bounded initial condition and the definition of the marker in [8] is the right end point of the support.…”