Fisher waves in the strong noise limit

TL;DR

This paper studies how strong stochastic fluctuations affect traveling wave solutions in the Fisher-Kolmogorov system, revealing new behaviors in wave velocity, profile structure, and front dynamics that differ from traditional deterministic models.

Contribution

It provides the first analysis of Fisher waves under strong noise, showing linear and square-root velocity dependence and rugged front structures with power-law size distributions.

Findings

Wave velocity depends linearly and on the square root of particle density in 1D and 2D.

Front profiles are composed of rugged kinks rather than smooth sigmoids.

Front size distribution follows a power-law tail.

Abstract

We investigate the effects of strong number fluctuations on traveling waves in the Fisher-Kolmogorov reaction-diffusion system. Our findings are in stark contrast to the commonly used deterministic and weak-noise approximations. We compute the wave velocity in one and two spatial dimensions, for which we find a linear and a square-root dependence of the speed on the particle density. Instead of smooth sigmoidal wave profiles, we observe fronts composed of a few rugged kinks that diffuse, annihilate, and rarely branch; this dynamics leads to power-law tails in the distribution of the front sizes.