Propagation of fronts in the Fisher-Kolmogorov equation with spatially varying diffusion

TL;DR

This paper develops an analytical framework to understand how fronts propagate in the Fisher-Kolmogorov equation with spatially varying diffusion, revealing complex behaviors near diffusion minima and supporting findings with numerical simulations.

Contribution

It introduces a novel analytical approach combining coordinate changes, WKB, and multiple scales analysis for spatially varying diffusion in the Fisher-Kolmogorov equation.

Findings

Front propagation halts near the diffusion minimum.

Standard traveling wave frames are inadequate for this scenario.

Numerical simulations confirm the analytical predictions.

Abstract

The propagation of fronts in the Fisher-Kolmogorov equation with spatially varying diffusion coefficients is studied. Using coordinate changes, WKB approximations, and multiple scales analysis, we provide an analytic framework that describes propagation of the front up to the minimum of the diffusion coefficient. We also present results showing the behavior of the front after it passes the minimum. In each case, we show that standard traveling coordinate frames do not properly describe front propagation. Lastly, we provide numerical simulations to support our analysis and to show, that around the minimum, the motion of the front is arrested on asymptotically significant timescales.