Proliferating L\'evy Walkers and Front Propagation

TL;DR

This paper develops kinetic equations for proliferating Lévy walkers, deriving conditions for maximum front propagation speed and analyzing how birth, death, and anomalous effects influence this velocity.

Contribution

It introduces a novel kinetic framework for proliferating Lévy walkers and derives Hamilton-Jacobi equations to analyze front propagation speeds.

Findings

Maximum propagation speed equals walker's speed under certain conditions.

Death rates can reduce propagation velocity below maximum.

Strong anomalous effects influence front speed dynamics.

Abstract

We develop non-linear integro-differential kinetic equations for proliferating L\'{e}vy walkers with birth and death processes. A hyperbolic scaling is applied directly to the general equations to get the Hamilton-Jacobi equations that will allow to determine the rate of front propagation. We found the conditions for switching, birth and death rates under which the propagation velocity reaches the maximum value, i.e. the walker's speed. In the strong anomalous case the death rate was found to influence the velocity of propagation to fall below the walker's maximum speed.