Speed-up of traveling waves by negative chemotaxis

TL;DR

This paper investigates how negative chemotaxis influences the speed of traveling waves in FKPP models, revealing asymptotic behaviors and establishing bounds and existence of waves across parameter regimes.

Contribution

It provides a comprehensive analysis of traveling wave speeds under chemotaxis, connecting different equations and extending wave existence results.

Findings

Asymptotic dependence of wave speed on chemotaxis parameters

Explicit lower bounds on minimal wave speed

Extended existence range for traveling waves

Abstract

We consider the traveling wave speed for Fisher-KPP (FKPP) fronts under the influence of repulsive chemotaxis and provide an almost complete picture of its asymptotic dependence on parameters representing the strength and length-scale of chemotaxis. Our study is based on the convergence to the porous medium FKPP traveling wave and a hyperbolic FKPP-Keller-Segel traveling wave in certain asymptotic regimes. In this way, it clarifies the relationship between three equations that have each garnered intense interest on their own. Our proofs involve a variety of techniques ranging from entropy methods and decay of oscillations estimates to a general description of the qualitative behavior to the hyperbolic FKPP-Keller-Segel equation. For this latter equation, we, as a part of our limiting arguments, establish an explicit lower bound on the minimal traveling wave speed and provide a new construction of traveling waves that extends the known existence range to all parameter values.