Slow and fast minimal speed traveling waves of the FKPP equation with chemotaxis

TL;DR

This paper investigates how chemotaxis influences the speed of traveling waves in a generalized FKPP model, revealing conditions where chemotaxis does not affect wave speed and regimes where it causes arbitrarily large speeds.

Contribution

It establishes thresholds for chemotactic influence on wave speed and demonstrates unbounded wave speeds under strong repulsive interactions.

Findings

Chemotaxis can be negligible for wave speed under certain localized interactions.

Weak interactions do not alter the minimal wave speed from the classical FKPP value of 2.

Strong repulsive interactions lead to arbitrarily large wave speeds.

Abstract

We examine a general model for the Fisher-KPP (FKPP) equation with nonlocal advection. The main interpretation of this model is as describing a diffusing and logistically growing population that is also influenced by intraspecific attraction or repulsion. For a particular choice of parameters, this specializes to the Keller-Segel-Fisher equation for chemotaxis. Our interest is in the effect of chemotaxis on the speed of traveling waves. We prove that there is a threshold such that, when interactions are weaker and more localized than this, chemotaxis, despite being non-trivial, does not influence the speed of traveling waves; that is, the minimal speed traveling wave has speed 2 as in the FKPP case. On the other hand, when the interaction is repulsive, we show that the minimal traveling wave speed is arbitrarily large in a certain asymptotic regime in which the interaction strength and length scale tend to infinity.