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We examine a general model for the Fisher-KPP (FKPP) equation with nonlocaladvection. The main interpretation of this model is as describing a diffusingand logistically growing population that is also influenced by intraspecificattraction or repulsion. For a particular choice of parameters, thisspecializes to the Keller-Segel-Fisher equation for chemotaxis. Our interest isin the effect of chemotaxis on the speed of traveling waves. We prove thatthere is a threshold such that, when interactions are weaker and more localizedthan this, chemotaxis, despite being non-trivial, does not influence the speedof traveling waves; that is, the minimal speed traveling wave has speed 2 as inthe FKPP case. On the other hand, when the interaction is repulsive, we showthat the minimal traveling wave speed is arbitrarily large in a certainasymptotic regime in which the interaction strength and length scale tend toinfinity.

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