Slow and fast minimal speed traveling waves of the FKPP equation with chemotaxis
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.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2021
- DOI:
- 10.48550/arXiv.2102.06065
- arXiv:
- arXiv:2102.06065
- Bibcode:
- 2021arXiv210206065H
- Keywords:
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- Mathematics - Analysis of PDEs
- E-Print:
- 31 pages, 1 figure, updated copy after referee comments