Propagation in a Fisher-KPP equation with non-local advection *

TL;DR

This paper studies how non-local advection affects propagation speeds in a Fisher-KPP equation, providing explicit bounds and characterizing front positions for various kernel functions.

Contribution

It generalizes the Fisher-KPP model with non-local advection, deriving explicit bounds and asymptotic behaviors for propagation speeds based on kernel properties.

Findings

Explicit bounds on propagation speed for kernels in L1(R).

Asymptotic front position estimates for kernels in Lp(R).

Characterization of front behavior depending on kernel decay and integrability.

Abstract

We investigate the influence of a general non-local advection term of the form K * u to propagation in the one-dimensional Fisher-KPP equation. This model is a generalization of the Keller-Segel-Fisher system. When K $\in$ L 1 (R), we obtain explicit upper and lower bounds on the propagation speed which are asymptotically sharp and more precise than previous works. When K $\in$ L p (R) with p > 1 and is non-increasing in (--$\infty$, 0) and in (0, +$\infty$), we show that the position of the "front" is of order O(t 1/p) if p < $\infty$ and O(e $\lambda$t) for some $\lambda$ > 0 if p = $\infty$ and K(+$\infty$) > 0. We use a wide range of techniques in our proofs.