Interplay between scales in the nonlocal FKPP equation

TL;DR

This paper explores how nonlocal interactions in a generalized FKPP model influence pattern formation, revealing that diffusion can sometimes promote rather than inhibit patterns depending on the influence functions.

Contribution

It introduces a nonlocal FKPP model with influence functions for growth, competition, and diffusion, analyzing their interplay in pattern formation through analytical and numerical methods.

Findings

Competition drives pattern formation.

Diffusion can promote patterns depending on influence functions.

Nonlocal effects significantly alter traditional FKPP dynamics.

Abstract

We consider a generalization of the FKPP equation for the evolution of the spatial density of a single-species population where all the terms are nonlocal. That is, the spatial extension of each process (growth, competition and diffusion) is ruled by an influence function, with a characteristic shape and range of action. Our purpose is to investigate the interference between these different components in pattern formation. We show that, while competition is the leading process behind patterns, the other two can act either constructively or destructively. For instance, diffusion that is commonly known to smooth out the concentration field can actually favor pattern formation depending on the shape and range of the dispersal kernel. The results are supported by analytical calculations accompanied by numerical simulations.