Quasiparticles for the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation

TL;DR

This paper develops quasiparticle solutions for the one-dimensional nonlocal FKPP equation using semiclassical methods, aiding in understanding population pattern dynamics with nonlocal interactions.

Contribution

It introduces a novel quasiparticle-based approach for the nonlocal FKPP equation, employing semiclassical techniques and auxiliary ODEs for asymptotic solutions.

Findings

Constructed quasiparticle solutions for the nonlocal FKPP equation.

Analyzed various spatial profiles of reproduction and competition.

Provided a framework for predicting population pattern dynamics.

Abstract

We construct quasiparticles-like solutions to the one-dimensional Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) with a nonlocal nonlinearity using the method of semiclassically concentrated states in the weak diffusion approximation. Such solutions are of use for predicting the dynamics of population patterns. The interaction of quasiparticles stems from nonlocal competitive losses in the FKPP model. We developed the formalism of our approach relying on ideas of the Maslov method. The construction of the asymptotic expansion of a solution to the original nonlinear evolution equation is based on solutions to an auxiliary dynamical system of ODEs. The asymptotic solutions for various specific cases corresponding to various spatial profiles of the reproduction rate and nonlocal competitive losses are studied within the framework of the approach proposed.