A slow pushed front in a Lotka-Volterra competition model
type: Item
instance of: scholarly article
zbMATH Open document ID: 1252.35106
DOI: 10.1088/0951-7715/25/7/2151
author: Matt Holzer
author: Arnd Scheel
published in: Nonlinearity
publication date: 2012-08-02T00:00:00Z
Mathematics Subject Classification ID: 35C07
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 35K45
Mathematics Subject Classification ID: 35K58
zbMATH DE Number: 6063068
zbMATH Keywords: wavespeed selection
zbMATH Keywords: geometric singular perturbation theory
zbMATH Keywords: two-species model
zbMATH Keywords: slowest monotone travelling front
zbMATH Keywords: marginally stable
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1088/0951-7715/25/7/2151
OpenAlex ID: W2130846836
title: A slow pushed front in a Lotka-Volterra competition model
review text: This paper studies a two-species Lotka-Volterra competition model, in which the two species diffuse on the real line, NEWLINE\[NEWLINEu_t=\epsilon^2 u_{xx}+(1-u-a_1 v)u,NEWLINE\]NEWLINE NEWLINE\[NEWLINEv_t=v_{xx} +r(1-a_2 u-v)v.NEWLINE\]NEWLINE Parameters are chosen so that \(a_1<1<a_2\) and, in the corresponding space-homogeneous model, the stable state is the \(u=1\), \(v=0\). It is also assumed that the diffusion rate \(\epsilon^2\) of the species \(u\) is sufficiently small. The question of interest is to understand the process of invasion of the species \(u\) into a homogeneous population of species \(v\), by studying travelling front connecting the unstable state \(u=0\), \(v=1\) to the stable state \(u=1\), \(v=0\). Using geometric singular perturbation theory, it is proved (for \(\epsilon\) sufficiently small) that there is a slowest monotone travelling front, which is marginally stable, and is thus the selected front. Of central interest here is the fact that the speed at which this front travels is slower than the linear spreading speed, that is the speed of waves obtained from the linearization of the equations around the unstable equilibrium. This contrasts with the more familiar phenomenon for scalar equations, whereby the speed of the selected front is higher than the linear spreading speed. An analysis of the resolvent of the linearized system shows that this effect is connected to the fact that the linear spreading speed is a simple pole of the resolvent rather than the a branch pole.
reviewed by: Guy Katriel
description: scientific article; zbMATH DE number 6063068