Traveling wave front for a two-component lattice dynamical system arising in competition models
type: Item
instance of: scholarly article
title: Traveling wave front for a two-component lattice dynamical system arising in competition models
zbMATH Open document ID: 1251.34018
author: Jong-Shenq Guo
author: Chang-Hong Wu
published in: Journal of Differential Equations
publication date: 2012-03-29T00:00:00Z
review text: The authors study the travelling front solutions for the following two-component lattice dynamical system \[ \begin{cases} \frac{d u_j}{dt} = (u_{j+1}+u_{j-1}-2u_j)+u_j(1-u_j-k v_j),\\ \frac{d v_j}{dt} = d(v_{j+1}+v_{j-1}-2v_j)+rv_j(1-v_j-h u_j), \end{cases} \] where \(t\in \mathbb{R}\), \(j\in \mathbb{Z}\), \(d,r\in (0,\infty)\) and \[ 0<k<1<h, \] which is known as the monostable case. The authors note that the other monostable case, \(0<h<1<k\), can be reduced to the previous one by exchanging the roles of \(u\) and \(v\). They show that there is a positive constant (the minimal wave speed) such that a travelling front exists if and only if its speed is above this minimal wave speed. Then, doing an analysis of asymptotic behaviour of wave tails, they show that: {\parindent=6mm\begin{itemize}\item[1)] All wave profiles are strictly monotone. \item[2)] If \(d\leq 1\), then the wave profile is unique up to translations for a given wave speed. \end{itemize}} Finally, they characterize the minimal wave speed by the parameters in the system.
reviewed by: Daniel Franco
Mathematics Subject Classification ID: 34A33
Mathematics Subject Classification ID: 34B40
Mathematics Subject Classification ID: 34C11
zbMATH DE Number: 6019795
zbMATH Keywords: travelling front
zbMATH Keywords: lattice dynamical system
zbMATH Keywords: competition model
zbMATH Keywords: minimal wave speed
zbMATH Keywords: wave profile
MaRDI profile type: Publication
full work available at URL: j.jde.2012.01.009
OpenAlex ID: W2013950926
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DOI: 10.1016/J.JDE.2012.01.009
description: scientific article; zbMATH DE number 6019795