Traveling wave front for a two-component lattice dynamical system arising in competition models
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.
scientific article; zbMATH DE number 6019795