\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 57, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/57\hfil Three-component lattice competition system] {Stability of traveling wavefronts for a three-component Lotka-Volterra competition system on a lattice} \author[T. Su, G.-B. Zhang \hfil EJDE-2018/57\hfilneg] {Tao Su, Guo-Bao Zhang} \address{Tao Su \newline College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China} \email{970913788@qq.com} \address{Guo-Bao Zhang (corresponding author) \newline College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China} \email{zhanggb2011@nwnu.edu.cn} \dedicatory{Communicated by Zhaosheng Feng} \thanks{Submitted September 8, 2017. Published March 1, 2018.} \subjclass[2010]{34A33, 34K20, 92D25} \keywords{Three-component competition system; lattice dynamical system; \hfill\break\indent traveling wavefronts; stability} \begin{abstract} This article concerns the stability of traveling wavefronts for a three-component Lotka-Volterra competition system on a lattice. By means of the weighted energy method and the comparison principle, it is proved that the traveling wavefronts with large speed are exponentially asymptotically stable, when the initial perturbation around the traveling wavefronts decays exponentially as $j+ct \to -\infty$, where $j\in\mathbb{Z}$, $t>0$ and $c>0$, but the initial perturbation can be arbitrarily large on other locations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} Consider the three-component Lotka-Volterra competition system on a lattice \begin{equation}\label{1.1} \begin{gathered} \frac{du_j(t)}{dt}=d_1\mathcal{D}[u_j](t)+r_1u_j(t)[1-u_j(t)-b_{12}v_j(t)],\\ \frac{dv_j(t)}{dt}=d_2\mathcal{D}[v_j](t)+r_2v_j(t)[1-b_{21}u_j(t)-v_j(t) -b_{23}w_j(t)],\\ \frac{dw_j(t)}{dt}=d_3\mathcal{D}[w_j](t)+r_3w_j(t)[1-b_{32}v_j(t)-w_j(t)], \end{gathered} \end{equation} with the initial data \[ u_j(0)=u_{j0},\quad v_j(0)=v_{j0},\quad w_j(0)=w_{j0}, \] where $j\in\mathbb{Z}$, $t>0$, $d_n>0$, $r_n>0$, $b_{nm}>0$, $m, n\in\{1,2,3\}$, $\mathcal{D}[z_j]=z_{j+1}+z_{j-1}-2z_j$ for $z=u, v, w$. Here, $u_j$, $v_j$ and $w_j$ are the population densities of three different species (call them as species 1, 2, 3) at site $j$ at time $t$, $d_n$ is the migration coefficient of species $n$, $r_n$ is the net birth rate of species $n$ and $b_{nm}$ is the competition coefficient of species $m$ to species $n$. Also, we have taken the scales so that the carrying capacity of each species is normalized to be $1$. Throughout this paper, we assume \begin{itemize} \item[(H1)] $b_{12}$, $b_{32}>1$, $b_{21}+b_{23}<1$, \end{itemize} which means that the species 1, 3 are weak competitors to the species 2. Therefore, it is expected that the species 2 shall win the competition eventually. It is easy to see that the system \eqref{1.1} has constant equilibria $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, $(0,0,1)$ and $(1,0,1)$. To understand the invading phenomenon between the residents $u$, $w$ and the invader $v$, the traveling wave solution connecting two equilibrium points $(1,0,1)$ and $(0,1,0)$ has been considered by many researchers \cite{GWWW,Wu0,Wu}. We note that a traveling wave solution of \eqref{1.1} is a special translation invariant solution of the form \[ u_j(t)=\varphi(\xi), \quad v_j(t)=\psi(\xi),\quad w_j(t)=\theta(\xi), \quad \xi=j+ct, \] where $c>0$ is the wave speed. If $\varphi$, $\psi$, $\theta$ are monotone, then $(\varphi, \psi,\theta)$ is called a traveling wavefront. Substituting $(\varphi(j+ct), \psi(j+ct), \theta(j+ct) )$ into \eqref{1.1}, we obtain the following wave profile system with the asymptotic boundary conditions \begin{align}\label{1.2} \begin{gathered} c\varphi'(\xi)=d_1\mathcal{D}[\varphi](\xi) +r_1\varphi(\xi)[1-\varphi(\xi)-b_{12}\psi(\xi)],\\ c\psi'(\xi)=d_2\mathcal{D}[\psi](\xi)+r_2\psi(\xi)[1-b_{21}\varphi(\xi) -\psi(\xi)-b_{23}\theta(\xi)],\\ c\theta'(\xi)=d_3\mathcal{D}[\theta](\xi) +r_3\theta(\xi)[1-b_{32}\psi(\xi)-\theta(\xi)],\\ (\varphi, \psi, \theta)(-\infty)=(1,0,1),\ (\varphi, \psi, \theta)(+\infty)=(0,1,0),\\ 0\le \varphi, \psi, \theta\le 1, \end{gathered} \end{align} where $\mathcal{D}[u](\xi)=u(\xi+1)+u(\xi-1)-2u(\xi)$. Clearly, when $\theta(\xi)=0$, system \eqref{1.2} reduces to the two-component Lotka-Volterra competition system, we refer to \cite{GW2,TZ}. For the three-component competition system \eqref{1.2}, by considering a truncated problem with the help of a super-solution, Guo et al. \cite{GWWW} showed that there exists a positive constant $c_{min}$ such that \eqref{1.2} has a strictly monotone solution if and only if $c\ge c_{min}$. At the same time, the linear determinacy for \eqref{1.2} was given in \cite{GWWW}. Later, Wu \cite{Wu} established the asymptotic behavior of solutions of \eqref{1.2} at infinity, and constructed some entire solutions of \eqref{1.1}. More recently, Wu \cite{Wu0} proved the monotonicity and uniqueness (up to translations) of solutions of \eqref{1.2} with speed $c\ge c_{min}$. A natural question is whether the traveling wavefronts of \eqref{1.1} (i.e., solutions of \eqref{1.2}) are stable for each admissible speed. In this paper, we give an answer to this question. The stability of traveling wave solutions for various evolution equations with or without delay has been extensively studied, for example, see \cite{CLW,LLLM,MD,MZ,MZ1,MLLS1,MOZ,SZ,TZ,YLW1,YY,YM,Z,ZL,ZM}. The main methods are the (technical) weighted energy method \cite{LLLM,ZM}, the sub- and supersolutions method and squeezing technique \cite{Chen,SZ}, and the combination of the comparison principle and the weighted energy method \cite{MLLS1,TZ,YY}. To the best our knowledge, for evolution systems, little has been done to establish the stability of traveling wave solutions. In 2011, Yang, Li and Wu \cite{YLW1,YLW2} considered a diffusive epidemic system with delay and established the stability of traveling wavefronts. Lv and Wang \cite{LW1} and Yu et al.\ \cite{YXZ} respectively investigated the stability of traveling wavefronts for two-component Lotka-Volterra cooperative and competitive systems with nonlocal dispersals. Encouraged by papers \cite{LW1,MLLS1,TZ,YLW1,YXZ}, in this paper, we take the weighted energy method together with the comparison principle to study the stability of traveling wavefronts for the three-component lattice competition system \eqref{1.1}. We first give a comparison principle and then prove that the traveling wavefronts of \eqref{1.1} are stable, when the difference between initial data and traveling wavefront decays exponentially as $j+ct\to -\infty$, but the initial data can be arbitrarily large on other locations. We should remark that although the main idea is same as that for two-component lattice competition system, some complexities and difficulties arise in the three component lattice competition system due to the coupling of the nonlinearities. The rest of this paper is organized as follows. In section $2$, we give the notations, the existence of traveling wavefronts, some necessary assumptions and the main theorem. Section $3$ is devoted to the proof of the stability theorem. \section{Preliminaries and main result} In this section, we first recall some known results, then define a weight function and state our main result. To study the stability of traveling wavefront of \eqref{1.1}, it is convenient to work on $(u_j^{*},v_j,w_j^{*})$, where $u_j^{*}=1-u_j,w_j^{*}=1-w_j$. For the sake of convenience, we drop the star. Then \eqref{1.1} is transformed into the system \begin{equation}\label{2.1} \begin{gathered} \frac{du_j(t)}{dt}=d_1\mathcal{D}[u_j](t)+r_1(1-u_j(t))[-u_j(t)+b_{12}v_j(t)],\\ \frac{dv_j(t)}{dt}=d_2\mathcal{D}[v_j](t)+r_2v_j(t)[1-b_{21}-b_{23}-v_j(t) +b_{21}u_j(t)+b_{23}w_j(t)],\\ \frac{dw_j(t)}{dt}=d_3\mathcal{D}[w_j](t)+r_3(1-w_j(t))[-w_j(t)+b_{32}v_j(t)], \end{gathered} \end{equation} with the initial data \begin{equation}\label{2.2} \begin{gathered} u_j(0)=1-u_{j0},\\ v_j(0)=v_{j0},\\ w_j(0)=1-w_{j0}. \end{gathered} \end{equation} Let $u_j(t)=\varphi(\xi)$, $v_j(t)=\psi(\xi)$, $w_j(t)=\theta(\xi)$, $\xi=j+ct$. Then the wave profile system of \eqref{2.1} is \begin{equation}\label{2.3} \begin{gathered} c\varphi'(\xi)=d_1\mathcal{D}[\varphi](\xi) +r_1(1-\varphi(\xi))[-\varphi(\xi)+b_{12}\psi(\xi)],\\ c\psi'(\xi)=d_2\mathcal{D}[\psi](\xi)+r_2\psi(\xi)[1-b_{21}-b_{23}-\psi(\xi) +b_{21}\varphi(\xi)+b_{23}\theta(\xi)],\\ c\varphi'(\xi)=d_3\mathcal{D}[\theta](\xi) +r_3(1-\theta(\xi))[-\theta(\xi)+b_{32}\psi(\xi)], \end{gathered} \end{equation} with the boundary condition \begin{equation} \label{2.3*} (\varphi,\psi,\theta)(-\infty)=(0,0,0) \quad\text{and} \quad (\varphi,\psi,\theta)(+\infty)=(1,1,1). \end{equation} The existence of traveling wavefront of \eqref{2.1} comes from Guo et al.\ \cite{GWWW}. \begin{proposition}[Existence]\label{201} Assume that {\rm (H1)} holds. Then there exists $c_{min}>0$ such that for any $c\geq c_{min}$, \eqref{2.1} admits a traveling wavefront $(\varphi(\xi),\psi(\xi),\theta(\xi))$ connecting $(0,0,0)$ and $(1,1,1)$, and satisfying $\varphi'(\cdot)>0$, $\psi'(\cdot)>0$ and $\theta'(\cdot)>0$ on $\mathbb{R}$. For any $c2+\frac{r_2b_{21}}{2r_1}$, $b_{32}>2+\frac{r_2b_{23}}{2r_3}$. \end{itemize} Define three functions on $\lambda$ as follows: \begin{gather*} \mathcal{M}_1(\lambda)=2d_1-4r_1+2r_1b_{12}-r_2b_{21}-d_1(e^{\lambda}+1),\\ \mathcal{M}_2(\lambda)=2d_2+2r_2-3r_2(b_{21}+b_{23})-d_2(e^{\lambda}+1),\\ \mathcal{M}_3(\lambda)=2d_3-4r_3+2r_3b_{32}-r_2b_{23}-d_3(e^{\lambda}+1). \end{gather*} From assumption (H2), we obtain \begin{gather*} \mathcal{M}_1(0)=-4r_1+2r_1b_{12}-r_2b_{21}>0,\\ \mathcal{M}_2(0)= 2r_2-3r_2(b_{21}+b_{23})>0,\\ \mathcal{M}_3(0)=-4r_3+2r_3b_{32}-r_2b_{23}>0. \end{gather*} Then by the continuity of $\mathcal{M}_1(\lambda)$, $\mathcal{M}_2(\lambda)$ and $\mathcal{M}_3(\lambda)$ with respect to $\lambda$, there exists $\lambda_0>0$ such that \begin{align}\label{lb} \mathcal{M}_1(\lambda_0)>0, \ \mathcal{M}_2(\lambda_0)>0,\ \mathcal{M}_2(\lambda_0)>0. \end{align} Furthermore, we define \begin{gather*} \mathcal{N}_1(\xi) =2d_1-4r_1+2r_1b_{12}\psi(\xi)+r_1b_{12}\varphi(\xi)-r_1b_{12} -r_2b_{21}-d_1(e^{\lambda_0}+1),\\ \begin{aligned} \mathcal{N}_2(\xi) &=2d_2-2r_2+4r_2\psi(\xi)-3r_2(b_{21}+b_{23})-r_1b_{12}-r_3b_{32} +r_1b_{12}\varphi(\xi)\\ &\quad +r_3b_{32}\theta(\xi)-d_2(e^{\lambda_0}+1), \end{aligned}\\ \mathcal{N}_3(\xi)=2d_3-4r_3+2r_3b_{32}\psi(\xi)+r_3b_{32}\theta(\xi) -r_3b_{32}-r_2b_{23}-d_3(e^{\lambda_0}+1), \end{gather*} where $(\varphi(\xi),\psi(\xi),\theta(\xi))$ is a traveling wavefront given in Proposition \ref{201}. By \eqref{2.3*}, we have \begin{gather*} \lim_{\xi\to+\infty}\mathcal{N}_1(\xi)=\mathcal{M}_1(\lambda_0)>0,\\ \lim_{\xi\to+\infty}\mathcal{N}_2(\xi)=\mathcal{M}_2(\lambda_0)>0,\\ \lim_{\xi\to+\infty}\mathcal{N}_3(\xi)=\mathcal{M}_3(\lambda_0)>0, \end{gather*} which imply that there exists a number $\xi_0>0$ large enough such that \begin{gather*} \mathcal{N}_1(\xi_0) =2d_1-4r_1+2r_1b_{12}\psi(\xi_0)+r_1b_{12}\varphi(\xi_0)-r_1b_{12} -r_2b_{21}-d_1(e^{\lambda_0}+1)>0,\\ \begin{aligned} \mathcal{N}_2(\xi_0) &=2d_2-2r_2+4r_2\psi(\xi_0)-3r_2(b_{21}+b_{23})-r_1b_{12}-r_3b_{32}+r_1b_{12}\varphi(\xi_0)\\ &\quad +r_3b_{32}\theta(\xi_0)-d_2(e^{\lambda_0}+1)>0, \end{aligned}\\ \mathcal{N}_3(\xi_0)=2d_3-4r_3+2r_3b_{32}\psi(\xi_0) +r_3b_{32}\theta(\xi_0)-r_3b_{32}-r_2b_{23}-d_3(e^{\lambda_0}+1)>0. \end{gather*} Define the weighted function \begin{equation}\label{2.5} w(\xi)= \begin{cases} e^{-\lambda_0(\xi-\xi_0)}, &\xi\leq \xi_0,\\ 1, &\xi> \xi_0, \end{cases} \end{equation} where $\lambda_0$ is defined by \eqref{lb}. Let \begin{gather} c_1=4r_1+r_1b_{12}+r_2b_{21}+d_1(e^{\lambda_0}+e^{-\lambda_0}+1),\label{2.6}\\ c_2=2r_2+3r_2(b_{21}+b_{23})+r_1b_{12}+r_3b_{32}+d_2(e^{\lambda_0}+e^{-\lambda_0}+1),\label{2.7}\\ c_3=4r_3+r_3b_{32}+r_2b_{23}+d_3(e^{\lambda_0}+e^{-\lambda_0}+1).\label{2.8} \end{gather} \begin{theorem}[Stability]\label{202} Assume that {\rm (H2)} holds. For any given traveling wavefront $(\varphi(\xi(t,j)),\psi(\xi(t,j)),\theta(\xi(t,j)))$ with the wave speed $c>\max\{c_{min},\tilde{c}\}$, where \begin{equation*} \tilde{c}=\frac{\max\{c_1,c_2,c_3\}}{\lambda_0}. \end{equation*} If the initial data satisfies \begin{equation*} (0,0,0)\leq(u_j(0),v_j(0),w_j(0))\leq(1,1,1), \quad j\in\mathbb{Z}, \end{equation*} and the initial perturbations satisfy \begin{gather*} u_j(0)-\varphi(j)\in l_w^2,\\ v_j(0)-\psi(j)\in l_w^2,\\ w_j(0)-\theta(j)\in l_w^2, \end{gather*} then the nonnegative solution of the Cauchy problem \eqref{2.1} and \eqref{2.2} uniquely exists and satisfies \begin{equation*} (0,0,0)\leq(u_j(t),v_j(t),w_j(t))\leq(1,1,1), \quad j\in\mathbb{Z},\ t>0, \end{equation*} and \begin{gather*} u_j(t)-\varphi(j+ct) \in C\left((0,+\infty);l^2_w\right),\\ v_j(t)-\psi(j+ct) \in C\left((0,+\infty);l^2_w\right),\\ w_j(t)-\theta(j+ct) \in C\left((0,+\infty);l^2_w\right), \end{gather*} where $w(\xi)$ is defined by \eqref{2.5}. Moreover, $(u_j(t),v_j(t),w_j(t))$ converges to the traveling wavefront $(\varphi(j+ct),\psi(j+ct),\theta(j+ct))$ exponentially in time $t$, i.e., \begin{gather*} \sup_{j\in\mathbb{Z}}|u_j(t)-\varphi(j+ct)|\leq Ce^{-\mu t}, \\ \sup_{j\in\mathbb{Z}}|v_j(t)-\psi(j+ct)|\leq Ce^{-\mu t}, \\ \sup_{j\in\mathbb{Z}}|w_j(t)-\theta(j+ct)|\leq Ce^{-\mu t}, \end{gather*} for all $t>0$, where $C$ and $\mu$ are some positive constants. \end{theorem} \section{Stability of traveling wavefronts} We first state the boundedness and the comparison principle for the Cauchy problem \eqref{2.1} and \eqref{2.2} and then prove the main theorem by using the weighted energy method combined with the comparison principle. \begin{lemma}[Boundedness]\label{*} Assume that {\rm (H1)} holds and that the initial data \break $(u_j(0),v_j(0),w_j(0))$ satisfy \begin{equation*} (0,0,0)\leq (u_j(0),v_j(0),w_j(0))\leq (1,1,1) \end{equation*} for $j\in\mathbb{Z}$. Then the solution $(u_j(t),v_j(t),w_j(t))$ of the Cauchy problem \eqref{2.1} and \eqref{2.2} exists and satisfies \begin{equation*} (0,0,0)\leq (u_j(t),v_j(t),w_j(t))\leq (1,1,1) \end{equation*} for $t\in (0,+\infty)$, $j\in\mathbb{Z}$. \end{lemma} \begin{lemma}[Comparison principle]\label{**} Assume that {\rm (H1)} holds. Let \\ $(u_j^{-}(t),v_j^{-}(t),w_j^{-}(t))$ and $(u_j^{+}(t),v_j^{+}(t),w_j^{+}(t))$ be the solution of \eqref{2.1} with the initial data $(u_j^{-}(0),v_j^{-}(0),w_j^{-}(0))$ and $(u_j^{+}(0),v_j^{+}(0),w_j^{+}(0))$, respectively. If \begin{equation*} (0,0,0)\leq (u_j^{-}(0),v_j^{-}(0),w_j^{-}(0))\leq(u_j^{+}(0),v_j^{+}(0),w_j^{+}(0))\leq (1,1,1) \end{equation*} for $j\in\mathbb{Z}$, then \begin{equation*} (0,0,0)\leq (u_j^{-}(t),v_j^{-}(t),w_j^{-}(t))\leq(u_j^{+}(t),v_j^{+}(t),w_j^{+}(t))\leq (1,1,1) \end{equation*} for $t\in (0,+\infty)$, $j\in\mathbb{Z}$. \end{lemma} Let the initial data $(u_j(0),v_j(0),w_j(0))$ be such that $$ (0,0,0)\le (u_j(0),v_j(0),w_j(0))\le (1,1,1) $$ for $j\in\mathbb{Z}$, and let \begin{gather*} u^{-}_j(0)=\min\{u_j(0),\varphi(j)\}, \quad j\in\mathbb{Z},\\ u^{+}_j(0)=\max\{u_j(0),\varphi(j)\}, \quad j\in\mathbb{Z},\\ v^{-}_j(0)=\min\{v_j(0),\psi(j)\}, \quad j\in\mathbb{Z},\\ v^{+}_j(0)=\max\{v_j(0),\psi(j)\}, \quad j\in\mathbb{Z},\\ w^{-}_j(0)=\min\{w_j(0),\theta(j)\},\quad j\in\mathbb{Z},\\ w^{+}_j(0)=\max\{w_j(0),\theta(j)\}, \quad j\in\mathbb{Z}. \end{gather*} Then we can easily get \begin{equation}\label{3.1*} \begin{gathered} 0\leq u^{-}_j(0)\leq u_j(0)\leq u^{+}_j(0)\leq 1, \quad j\in\mathbb{Z},\\ 0\leq u^{-}_j(0)\leq \varphi(j)\leq u^{+}_j(0)\leq 1, \quad j\in\mathbb{Z},\\ 0\leq v^{-}_j(0)\leq v_j(0)\leq v^{+}_j(0)\leq 1, \quad j\in\mathbb{Z},\\ 0\leq v^{-}_j(0)\leq \psi(j)\leq v^{+}_j(0)\leq 1, \quad j\in\mathbb{Z},\\ 0\leq w^{-}_j(0)\leq w_j(0)\leq w^{+}_j(0)\leq 1, \quad j\in\mathbb{Z},\\ 0\leq w^{-}_j(0)\leq \theta(j)\leq w^{+}_j(0)\leq 1, \quad j\in\mathbb{Z}. \end{gathered} \end{equation} Define $u^+_j(t)$, $u^-_j(t)$, $v^+_j(t)$, $v^-_j(t)$, $w^+_j(t)$, $w^-_j(t)$ as the corresponding solutions of \eqref{2.1} with the initial data $u^+_j(0)$, $u^-_j(0)$, $v^+_j(0)$, $v^-_j(0)$, $w^+_j(0)$, $w^-_j(0)$ respectively. Then by the comparison principle in Lemma \ref{**}, we obtain \begin{equation}\label{3.2*} \begin{gathered} 0\leq u^{-}_j(t)\leq u_j(t)\leq u^{+}_j(t)\leq 1, \quad t\in (0,+\infty),\; j\in\mathbb{Z},\\ 0\leq u^{-}_j(t)\leq \varphi(j+ct)\leq u^{+}_j(t)\leq 1, \quad t\in (0,+\infty),\; j\in\mathbb{Z},\\ 0\leq v^{-}_j(t)\leq v_j(t)\leq v^{+}_j(t)\leq 1, \quad t\in (0,+\infty), \; j\in\mathbb{Z},\\ 0\leq v^{-}_j(t)\leq \psi(j+ct)\leq v^{+}_j(t)\leq 1, \quad t\in (0,+\infty), \; j\in\mathbb{Z},\\ 0\leq w^{-}_j(t)\leq w_j(t)\leq w^{+}_j(t)\leq 1, \quad t\in (0,+\infty), \; j\in\mathbb{Z},\\ 0\leq w^{-}_j(t)\leq \theta(j+ct)\leq w^{+}_j(t)\leq 1, \quad t\in (0,+\infty), \; j\in\mathbb{Z}. \end{gathered} \end{equation} Let \begin{gather*} U_j(t)=u^{+}_j(t)-\varphi(j+ct), \quad U_{j0}(0)=u^{+}_j(0)-\varphi(j), \\ V_j(t)=v^{+}_j(t)-\psi(j+ct), \quad V_{j0}(0)=v^{+}_j(0)-\psi(j), \\ W_j(t)=w^{+}_j(t)-\theta(j+ct), \quad W_{j0}(0)=w^{+}_j(0)-\theta(j), \end{gather*} where $t\in (0,+\infty)$, $j\in\mathbb{Z}$. Then by \eqref{2.1} and \eqref{2.3}, $(U_j(t),V_j(t),W_j(t))$ satisfies \begin{equation}\label{3.4} \begin{gathered} \begin{aligned} \frac{dU_j(t)}{dt} &=d_1[U_{j+1}(t)+U_{j-1}(t)-2U_j(t)]+U_j(t)[2r_1\varphi(\xi(t,j))-r_1 -r_1b_{12}V_j(t) \\ &\quad -r_1b_{12}\psi(\xi(t,j))]+r_1U_j^2(t) +r_1b_{12}(1-\varphi(\xi(t,j))) V_j(t), \end{aligned}\\ \begin{aligned} \frac{dV_j(t)}{dt} &=d_2[V_{j+1}(t)+V_{j-1}(t)-2V_j(t)]+V_j(t)[r_2-r_2b_{21}-r_2b_{23}- 2r_2\psi(\xi(t,j))\\ &\quad +r_2b_{21}U_j(t)+r_2b_{23}W_j(t)+r_2b_{21}\varphi(\xi(t,j)) +r_2b_{23}\theta(\xi(t,j))]\\ &\quad -r_2V_j^2(t)+[r_2b_{21}U_j(t)+r_2b_{23}W_j(t)]\psi(\xi(t,j)), \end{aligned}\\ \begin{aligned} \frac{dW_j(t)}{dt} &=d_3[W_{j+1}(t)+W_{j-1}(t)-2W_j(t)]+W_j(t)[2r_3\theta(\xi(t,j))-r_3 -r_3b_{32}V_j(t)\\ &\quad -r_3b_{32}\psi(\xi(t,j))]+r_3W_j^2(t)+r_3b_{32}(1-\theta(\xi(t,j))) V_j(t), \end{aligned} \end{gathered} \end{equation} with the initial data $U_j(0)=U_{j0}(0)$, $V_j(0)=V_{j0}(0)$, $W_j(0)=W_{j0}(0)$, $j\in\mathbb{Z}$. It follows from \eqref{3.1*} and \eqref{3.2*} that \begin{gather*} (0,0,0)\leq (U_j(t),V_j(t),W_j(t))\leq(1,1,1),\\ (0,0,0)\leq (U_{j0}(0),V_{j0}(0),V_{j0}(0))\leq(1,1,1). \end{gather*} We define \begin{gather}\label{3.1} B^{i}_{\mu, w}(t,j)=A^{i}_w(t,j)-2\mu, \quad i=1,2,3, \end{gather} where \begin{align*} A^{1}_w(t,j) &= 2\Big(2d_1-\frac{c}{2}\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))} -2r_1\varphi(\xi(t,j))+r_1+r_1b_{12}V_j(t) \\ &\quad +r_1b_{12}\psi(\xi(t,j))\Big) -2r_1U_j(t)-r_1b_{12}(1-\varphi(\xi(t,j))-r_2b_{21}\psi(\xi(t,j)) \\ &\quad -d_1\Big(2+\frac{w(\xi(t,j-1))}{w(\xi(t,j))} +\frac{w(\xi(t,j+1))}{w(\xi(t,j))}\Big), \end{align*} \begin{align*} A^2_w(t,j) &=2\Big(2d_2-\frac{c}{2}\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))}-r_2+r_2b_{21} +r_2b_{23} +2r_2\psi(\xi(t,j)) \\ &-r_2b_{21}U_j(t) -r_2b_{21}\varphi(\xi(t,j))-r_2b_{23}W_j(t)-r_2b_{23}\theta(\xi(t,j))\Big) +2r_2V_j(t) \\ &\quad -r_2b_{21}\psi(\xi(t,j))-r_2b_{23}\psi(\xi(t,j))-r_1b_{12}(1-\varphi(\xi(t,j)) \\ &\quad -r_3b_{32}(1-\theta(\xi(t,j)))-d_2\Big(2+\frac{w(\xi(t,j-1))}{w(\xi(t,j))} +\frac{w(\xi(t,j+1))}{w(\xi(t,j))}\Big), \end{align*} and \begin{align*} A^{3}_w(t,j) &= 2\Big(2d_3-\frac{c}{2}\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))}-2r_3\varphi(\xi(t,j)) +r_3 +r_3b_{32}V_j(t) \\ &\quad +r_3b_{32}\psi(\xi(t,j))\Big) -2r_3W_j(t)-r_3b_{32}(1-\theta(\xi(t,j))-r_2b_{23}\psi(\xi(t,j)) \\ &\quad -d_3\Big(2+\frac{w(\xi(t,j-1))}{w(\xi(t,j))} +\frac{w(\xi(t,j+1))}{w(\xi(t,j))}\Big). \end{align*} Clearly, $\xi(t,j+1)=\xi(t,j)+1$ and $\xi(t,j-1)=\xi(t,j)-1$. Now we establish some key inequalities. \begin{lemma}\label{301} Assume that {\rm (H2)} holds. For any $c>\max\{c_{min}, \tilde{c}\}$, there exist some positive constants $C_{i}$ such that \begin{gather*} A^{i}_w(t,j)\geq C_{i}, \quad i=1, 2,3, \end{gather*} for all $t> 0$ and $j\in\mathbb{Z}$. \end{lemma} \begin{proof} Since $c>\max\{c_{min},\tilde{c}\}$, we obtain $c\lambda_0>c_1$, $c\lambda_0>c_2$, and $c\lambda_0>c_3$, where $c_1$, $c_2$ and $c_3$ can be seen in \eqref{2.6}, \eqref{2.7} and \eqref{2.8}. That is, \begin{gather*} c\lambda_0-4r_1-r_1b_{12}-r_2b_{21}-d_1(e^{\lambda_0}+e^{-\lambda_0}+1)>0,\\ c\lambda_0-2r_2-3r_2(b_{21}+b_{23})-r_1b_{12}-r_3b_{32}-d_2(e^{\lambda_0} +e^{-\lambda_0}+1)>0, \\ c\lambda_0-4r_3-r_3b_{32}-r_2b_{23}-d_3(e^{\lambda_0}+e^{-\lambda_0}+1)>0. \end{gather*} Firstly, we prove that $A^{1}_w(t,j)\geq C_1$ for some positive constant $C_1$. \smallskip \noindent\textbf{Case 1: $\xi(t,j)<\xi_0-1$.} It is clear that $\xi(t,j)<\xi_0$, $\xi(t,j+1)<\xi_0$ and $\xi(t,j-1)<\xi_0$. Then $w(\xi(t,j))=e^{-\lambda_0(\xi(t,j)-\xi_0)}$, $w(\xi(t,j-1))=e^{-\lambda_0(\xi(t,j)-1-\xi_0)}$ and $w(\xi(t,j+1))=e^{-\lambda_0(\xi(t,j)+1-\xi_0)}$. Thus, we have \begin{align*} A^{1}_w(t,j) =&4d_1-c\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))}-4r_1\varphi(\xi(t,j)) +2r_1+2r_1b_{12}V_j(t)+2r_1b_{12}\psi(\xi(t,j))\\ &-2r_1U_j(t)-r_1b_{12}(1-\varphi(\xi(t,j)))-r_2b_{21}\psi(\xi(t,j))\\ &-d_1\Big(2+\frac{w(\xi(t,j-1))}{w(\xi(t,j))}+\frac{w(\xi(t,j+1))}{w(\xi(t,j))} \Big)\\ > &2d_1+c\lambda_0-4r_1-r_1b_{12}-r_2b_{21}-d_1(e^{\lambda_0}+e^{-\lambda_0})\\ = &c\lambda_0-4r_1-r_1b_{12}-r_2b_{21}-d_1(e^{\lambda_0}+e^{-\lambda_0}+1)+3d_1\\ > &3d_1>0. \end{align*} \smallskip \noindent\textbf{Case 2: $\xi_0-1 \leq \xi(t,j) \leq \xi_0$.} In this case, $\xi(t,j-1)<\xi_0$ and $\xi(t,j+1)\geq\xi_0$. Then $w(\xi(t,j))=e^{-\lambda_0(\xi(t,j)-\xi_0)}$, $w(\xi(t,j-1))=e^{-\lambda_0(\xi(t,j)-1-\xi_0)}$ and $w(\xi(t,j+1))=1$. Hence, we obtain \begin{align*} A^{1}_w(t,j)= &4d_1-c\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))}-4r_1\varphi(\xi(t,j))+2r_1 +2r_1b_{12}V_j(t)+2r_1b_{12}\psi(\xi(t,j))\\ &-2r_1U_j(t)-r_1b_{12}(1-\varphi(\xi(t,j))-r_2b_{21}\psi(\xi(t,j))\\ &-d_1\Big(2+\frac{w(\xi(t,j-1))}{w(\xi(t,j))}+\frac{w(\xi(t,j+1))}{w(\xi(t,j))} \Big)\\ > &2d_1+c\lambda_0-4r_1-r_1b_{12}-r_2b_{21}-d_1 \big(e^{\lambda_0}+e^{\lambda_0(\xi(t,j)-\xi_0)}\big)\\ \geq &c\lambda_0-4r_1-r_1b_{12}-r_2b_{21}-d_1(e^{\lambda_0} +1+e^{-\lambda_0})+d_1e^{-\lambda_0}+2d_1\\ > &d_1e^{-\lambda_0}+2d_1 > 0. \end{align*} \smallskip \noindent\textbf{Case 3: $ \xi_0 <\xi(t,j) \leq \xi_0+1$.} In this case, $\xi(t,j-1)\leq\xi_0$ and $\xi(t,j+1)>\xi_0$. Then $w(\xi(t,j-1))=e^{-\lambda_0(\xi(t,j)-1-\xi_0)}$ and $w(\xi(t,j))=w(\xi(t,j+1))=1$. Thus, one has \begin{align*} A^{1}_w(t,j)= &4d_1-c\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))}-4r_1\varphi(\xi(t,j))+2r_1 +2r_1b_{12}V_j(t)+2r_1b_{12}\psi(\xi(t,j))\\ &-2r_1U_j(t)-r_1b_{12}(1-\varphi(\xi(t,j))-r_2b_{21}\psi(\xi(t,j))\\ &-d_1\Big(2+\frac{w(\xi(t,j-1))}{w(\xi(t,j))}+\frac{w(\xi(t,j+1))}{w(\xi(t,j))}\Big) \\ > &2d_1-4r_1+2r_1b_{12}\psi(\xi_0)+r_1b_{12}\varphi(\xi_0)-r_1b_{12} -r_2b_{21}\\ &-d_1(e^{-\lambda_0(\xi(t,j)-1-\xi_0)}+1)\\ > &2d_1-4r_1+2r_1b_{12}\psi(\xi_0)+r_1b_{12}\varphi(\xi_0)-r_1b_{12} -r_2b_{21}-d_1(e^{\lambda_0}+1)\\ = &\mathcal{N}_1(\xi_0)>0. \end{align*} \smallskip \noindent\textbf{Case 4: $\xi(t,j) > \xi_0+1$.} In this case, $\xi(t,j)>\xi_0$, $\xi(t,j+1)>\xi_0$ and $\xi(t,j-1)>\xi_0$. Then $w(\xi(t,j))=w(\xi(t,j-1))=w(\xi(t,j+1))=1$. Hence, we obtain \begin{align*} A^{1}_w(t,j)= &4d_1-c\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))}-4r_1\varphi(\xi(t,j))+2r_1+2r_1b_{12}V_j(t)+2r_1b_{12}\psi(\xi(t,j))\\ &-2r_1U_j(t)-r_1b_{12}(1-\varphi(\xi(t,j))-r_2b_{21}\psi(\xi(t,j))\\ &-d_1\Big(2+\frac{w(\xi(t,j-1))}{w(\xi(t,j))}+\frac{w(\xi(t,j+1))}{w(\xi(t,j))}\Big)\\ > &-4r_1+2r_1b_{12}\psi(\xi_0)+r_1b_{12}\varphi(\xi_0)-r_1b_{12}-r_2b_{21}\\ = &\mathcal{N}_1(\xi_0)+d_1(e^{\lambda_0}+1)-2d_1\\ > &d_1(e^{\lambda_0}-1)>0. \end{align*} Therefore, we can obtain $A^{1}_w(t,j)\geq C_1>0$ by choosing a suitable $C_1$ small enough. Secondly, we show $A^2_w(t,j)\geq C_2$ for some positive constant $C_2$. \smallskip \noindent\textbf{Case 1: $\xi(t,j)<\xi_0-1$.} It is clear that $\xi(t,j)<\xi_0$, $\xi(t,j+1)<\xi_0$ and $\xi(t,j-1)<\xi_0$. Hence, $w(\xi(t,j))=e^{-\lambda_0(\xi(t,j)-\xi_0)}$, $w(\xi(t,j-1))=e^{-\lambda_0(\xi(t,j)-1-\xi_0)}$ and $w(\xi(t,j+1))=e^{-\lambda_0(\xi(t,j)+1-\xi_0)}$. Then one has \begin{align*} A^2_w(t,j) =&4d_2-c\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))}-2r_2+2r_2b_{21}+2r_2b_{23} +4r_2\psi(\xi(t,j))+2r_2V_j(t)\\ &-2r_2b_{21}U_j(t)-2r_2b_{21}\varphi(\xi(t,j))-2r_2b_{23}W_j(t) -2r_2b_{23}\theta(\xi(t,j))\\ &-r_2b_{21}\psi(\xi(t,j))-r_2b_{23}\psi(\xi(t,j)) -r_1b_{12}(1-\varphi(\xi(t,j)))\\ &-r_3b_{32}(1-\theta(\xi(t,j)))-d_2\Big(2+\frac{w(\xi(t,j-1))}{w(\xi(t,j))} +\frac{w(\xi(t,j+1))}{w(\xi(t,j))}\Big)\\ > &2d_2+c\lambda_0-2r_2-3r_2(b_{21}+b_{23})-r_1b_{12}-r_3b_{32} -d_2(e^{\lambda_0}+e^{-\lambda_0})\\ = &c\lambda_0-2r_2-3r_2(b_{21}+b_{23})-r_1b_{12}-r_3b_{32} -d_2(e^{\lambda_0}+e^{-\lambda_0}+1)+3d_2 \\ > &3d_2>0. \end{align*} \smallskip \noindent\textbf{Case 2: $\xi_0-1 \leq \xi(t,j) \leq \xi_0$.} In this case, $\xi(t,j-1)<\xi_0$ and $\xi(t,j+1)\geq\xi_0$. Then $w(\xi(t,j))=e^{-\lambda_0(\xi(t,j)-\xi_0)}$, $w(\xi(t,j-1))=e^{-\lambda_0(\xi(t,j)-1-\xi_0)}$ and $w(\xi(t,j+1))=1$. Hence, we obtain \begin{align*} A^2_w(t,j) =& 4d_2-c\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))}-2r_2+2r_2b_{21}+2r_2b_{23} +4r_2\psi(\xi(t,j))+2r_2V_j(t)\\ &-2r_2b_{21}U_j(t)-2r_2b_{21}\varphi(\xi(t,j))-2r_2b_{23}W_j(t) -2r_2b_{23}\theta(\xi(t,j))\\ &-r_2b_{21}\psi(\xi(t,j))-r_2b_{23}\psi(\xi(t,j))-r_1b_{12}(1-\varphi(\xi(t,j))\\ &-r_3b_{32}(1-\theta(\xi(t,j)))-d_2\Big(2+\frac{w(\xi(t,j-1))}{w(\xi(t,j))} +\frac{w(\xi(t,j+1))}{w(\xi(t,j))}\Big)\\ > &2d_2+c\lambda_0-2r_2-3r_2(b_{21}+b_{23})-r_1b_{12}-r_3b_{32}\\ &-d_2(e^{\lambda_0}+e^{\lambda_0(\xi(t,j)-\xi_0)})\\ \geq &c\lambda_0-2r_2-3r_2(b_{21}+b_{23})-r_1b_{12}-r_3b_{32} -d_2(e^{\lambda_0}+1+e^{-\lambda_0})\\ &+d_2e^{-\lambda_0}+2d_2\\ > &d_2e^{-\lambda_0}+2d_2>0. \end{align*} \smallskip \noindent\textbf{Case 3: $ \xi_0 <\xi(t,j) \leq \xi_0+1$.} In this case, $\xi(t,j-1)\leq\xi_0$ and $\xi(t,j+1)>\xi_0$. Then $w(\xi(t,j-1))=e^{-\lambda_0(\xi(t,j)-1-\xi_0)}$ and $w(\xi(t,j))=w(\xi(t,j+1))=1$. Thus, we obtain \begin{align*} A^2_w(t,j)= &4d_2-c\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))}-2r_2+2r_2b_{21}+2r_2b_{23} +4r_2\psi(\xi(t,j))+2r_2V_j(t)\\ &-2r_2b_{21}U_j(t)-2r_2b_{21}\varphi(\xi(t,j))-2r_2b_{23}W_j(t) -2r_2b_{23}\theta(\xi(t,j))\\ &-r_2b_{21}\psi(\xi(t,j))-r_2b_{23}\psi(\xi(t,j))-r_1b_{12}(1-\varphi(\xi(t,j))\\ &-r_3b_{32}(1-\theta(\xi(t,j)))-d_2\Big(2+\frac{w(\xi(t,j-1))}{w(\xi(t,j))} +\frac{w(\xi(t,j+1))}{w(\xi(t,j))}\Big)\\ > &2d_2-2r_2+4r_2\psi(\xi_0)-3r_2(b_{21}+b_{23})-r_1b_{12}-r_3b_{32} +r_1b_{12}\varphi(\xi_0)\\ &+r_3b_{32}\theta(\xi_0)-d_2(e^{-\lambda_0(\xi(t,j)-1-\xi_0)}+1)\\ > &2d_2-2r_2+4r_2\psi(\xi_0)-3r_2(b_{21}+b_{23})-r_1b_{12}-r_3b_{32} +r_1b_{12}\varphi(\xi_0)\\ &+r_3b_{32}\theta(\xi_0)-d_2(e^{\lambda_0}+1)\\ = &\mathcal{N}_2(\xi_0)>0. \end{align*} \smallskip \noindent\textbf{Case 4: $\xi(t,j) > \xi_0+1$.} In this case, $\xi(t,j)>\xi_0$, $\xi(t,j+1)>\xi_0$ and $\xi(t,j-1)>\xi_0$. Then $w(\xi(t,j))=w(\xi(t,j-1))=w(\xi(t,j+1))=1$. Hence, we have \begin{align*} A^2_w(t,j)= &4d_2-c\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))}-2r_2+2r_2b_{21}+2r_2b_{23} +4r_2\psi(\xi(t,j))+2r_2V_j(t)\\ &-2r_2b_{21}U_j(t)-2r_2b_{21}\varphi(\xi(t,j))-2r_2b_{23}W_j(t) -2r_2b_{23}\theta(\xi(t,j))\\ &-r_2b_{21}\psi(\xi(t,j))-r_2b_{23}\psi(\xi(t,j))-r_1b_{12}(1-\varphi(\xi(t,j))\\ &-r_3b_{32}(1-\theta(\xi(t,j)))-d_2\Big(2+\frac{w(\xi(t,j-1))}{w(\xi(t,j))} +\frac{w(\xi(t,j+1))}{w(\xi(t,j))}\Big)\\ > &-2r_2+4r_2\psi(\xi_0)-3r_2(b_{21}+b_{23})-r_1b_{12}-r_3b_{32} +r_1b_{12}\varphi(\xi_0)\\ &+r_3b_{32}\theta(\xi_0)\\ = &\mathcal{N}_2(\xi_0)+d_2(e^{\lambda_0}+1)-2d_2\\ > &d_2(e^{\lambda_0}-1)>0. \end{align*} Therefore, we obtain $A^2_w(t,j)\geq C_2>0$ by choosing a suitable $C_2$ small enough. Thirdly, we prove $A^{3}_w(t,j)\geq C_3$ for some positive constant $C_3$. \smallskip \noindent\textbf{Case 1: $\xi(t,j)<\xi_0-1$.} It is clear that $\xi(t,j)<\xi_0$, $\xi(t,j+1)<\xi_0$ and $\xi(t,j-1)<\xi_0$. Hence, $w(\xi(t,j))=e^{-\lambda_0(\xi(t,j)-\xi_0)}$, $w(\xi(t,j-1))=e^{-\lambda_0(\xi(t,j)-1-\xi_0)}$ and $w(\xi(t,j+1))=e^{-\lambda_0(\xi(t,j)+1-\xi_0)}$. Then one has \begin{align*} A^{3}_w(t,j)= &4d_3-c\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))}-4r_3\theta(\xi(t,j))+2r_3+2r_3b_{32}V_j(t)+2r_3b_{32}\psi(\xi(t,j))\\ &-2r_3W_j(t)-r_3b_{32}(1-\theta(\xi(t,j))-r_2b_{23}\psi(\xi(t,j))\\ &-d_3\Big(2+\frac{w(\xi(t,j-1))}{w(\xi(t,j))}+\frac{w(\xi(t,j+1))}{w(\xi(t,j))} \Big)\\ > &2d_3+c\lambda_0-4r_3-r_3b_{32}-r_2b_{23}-d_3(e^{\lambda_0}+e^{-\lambda_0})\\ = &c\lambda_0-4r_3-r_3b_{32}-r_2b_{23}-d_3(e^{\lambda_0}+e^{-\lambda_0}+1)+3d_3\\ > &3d_3>0. \end{align*} \textbf{Case 2: $\xi_0-1 \leq \xi(t,j) \leq \xi_0$.} In this case, $\xi(t,j-1)<\xi_0$ and $\xi(t,j+1)\geq\xi_0$. Then $w(\xi(t,j))=e^{-\lambda_0(\xi(t,j)-\xi_0)}$, $w(\xi(t,j-1))=e^{-\lambda_0(\xi(t,j)-1-\xi_0)}$ and $w(\xi(t,j+1))=1$. Hence, we obtain \begin{align*} A^{3}_w(t,j)= &4d_3-c\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))}-4r_3\theta(\xi(t,j)) +2r_3+2r_3b_{32}V_j(t)+2r_3b_{32}\psi(\xi(t,j))\\ &-2r_3W_j(t)-r_3b_{32}(1-\theta(\xi(t,j))-r_2b_{23}\psi(\xi(t,j))\\ &-d_3\Big(2+\frac{w(\xi(t,j-1))}{w(\xi(t,j))} +\frac{w(\xi(t,j+1))}{w(\xi(t,j))}\Big)\\ > &2d_3+c\lambda_0-4r_3-r_3b_{32}-r_2b_{23} -d_3\big(e^{\lambda_0}+e^{\lambda_0(\xi(t,j)-\xi_0)}\big)\\ \geq &c\lambda_0-4r_3-r_3b_{32}-r_2b_{23}-d_3(e^{\lambda_0}+1+e^{-\lambda_0}) +d_3e^{-\lambda_0}+2d_3\\ > &d_3e^{-\lambda_0}+2d_3 > 0. \end{align*} \smallskip \noindent\textbf{Case 3: $ \xi_0 <\xi(t,j) \leq \xi_0+1$.} In this case, $\xi(t,j-1)\leq\xi_0$ and $\xi(t,j+1)>\xi_0$. Then $w(\xi(t,j-1))=e^{-\lambda_0(\xi(t,j)-1-\xi_0)}$ and $w(\xi(t,j))=w(\xi(t,j+1))=1$. Thus, we have \begin{align*} A^{3}_w(t,j)= &4d_3-c\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))}-4r_3\theta(\xi(t,j)) +2r_3+2r_3b_{32}V_j(t)+2r_3b_{32}\psi(\xi(t,j))\\ &-2r_3W_j(t)-r_3b_{32}(1-\theta(\xi(t,j))-r_2b_{23}\psi(\xi(t,j))\\ &-d_3\Big(2+\frac{w(\xi(t,j-1))}{w(\xi(t,j))} +\frac{w(\xi(t,j+1))}{w(\xi(t,j))}\Big)\\ > &2d_3-4r_3+2r_3b_{32}\psi(\xi_0)+r_3b_{32}\theta(\xi_0)-r_3b_{32} -r_2b_{23}\\ &-d_3(e^{-\lambda_0(\xi(t,j)-1-\xi_0)}+1)\\ > &2d_3-4r_3+2r_3b_{32}\psi(\xi_0)+r_3b_{32}\theta(\xi_0)-r_3b_{32} -r_2b_{23}-d_3(e^{\lambda_0}+1)\\ = &\mathcal{N}_3(\xi_0)>0. \end{align*} \smallskip \noindent\textbf{Case 4: $\xi(t,j) > \xi_0+1$.} In this case, $\xi(t,j)>\xi_0$, $\xi(t,j+1)>\xi_0$ and $\xi(t,j-1)>\xi_0$. Then $w(\xi(t,j))=w(\xi(t,j-1))=w(\xi(t,j+1))=1$. Hence, we have \begin{align*} A^{3}_w(t,j)= &4d_3-c\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))}-4r_3\theta(\xi(t,j)) +2r_3+2r_3b_{32}V_j(t)+2r_3b_{32}\psi(\xi(t,j))\\ &-2r_3W_j(t)-r_3b_{32}(1-\theta(\xi(t,j))-r_2b_{23}\psi(\xi(t,j))\\ &-d_3\Big(2+\frac{w(\xi(t,j-1))}{w(\xi(t,j))}+\frac{w(\xi(t,j+1))}{w(\xi(t,j))} \Big)\\ > &-4r_3+2r_3b_{32}\psi(\xi_0)+r_3b_{32}\theta(\xi_0)-r_3b_{32}-r_2b_{23}\\ = &\mathcal{N}_3(\xi_0)+d_3(e^{\lambda_0}+1)-2d_3\\ > &d_3(e^{\lambda_0}-1)>0. \end{align*} Therefore, we show $A^{3}_w(t,j)\geq C_3>0$ by choosing a suitable $C_3$ small enough. The proof is complete. \end{proof} \begin{lemma}\label{302} Assume that {\rm (H2)} holds. For any $c>\max\{c_{min}, \tilde{c}\}$, there exist some positive constants $C_{i}$ such that \begin{gather*} B^{i}_{\mu,w}(t,j)\geq C_{i}, \quad i=1, 2,3, \end{gather*} for all $t> 0$, $j\in\mathbb{Z}$ and $0<\mu<\frac{\min_{i=1, 2, 3}\{C_{i}\}}{2}$. \end{lemma} The proof of the above lemma can be easily obtained by Lemma \ref{301}, so we omit here. Next, we will give the energy estimates. \begin{lemma}\label{303} Assume that {\rm (H2)} hold. For any $c>\max\{c_{min}, \tilde{c}\}$, it holds \begin{equation} \label{3.5} \begin{aligned} & \|U_j(t)\|^2_{l^2_w}+\|V_j(t)\|^2_{l^2_w} +\|W_j(t)\|^2_{l^2_w} \\ &+\int^{t}_0e^{-2\mu(t-s)}\Big(\|U_j(s)\|^2_{l^2_w} +\|V_j(s)\|^2_{l^2_w}+\|W_j(t)\|^2_{l^2_w}\Big)ds \\ &\leq Ce^{-2\mu t}\Big(\|U_{j0}(0)\|^2_{l^2_w}+\|V_{j0}(0)\|^2_{l^2_w} +\|W_{j0}(0)\|^2_{l^2_w}\Big) \end{aligned} \end{equation} for some positive constant $C$. \end{lemma} \begin{proof} Multiplying the equations in \eqref{3.4} by $e^{2\mu t}w(\xi(t,j))U_j(t)$, $e^{2\mu t}w(\xi(t,j))V_j(t)$ and $e^{2\mu t}w(\xi(t,j))W_j(t)$ respectively, where $\mu>0$ is defined in Lemma \ref{302}, we obtain \begin{gather} \label{3.6} \begin{aligned} &\Big(\frac{1}{2}e^{2\mu t}w(\xi(t,j))U_j^2(t)\Big)_{t}-d_1e^{2\mu t}w(\xi(t,j))U_j(t)(U_{j+1}(t)+U_{j-1}(t)) \\ &+\Big( 2d_1-\frac{c}{2}\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))} -\mu-2r_1\varphi(\xi(t,j))+r_1+r_1b_{12}V_j(t)\\ &+r_1b_{12}\psi(\xi(t,j))\Big) e^{2\mu t}w(\xi(t,j))U_j^2(t) \\ &=r_1e^{2\mu t}w(\xi(t,j))U_j^{3}(t)+r_1b_{12}(1-\varphi(\xi(t,j))) e^{2\mu t}w(\xi(t,j))U_j(t)V_j(t), \end{aligned} \\ \label{3.7} \begin{aligned} &\Big(\frac{1}{2}e^{2\mu t}w(\xi(t,j))V_j^2(t)\Big)_{t} -d_2e^{2\mu t}w(\xi(t,j))V_j(t)(V_{j+1}(t)+V_{j-1}(t)) \\ &+\Big(2d_2-\frac{c}{2}\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))}-\mu-r_2+r_2b_{21} +r_2b_{23}+2r_2\psi(\xi(t,j))-r_2b_{21}U_j(t) \\ &-r_2b_{23}W_j(t)-r_2b_{21}\varphi(\xi(t,j))-r_2b_{23}\theta(\xi(t,j))\Big) e^{2\mu t}w(\xi(t,j))V_j^2(t) \\ &= -r_2e^{2\mu t}w(\xi(t,j))V_j^{3}(t)+r_2b_{21}\psi(\xi(t,j)) e^{2\mu t}w(\xi(t,j))U_j(t)V_j(t) \\ &\quad +r_2b_{23}\psi(\xi(t,j))e^{2\mu t}w(\xi(t,j))W_j(t)V_j(t), \end{aligned} \\ \label{3.8} \begin{aligned} &\Big(\frac{1}{2}e^{2\mu t}w(\xi(t,j))W_j^2(t)\Big)_{t}-d_3e^{2\mu t} w(\xi(t,j))W_j(t)(W_{j+1}(t)+W_{j-1}(t)) \\ &+\Big(2d_3-\frac{c}{2}\frac{w'_\xi(\xi(t,j))}{w(\xi(t,j))} -\mu-2r_3\theta(\xi(t,j))+r_3+r_3b_{32}V_j(t)\\ &+r_3b_{32}\psi(\xi(t,j))\Big) e^{2\mu t}w(\xi(t,j))W_j^2(t) \\ &= r_3e^{2\mu t}w(\xi(t,j))W_j^{3}(t)+r_3b_{32}(1-\theta(\xi(t,j))) e^{2\mu t}w(\xi(t,j))W_j(t)V_j(t). \end{aligned} \end{gather} By the Cauchy-Schwarz inequality $2ab\leq a^2+b^2$, we obtain \begin{gather*} 2U_{j+1}(t)U_j(t)\leq U_{j+1}^2(t)+U_j^2(t),\\ 2V_{j+1}(t)V_j(t)\leq V_{j+1}^2(t)+V_j^2(t),\\ 2W_{j+1}(t)W_j(t)\leq W_{j+1}^2(t)+W_j^2(t). \end{gather*} By summing over all $j\in\mathbb{Z}$ for \eqref{3.6}, \eqref{3.7} and \eqref{3.8}, then integrating over $[0,t]$, one has \begin{equation} \label{3.9} \begin{aligned} &e^{2\mu t}\|U_j(t)\|^2_{l^2_w} +\int^{t}_0\sum_j\Big[2\Big( 2d_1-\frac{c}{2}\frac{w'_\xi(\xi(s,j))}{w(\xi(s,j))} -\mu-2r_1\varphi(\xi(s,j)) \\ &+r_1+r_1b_{12}V_j(s) +r_1b_{12}\psi(\xi(s,j))\Big) -d_1\frac{w(\xi(s,j+1))}{w(\xi(s,j))} \\ &-d_1\frac{w(\xi(s,j-1))}{w(\xi(s,j))}-2d_1\Big] e^{2\mu s}w(\xi(s,j))U_j^2(s)ds \\ &\leq \|U_{j0}(0)\|^2_{l^2_w}+2r_1\int^{t}_0 \sum_je^{2\mu s}w(\xi(s,j))U_j(s)U_j^2(s)ds \\ &\quad +\int^{t}_0\sum_jr_1b_{12}(1-\varphi(\xi(s,j))) e^{2\mu s} w(\xi(s,j))(U_j^2(s)+V_j^2(s))ds, \end{aligned} \end{equation} \begin{align} &e^{2\mu t}\|V_j(t)\|^2_{l^2_w} +\int^{t}_0\sum_j\Big[2\Big( 2d_2-\frac{c}{2}\frac{w'_\xi(\xi(s,j))}{w(\xi(s,j))} -\mu-r_2+r_2b_{21} +r_2b_{23} \nonumber \\ &+2r_2\psi(\xi(s,j)) -r_2b_{21}U_j(s)-r_2b_{23}W_j(s)-r_2b_{21}\varphi(\xi(s,j)) -r_2b_{23}\theta(\xi(s,j))\Big) \nonumber\\ &-d_2\frac{w(\xi(s,j+1))}{w(\xi(s,j))}-d_2\frac{w(\xi(s,j-1))}{w(\xi(s,j))} -2d_2\Big] e^{2\mu s}w(\xi(s,j))V_j^2(s)ds \nonumber\\ &\leq \|V_{j0}(0)\|^2_{l^2_w}-2r_2\int^{t}_0\sum_je^{2\mu s}w(\xi(s,j)) V_j(s)V_j^2(s)ds \nonumber \\ &+\int^{t}_0\sum_jr_2b_{21}\psi(\xi(s,j)))e^{2\mu s}w(\xi(s,j))(U_j^2(s) +V_j^2(s))ds \nonumber \\ &+\int^{t}_0\sum_jr_2b_{23}\psi(\xi(s,j)))e^{2\mu s}w(\xi(s,j))(W_j^2(s)+V_j^2(s))ds, \label{3.10} \end{align} \begin{equation}\label{3.11} \begin{aligned} &e^{2\mu t}\|W_j(t)\|^2_{l^2_w} +\int^{t}_0\sum_j\Big[2\Big( 2d_3-\frac{c}{2}\frac{w'_\xi(\xi(s,j))}{w(\xi(s,j))} -\mu-2r_3\theta(\xi(s,j)) \\ &+r_3+r_3b_{32}V_j(s) \Big) +r_3b_{32}\psi(\xi(s,j)) -d_3\frac{w(\xi(s,j+1))}{w(\xi(s,j))} \\ &-d_3\frac{w(\xi(s,j-1))}{w(\xi(s,j))}-2d_3\Big] e^{2\mu s}w(\xi(s,j))W_j^2(s)ds \\ &\leq \|W_{j0}(0)\|^2_{l^2_w}+2r_3\int^{t}_0\sum_je^{2\mu s}w(\xi(s,j)) W_j(s)W_j^2(s)ds \\ &\quad +\int^{t}_0\sum_jr_3b_{32}(1-\theta(\xi(s,j))) e^{2\mu s} w(\xi(s,j))(W_j^2(s)+V_j^2(s))ds. \end{aligned} \end{equation} Adding the inequalities \eqref{3.9}, \eqref{3.10}, \eqref{3.11}, we have \begin{align*} &e^{2\mu t}\Big( \|U_j(t)\|^2_{l^2_w}+\|V_j(t)\|^2_{l^2_w} +\|W_j(t)\|^2_{l^2_w}\Big) +\int^{t}_0\sum_je^{2\mu s}\Big(B^{1}_{\mu,w}(s,j)U_j^2(s) \\ &+B^2_{\mu,w}(s,j)V_j^2(s)+B^{3}_{\mu,w}(s,j)W_j^2(s)\Big)w(\xi(s,j))ds \\ &\leq \|U_{j0}(0)\|^2_{l^2_w}+\|V_{j0}(0)\|^2_{l^2_w} +|W_{j0}(0)\|^2_{l^2_w}, \end{align*} where $B^{1}_{\mu,w}(t,j)$ , $B^2_{\mu,w}(t,j)$ and $B^2_{\mu,w}(t,j)$ are defined in \eqref{3.1}. By Lemma \ref{302}, we obtain \eqref{3.5}, i.e., \begin{equation} \label{3.12} \begin{aligned} &\|U_j(t)\|^2_{l^2_w}+\|V_j(t)\|^2_{l^2_w}+\|W_j(t)\|^2_{l^2_w} \\ & +\int^{t}_0e^{-2\mu(t-s)}\Big( \|U_j(s)\|^2_{l^2_w}+\|V_j(s)\|^2_{l^2_w} +\|W_j(s)\|^2_{l^2_w}\Big)ds \\ &\leq Ce^{-2\mu t}\Big(\|U_{j0}(0)\|^2_{l^2_w}+\|V_{j0}(0)\|^2_{l^2_w} +\|W_{j0}(0)\|^2_{l^2_w}\Big) \end{aligned} \end{equation} for some positive constant $C$. The proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{202}] Since $w(\xi)\geq 1$ defined by \eqref{2.5}, we obtain $\|\cdot\|_{l^2}\leq \|\cdot\|_{l^2_w}$. Furthermore, by the Sobolev's embedding inequality $l^2\hookrightarrow l^{\infty}$, one has \begin{gather*} \sup_{j\in\mathbb{Z}}|U_j(t)|\leq C\|U_j(t)\|_{l^2}\leq C\|U_j(t)\|_{l^2_w},\\ \sup_{j\in\mathbb{Z}}|V_j(t)|\leq C\|V_j(t)\|_{l^2}\leq C\|V_j(t)\|_{l^2_w},\\ \sup_{j\in\mathbb{Z}}|W_j(t)|\leq C\|W_j(t)\|_{l^2}\leq C\|W_j(t)\|_{l^2_w}. \end{gather*} Then by \eqref{3.12}, we obtain \begin{gather*} \sup_{j\in\mathbb{Z}}|u^{+}_j(t)-\varphi(j+ct)| =\sup_{j\in\mathbb{Z}}|U_j(t)|\leq Ce^{-\mu t},\\ \sup_{j\in\mathbb{Z}}|v^{+}_j(t)-\psi(j+ct)| =\sup_{j\in\mathbb{Z}}|V_j(t)|\leq Ce^{-\mu t},\\ \sup_{j\in\mathbb{Z}}|w^{+}_j(t)-\theta(j+ct)| =\sup_{j\in\mathbb{Z}}|W_j(t)|\leq Ce^{-\mu t}, \end{gather*} where $t>0$. Similarly, we can obtain \begin{gather*} \sup_{j\in\mathbb{Z}}|u^{-}_j(t)-\varphi(j+ct)| =\sup_{j\in\mathbb{Z}}|U_j(t)|\leq Ce^{-\mu t},\\ \sup_{j\in\mathbb{Z}}|v^{-}_j(t)-\psi(j+ct)| =\sup_{j\in\mathbb{Z}}|V_j(t)|\leq Ce^{-\mu t},\\ \sup_{j\in\mathbb{Z}}|w^{-}_j(t)-\theta(j+ct)| =\sup_{j\in\mathbb{Z}}|W_j(t)|\leq Ce^{-\mu t}. \end{gather*} Then the squeezing technique yields \begin{gather*} \sup_{j\in\mathbb{Z}}|u_j(t)-\varphi(j+ct)|\leq Ce^{-\mu t}, \quad t>0,\\ \sup_{j\in\mathbb{Z}}|v_j(t)-\psi(j+ct)|\leq Ce^{-\mu t}, \quad t>0,\\ \sup_{j\in\mathbb{Z}}|w_j(t)-\theta(j+ct)|\leq Ce^{-\mu t}, \quad t>0. \end{gather*} This completes the proof. \end{proof} \subsection*{Acknowledgments} G.-B. Zhang was supported by NSF of China (11401478). \begin{thebibliography}{00} \bibitem{Chen} X. Chen; \emph{Existence, uniqueness and asymptotic stability of traveling waves in non-local evolution equations, Adv. Differential Equations}, Adv. Differential Equations, 2 (1997), 125-160. \bibitem{CLW} C. Cheng, W.-T. Li, Z.-C. Wang; \emph{Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two dimensional spatial lattice}, Discrete Contin. Dyn. Syst. Ser. B., 13 (2010), 559-575. \bibitem{GW2} J.-S. Guo, C.-H. Wu; \emph{Traveling wave front for a two-component lattice dynamical system arising in competition models}, J. Differential Equations, 252 (2012), 4357-4391. \bibitem{GWWW}J.-S. Guo, Y. Wang, C.-H. Wu, C.-C. Wu; \emph{The minimal speed of traveling wave solutions for a diffusive three species competition system}, Taiwan. J. Math., 19 (2015), 1805-1829. \bibitem{LLLM} C.-K. Lin, C.-T. Lin, Y.-P. Lin, M. Mei; \emph{Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation}, SIAM J. Math. Anal., 46 (2014), 1053-1084. \bibitem{LW1} G. Lv, X. Wang; \emph{Stability of traveling wave fronts for nonlocal delayed reaction diffusion systems}, Z. Anal. Anwend. J. Anal. Appl., 33 (2014), 463-480. \bibitem{MD} S. Ma, Y.-R. Duan; \emph{Asymptotic stability of traveling waves in a discrete convolution model for phase transitions}, J. Math. Anal. Appl., 308 (2005), 240-256. \bibitem{MZ} S. Ma, X.-Q. Zhao; \emph{Global asymptotic stability of minimal fronts in monostable lattice equations}, Discrete Contin. Dyn. Syst., 21 (2008), 259-275. \bibitem{MZ1} S. Ma, X. Zou; \emph{Existence, uniqueness and stability of traveling waves in a discrete reaction-diffusion monostable equation with delay}, J. Differentil Equations, 217 (2005), 54-87. \bibitem{MLLS1} M. Mei, C.-K. Lin, C.-T. Lin, J.-W.-H. So; \emph{Traveling wavefronts for time-delayed reaction-diffusion equation: (II) nonlocal nonlinearity}, J. Differential Equations, 247 (2009), 511-529. \bibitem{MOZ} M. Mei, C. Ou, X.-Q. Zhao; \emph{Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations}, SIAM J. Math. Anal., 42 (6) (2010), 2762-2790. \bibitem{SZ} H.-L. Smith, X.-Q. Zhao; \emph{Global asymptotic stability of traveling waves in delayed recation-diffusion equations}, SIAM J. Math. Anal., 31 (2000), 514-534. \bibitem{TZ} G. Tian, G.-B. Zhang; \emph{Stabilty of traveling wavefronts for a discrete diffusive Lotka-Volterra competition system}, J. Math. Anal. Appl., 447 (2017), 222-242. \bibitem{Wu0} C.-C. Wu; \emph{ Monotonicity and uniqueness of wave profiles for a three components lattice dynamical system}, Discrete Contin. Dyn. Syst., 37 (2017), 2813-2827. \bibitem{Wu} C.-H. Wu; \emph{A general approach to the asymptotic behavior of traveling waves in a class of three-component lattice dynamical systems}, J. Dynam. Differential Equations, 28 (2016), 317-338. \bibitem{YLW1} Y.-R. Yang, W.-T. Li, S.-L. Wu; \emph{Exponential stability of traveling fronts in a diffusion epidemic system with delay}, Nonlinear Anal. Real World Appl., 12 (2011), 1223-1234. \bibitem{YLW2} Y.-R. Yang, W.-T. Li , S.-L. Wu; \emph{ Stability of traveling waves in a monostable delayed system without quasi-monotonicity}, Nonlinear Anal. Real World Appl., 3 (2013), 1511-1526. \bibitem{YY} Z.-X. Yu, R. Yuan; \emph{Nonlinear stability of wavefronts for a delayed stage-structured population model on 2-D lattice}, Osaka J. Math., 50 (4) (2013), 963-976. \bibitem{YM} Z.-X. Yu, M. Mei; \emph{Uniqueness and stability of traveling waves for cellular neural networks with multiple delays}, J. Differential Equations, 260 (2016), 241-267. \bibitem{YXZ} Z. Yu, F. Xu, W. G. Zhang; \emph{Stability of invasion traveling waves for a competition system with nonlocal dispersals}, Appl. Anal., 96 (2017), 1107-1125. \bibitem{Z} G.-B. Zhang; \emph{Global stability of traveling wave fronts for non-local delayed lattice differential equations}, Nonlinear Anal. Real World Appl., 13 (2012), 1790-1801. \bibitem{ZL}G.-B. Zhang, W.-T. Li; \emph{Nonlinear stability of traveling wavefronts in an age-structured population model with nonlocal dispersal and delay}, Z. Angew. Math. Phys., 64 (2013), 1643-1659. \bibitem{ZM} G.-B. Zhang, R. Ma; \emph{Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution type crossing-monostable nonlinearity}, Z. Angew. Math. Phys., 65 (2014), 819-844. \end{thebibliography} \end{document}