1 Introduction

In this paper, we consider the following two-dimensional (2D) lattice differential system:

$$ \textstyle\begin{cases} \frac{\mathrm{d}u_{i,j}(t)}{\mathrm{d}t}=(\Delta _{2}u)_{i,j}+f(u_{i,j}(t))- \gamma _{1}u_{i,j}(t)+\gamma _{2}v_{i,j}(t), \\ \frac{\mathrm{d}v_{i,j}(t)}{\mathrm{d}t}=\gamma _{1}u_{i,j}(t)-\gamma _{2}v_{i,j}(t), \end{cases} $$
(1.1)

where \(u_{i,j}(t)\in \mathbb{R}\), \(v_{i,j}(t)\in \mathbb{R}\), \(i,j\in \mathbb{Z}\), \(t>0\), and

$$ (\Delta _{2}u)_{i,j}:= d[u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1}-4u _{i,j}]. $$

The system models a species with mobile and stationary subpopulations in an environment in which the habitat is two-dimensional and divided into countable niches (see [4]). Such behavior is typical for invertebrates living in small ponds in arid climates, which dry up and reappear subject to rainfall (Hadeler and Lewis [4]). Here \(u_{i,j}(t)\) and \(v_{i,j}(t)\) represent the densities of mobile and stationary subpopulations at a point \((i,j)\) and time t, respectively, \(d>0\) is the diffusion coefficient of the mobile subpopulation, and f is the reproduction function. The rates of transfer between two states are assumed to be \(\gamma _{1}>0\) and \(\gamma _{2}>0\).

In fact, system (1.1) is a spatially discrete version of the following continuous model with a quiescent stage (see Hadeler and Lewis [4]):

$$ \textstyle\begin{cases} \partial _{t}u(x,t)=d \Delta u(x,t)+ f(u(x,t))- \gamma _{1} u(x,t)+ \gamma _{2} v(x,t), \\ \partial _{t}v(x,t)= \gamma _{1} u(x,t)-\gamma _{2} v(x,t). \end{cases} $$
(1.2)

However, in some cases, a lattice differential model is more suitable than the continuous one. For example, the ponds, the living environment of frogs, which are scattered in the grassland, cannot be regarded as a continuous environment. Actually, lattice dynamical systems have been extensively used to model biological and epidemic problems, and some new phenomenon has been observed in discrete models, such as occurrence of “propagation failure” or “pinning”; we refer to the survey paper [3] and book [14].

The traveling wave and related problems of systems (1.1) and (1.2) have been widely studied. Zhang and Zhao [20] established the existence of spreading speed of (1.2) and showed that it coincides with the minimal wave speed for traveling wavefronts. Zhang and Li [21] further considered the monotonicity and uniqueness of the traveling wave solutions of (1.2). Recently, Zhao and Wu [22] studied the existence, asymptotic behavior, monotonicity, and uniqueness of traveling wavefronts of the 2D discrete model (1.1). The effects of the direction of propagation on the minimal wave speed \(c_{*}(\theta )\) is also investigated. We found that: (i) \(c_{*}(\theta )\) is a periodic function with period \(\frac{\pi }{2}\) and has the symmetry in \(\theta = \frac{\pi }{4}\); (ii) the 1D front (i.e., \(\theta =0,\frac{\pi }{2}\)) is the fastest, and the diagonal front (i.e., \(\theta = \frac{\pi }{4}\)) is the slowest; (iii) \(c_{*}(\theta )\) is monotonically decreasing and increasing in \(\theta \in [0,\frac{\pi }{4}]\) and \(\theta \in [\frac{ \pi }{4},\frac{\pi }{2}]\), respectively. However, the stability of the traveling wavefronts for systems (1.1) and (1.2) remains unsolved. This constitutes the purpose of this paper.

In the past decades, there are many techniques developed to deal with the stability of the traveling waves (see [1, 2, 5, 6, 9,10,11,12,13, 16, 17, 19]), such as the spectral analysis method ([1, 19]), the weighted-energy method ([12]), the sub- and supersolutions combining squeezing technique ([2, 17]), and the combination of the comparison principle and the weighted-energy method ([9, 11]). More recently, Ouyang and Ou [13] used a new method to prove the asymptotic stability of traveling waves of a nonlocal reaction–diffusion equation in periodic media. In this paper, we extend this method to systems (1.1) and (1.2). We focus on the stability of the traveling wavefronts for 2D discrete system (1.1), since the same issue for the continuous model (1.2) can be treated similarly. More precisely, by establishing a comparison theorem for a related initial-boundary value problem (see Lemma 3.1) and giving a variational characterization of the minimal wave speed (see Lemma 2.4) we show that any traveling wavefront of (1.1) with given direction of propagation \(\theta \in [0,2 \pi ]\) and wave speed \(c>c_{*}(\theta )\) is exponentially stable in time (see Theorem 3.2). In particular, we obtain the exponential convergence rate.

Throughout this paper, we make the following assumptions on the function f:

\(( A_{1})\) :

\(f\in C^{2}[0,1]\), \(f(0)=f(1)=0\), \(f'(0)>0\), \(f'(1)<0\), and \(f(u)>0\) for all \(u\in (0,1)\).

\(( A_{2})\) :

\(f'(u)\leq f'(0)\) for all \(u\in [0,1]\).

From \((A_{1})\) we see that (1.1) has two equilibria \(\mathbf{0}:=(0,0)\) and \(\mathbf{K}:=(1,K)\), where \(K:=\gamma _{1}/ \gamma _{2}\). We are interested in traveling wave solutions that connect the two equilibria 0 and K. A solution \(w_{i,j}(t)=(u_{i,j}(t),v_{i,j}(t))\) of (1.1) is called a traveling wave solution of (1.1) with speed c and direction θ if there exist constants \(\theta \in \mathbb{R}\) and \(c>0\) and a differentiable function \(W=(U, V ):\mathbb{R}\rightarrow \mathbb{R} ^{2}\) such that

$$ \bigl(u_{i,j}(t),v_{i,j}(t) \bigr)= \bigl(U(i\cos \theta +j\sin \theta + ct), V (i\cos \theta +j\sin \theta + ct) \bigr),\quad \forall i,j \in \mathbb{Z}, t\in \mathbb{R}. $$

Moreover, if \(W:\mathbb{R}\rightarrow \mathbb{R}^{2}\) is monotone, then we say that W is a traveling wavefront.

The rest of this paper is organized as follows. In Sect. 2, we first recall a result on the existence of the minimal wave speed \(c_{*}( \theta )\). Then we give a variational characterization of \(c_{*}( \theta )\). Section 3 is devoted to the stability of noncritical traveling wavefronts.

2 Characterization of minimal wave speed

In this section, we first recall some results on the existence of the minimal wave speed \(c_{*}(\theta )\). Then we give a variational characterization of \(c_{*}(\theta )\), which will be used in proving our stability result.

For \(c\geq 0\) and \(\lambda \in \mathbb{C}\) with \(\lambda \neq -\frac{ \gamma _{2}}{c}\), we define the characteristic function as follows:

$$\begin{aligned} \triangle (c,\lambda )=c\lambda - d\bigl[e^{\lambda \cos \theta }+e^{- \lambda \cos \theta }+e^{\lambda \sin \theta } +e^{-\lambda \sin \theta }-4 \bigr]-f'(0)+\gamma _{1}- \frac{\gamma _{1}\gamma _{2}}{c\lambda +\gamma _{2}}. \end{aligned}$$

We have the following results on the existence of the minimal wave speed.

Lemma 2.1

([22, Lemma 2.5])

For any fixed \(\theta \in \mathbb{R}\), there exists a positive number \(c_{*}(\theta )\) such that

  1. (i)

    if \(c\geq c_{*}(\theta )\), then \(\triangle (c, \lambda )=0\) has two roots \(\lambda _{1}(c)\), \(\lambda _{2}(c)\) with \(0<\lambda _{1}(c)\leq \lambda _{2}(c)\);

  2. (ii)

    if \(c=c_{*}(\theta )\), then \(\lambda _{1}(c_{*})= \lambda _{2}(c_{*}):=\lambda _{*}(\theta )\), and if \(c>c_{*}(\theta )\), then \(\lambda _{1}(c)<\lambda _{*}(\theta )<\lambda _{2}(c)\), and \(\triangle (c,\cdot )<0 \) in \(\mathbb{R}\setminus (\lambda _{1}(c), \lambda _{2}(c) )\), \(\triangle (c,\cdot )>0\) in \((\lambda _{1}(c),\lambda _{2}(c) )\).

Lemma 2.2

([22, Theorem 2.7])

Let \(\theta \in \mathbb{R}\). Then:

  1. (i)

    for each \(c\geq c_{*}(\theta )\), (1.1) has a traveling wavefront \(W(\xi ) \), \(\xi =i \cos \theta +j\sin \theta +ct\), with direction of propagation θ and wave speed c;

  2. (ii)

    for any \(0< c< c_{*}(\theta )\), (1.1) has no traveling wave solutions.

Now, we give a variational characterization of \(c_{*}(\theta )\). Substituting

$$\bigl(u_{i,j}(t),v_{i,j}(t)\bigr)=\mathrm{e}^{\lambda (i \cos \theta +j\sin \theta )}{ \bigl(u(t),v(t)\bigr)},\quad \lambda \in \mathbb{R}, $$

into the linearized system of (1.1) at \((0,0)\), we obtain

$$ \textstyle\begin{cases} u'(t)=d[e^{\lambda \cos \theta }+e^{-\lambda \cos \theta }+e^{\lambda \sin \theta } +e^{-\lambda \sin \theta }-4 ]u(t)+[f'(0) -\gamma _{1}]u(t)+\gamma _{2}v(t), \\ v'(t)=\gamma _{1}u(t)-\gamma _{2}v(t). \end{cases} $$

Clearly, the eigenvalue problem of this system has the following from:

$$ \textstyle\begin{cases} M(\lambda )\nu _{1}=d[e^{\lambda \cos \theta }+e^{-\lambda \cos \theta }+e^{\lambda \sin \theta } +e^{-\lambda \sin \theta }-4 ]\nu _{1} +[f'(0) -\gamma _{1}]\nu _{1}+\gamma _{2}\nu _{2}, \\ M(\lambda )\nu _{2}=\gamma _{1}\nu _{1}-\gamma _{2}\nu _{2}. \end{cases} $$
(2.1)

From [15, Theorem 5.5.1], we see that problem (2.1) has a principal eigenvalue \(M(\lambda )\) with a strictly positive eigenfunction \(\nu (\lambda )=(\nu _{1}(\lambda ), \nu _{2}(\lambda ))\).

Lemma 2.3

Let \(\varPhi (\lambda )=M(\lambda )/\lambda \). Then:

  1. (i)

    \(\lim_{\lambda \rightarrow 0+}\varPhi (\lambda ) \rightarrow \infty \) and \(\lim_{\lambda \rightarrow \infty }\varPhi ( \lambda )=\infty \).

  2. (ii)

    \(\varPhi (\lambda ) \) is decreasing near 0, and \(\varPhi '(\lambda )\) changes the sign at most once on \((0,\infty )\).

  3. (iii)

    There exists \(\lambda ^{*}=\lambda ^{*}(\theta )\in (0,\infty )\) such that \(c^{*}(\theta ):=\varPhi (\lambda ^{*})= \inf_{\lambda >0} \varPhi (\lambda )\).

Proof

It is clear that \(M(\lambda )\) is the spectral radius of a matrix and

$$\begin{aligned} M(\lambda )={} &\frac{{H(\lambda )+}\sqrt{H^{2}( \lambda )+4 \{d[e^{\lambda \cos \theta }+e^{-\lambda \cos \theta }+e^{\lambda \sin \theta } +e^{-\lambda \sin \theta }-4]+f'(0) \}\gamma _{2} }}{2} \\ \geq{} & \sqrt{ f'(0) \gamma _{2}}, \end{aligned}$$

where

$$ H(\lambda )=d\bigl[e^{\lambda \cos \theta }+e^{-\lambda \cos \theta }+e ^{\lambda \sin \theta } +e^{-\lambda \sin \theta }-4 \bigr]+f'(0)-\gamma _{1}-\gamma _{2}. $$

Hence \(\lim_{\lambda \rightarrow 0+}\varPhi (\lambda )\rightarrow \infty \). Moreover, by the first equation of (2.1) we get

$$ \varPhi (\lambda )\geq \frac{d[e^{\lambda \cos \theta }+e^{-\lambda \cos \theta }+e^{\lambda \sin \theta } +e^{-\lambda \sin \theta }-4 ]}{\lambda }+\frac{f'(0)-\gamma _{1} }{\lambda } . $$

Since \(\cos ^{2}\theta +\sin ^{2}\theta =1\), it is clear that \(\lim_{\lambda \rightarrow \infty }\varPhi (\lambda )=\infty \). Then the lemma directly follows from Liang and Zhao [7, Lemma 3.8]. □

The following result says that \(c^{*}(\theta )\) equals the minimal wave speed \(c_{*}(\theta )\).

Lemma 2.4

For any given \(\theta \in \mathbb{R}\), \((c^{*}(\theta ),\lambda ^{*}( \theta ))=(c_{*}(\theta ),\lambda _{*}(\theta )) \).

Proof

For convenience, we denote \((c^{*}(\theta ),\lambda ^{*}(\theta ))\) and \((c_{*}(\theta ),\lambda _{*}(\theta )) \) by \((c^{*},\lambda ^{*})\) and \((c_{*},\lambda _{*}) \), respectively. By Lemma 2.1 we see that \((c_{*},\lambda _{*})\) is a unique solution of the system

$$\begin{aligned} \triangle (c,\lambda )=0\quad \text{and}\quad \frac{\partial }{\partial \lambda } \triangle (c,\lambda )=0. \end{aligned}$$
(2.2)

From Lemma 2.3 we have \(\varPhi (\lambda ^{*})=M(\lambda ^{*})/ \lambda ^{*}=c^{*}\) and \(\varPhi '(\lambda ^{*})=0\). Hence

$$M\bigl(\lambda ^{*}\bigr)=c^{*}\lambda ^{*} \quad \mbox{and}\quad M'\bigl(\lambda ^{*}\bigr) =\frac{M( \lambda ^{*})}{\lambda ^{*}} =c^{*}. $$

Letting \(\nu _{1}=1\), it then follows from (2.1) that \(\nu _{2}= \gamma _{1}/(c^{*}\lambda ^{*}+\gamma _{2})\) and

$$ \begin{aligned}c^{*}\lambda ^{*}&= M\bigl(\lambda ^{*} \bigr) \\ & =d\bigl[e^{\lambda ^{*}\cos \theta }+e^{- \lambda ^{*}\cos \theta }+e^{\lambda ^{*}\sin \theta } +e^{-\lambda ^{*} \sin \theta }-4 \bigr] +\bigl[f'(0) -\gamma _{1}\bigr] + \frac{\gamma _{1} \gamma _{2} }{c^{*}\lambda ^{*}+\gamma _{2}}, \end{aligned}$$

that is, \(\triangle (c^{*},\lambda ^{*})=0\). Moreover, from (2.1) we obtain

$$ M(\lambda ) =d\bigl[e^{\lambda \cos \theta }+e^{-\lambda \cos \theta }+e ^{\lambda \sin \theta } +e^{-\lambda \sin \theta }-4 \bigr] +\bigl[f'(0) - \gamma _{1}\bigr] + \frac{\gamma _{1} \gamma _{2} }{M(\lambda )+\gamma _{2}}. $$

Differentiating with respect to λ this equality, we get

$$\begin{aligned} c^{*}&=M'\bigl(\lambda ^{*}\bigr) \\ & = d\bigl[\cos \theta e^{\lambda ^{*}\cos \theta }- \cos \theta e^{-\lambda ^{*}\cos \theta }+\sin \theta e^{\lambda ^{*} \sin \theta } - \sin \theta e^{-\lambda ^{*}\sin \theta } \bigr]- \frac{ \gamma _{1} \gamma _{2}M'(\lambda ^{*}) }{[M(\lambda ^{*})+\gamma _{2}]^{2}} \\ &= d\bigl[\cos \theta e^{\lambda ^{*}\cos \theta }- \cos \theta e^{-\lambda ^{*}\cos \theta }+\sin \theta e^{\lambda ^{*}\sin \theta } - \sin \theta e^{-\lambda ^{*}\sin \theta } \bigr]- \frac{\gamma _{1} \gamma _{2}c ^{*} }{[c^{*}\lambda ^{*}+\gamma _{2}]^{2}}, \end{aligned}$$

which implies that \(\frac{\partial }{\partial \lambda }\triangle (c ^{*},\lambda ) |_{\lambda =\lambda ^{*}}=0\). Hence \((c^{*},\lambda ^{*})\) is also a solution of system (2.2). By the uniqueness of solutions of (2.2) we deduce that \((c^{*},\lambda ^{*})=(c_{*}, \lambda _{*})\). This completes the proof. □

By Lemmas 2.3–2.4, for each \(c>c_{*}(\theta )\), \(c\lambda _{1}(c)=M(\lambda _{1}(c))\), and \(c\lambda >M(\lambda )\) for all \(\lambda \in (\lambda _{1}(c),\lambda _{*})\). Let \(\nu (\lambda _{1})=( \nu _{1}(\lambda _{1}), \nu _{2}(\lambda _{1}))\) be the eigenvector associated with \(M(\lambda _{1}(c ))\).

3 Stability of traveling wavefronts

In this section, we consider the stability of the noncritical traveling wavefronts of (1.1) with given wave direction and speed. We always assume that \(\varPhi (\xi )=(\phi (\xi ), \psi (\xi ))\), \(\xi =i \cos \theta +j\sin \theta +ct\), is a traveling wavefront of (1.1) with speed \(c>c_{*}(\theta )\) and direction \(\theta \in [0,2\pi ]\) connecting 0 and K. Without loss of generality, we extend f satisfying \(( A_{1})\) and \(( A_{2})\) to \(\hat{f}:[0,\infty )\rightarrow \mathbb{R} \) by

$$ \hat{f}_{j}(u):=\textstyle\begin{cases} f_{j}(u),& u\in [0,1], \\ f_{j}'(1)(u-1),& u\in [1,+\infty ). \end{cases} $$

For convenience, we still denote f̂ by f in the rest of this section.

We first prove the following comparison theorem for a related initial-boundary problem of (1.1), which will play an important role in proving our main result. Given any \(X_{0}\in \mathbb{R}\cup \{-\infty \}\), define the domain \(\varOmega _{X_{0}}\) and its boundary \(\partial \varOmega _{X_{0}}\) as follows:

$$ \varOmega _{X_{0}}:=\bigl\{ (i,j,t)\in \mathbb{Z}^{2}\times \mathbb{R}_{+} |i\cos \theta +j\sin \theta +ct>X_{0}, t>0\bigr\} $$

and

$$ \begin{aligned}\partial \varOmega _{X_{0}}:={}&\bigl\{ (i,j, t)\in \mathbb{Z}^{2}\times \mathbb{R}_{+}|i\cos \theta +j\sin \theta +ct>X_{0}, \\ &{} t=0\text{ or }i\cos \theta +j\sin \theta +ct=X_{0}, t\geq 0\bigr\} . \end{aligned} $$

Lemma 3.1

Given any \(\beta \in \mathbb{R}\), assume that \(W^{\pm }_{i,j}(t)= (u^{\pm }_{i,j}(t), v^{\pm }_{i,j}(t) ):\mathbb{Z}^{2}\times \mathbb{R}_{+}\rightarrow \mathbb{R}^{2} \) are two functions satisfying

  1. (i)

    \(W^{+}_{i,j}(t)\geq {\mathbf{0}}\) and \(W^{-}_{i,j}(t)\leq {\mathbf{K}}\) for all \((i,j,t)\in \mathbb{Z}^{2} \times \mathbb{R}^{+}\);

  2. (ii)

    \(W^{+}_{i,j}(t)\geq W^{-}_{i,j}(t)\) for all \((i,j,t)\in ( \mathbb{Z}^{2}\times \mathbb{R}_{+}) \setminus \varOmega _{X_{0}}\);

  3. (iii)

    For \((i,j,t)\in \varOmega _{X_{0}} \), we have

    $$ \textstyle\begin{cases} \frac{\mathrm{d}u_{i,j}^{+}(t)}{\mathrm{d}t}\geq (\Delta _{2}u^{+})_{i,j}+ \beta u_{i,j}^{+}(t)+\gamma _{2}v_{i,j}^{+}(t), \\ \frac{\mathrm{d}v_{i,j}^{+}(t)}{\mathrm{d}t}\geq \gamma _{1}u_{i,j} ^{+}(t)-\gamma _{2}v_{i,j}^{+}(t), \end{cases} $$

    and

    $$ \textstyle\begin{cases} \frac{\mathrm{d}u_{i,j}^{-}(t)}{\mathrm{d}t}\leq (\Delta _{2}u^{-})_{i,j}+ \beta u_{i,j}^{-}(t)+\gamma _{2}v_{i,j}^{-}(t), \\ \frac{\mathrm{d}v_{i,j}^{-}(t)}{\mathrm{d}t}\leq \gamma _{1}u_{i,j} ^{-}(t)-\gamma _{2}v_{i,j}^{-}(t). \end{cases} $$

Then \(W^{+}_{i,j}(t)\geq W^{-}_{i,j}(t)\) for all \((i,j,t)\in \mathbb{Z}^{2}\times \mathbb{R}^{+}\).

Proof

Take

$$W_{i,j}(t)=\bigl(W_{i,j}^{(1)}(t),W_{i,j}^{(2)}(t) \bigr):=W^{+}_{i,j}(t)-W^{-} _{i,j}(t),\quad \forall (i,j)\in \mathbb{Z}^{2}, t\geq 0. $$

By our assumptions we see that \(W_{i,j}(t)\) is bounded from below by −K and \(W_{i,j}(t)\geq {\mathbf{0}}\) for all \((i,j,t) \in ( \mathbb{Z}^{2}\times \mathbb{R}_{+}) \setminus \varOmega _{X_{0}}\). Define the function

$$ \bar{ W}(t)=\min_{l=1, 2}\inf_{(i,j,t)\in \overline{\varOmega }_{X_{0}}}W _{i,j}^{(l)}(t)\quad \text{ for }t\geq 0. $$

It is clear that \(\bar{ W}(0)\geq 0\) and \(\bar{ W}(t)\) is a continuous function on \([0,\infty )\).

Fix \(\delta >\max \{\beta +\gamma _{2},\gamma _{1}\}\). Note that

$$\liminf_{t\rightarrow \infty }\bar{ W}(t)e^{-\delta t}\geq - \liminf _{t\rightarrow \infty }\min \{1,K\}e^{-\delta t}=0. $$

We claim that \(\bar{ W}(t)\geq 0\) for all \(t\geq 0\). If this claim is not true, then there exists \(t_{0}>0\) such that \(\bar{ W}(t_{0})<0\) and

$$ \bar{ W}(t_{0})e^{-\delta t_{0}}=\min_{t\in [0,t_{0}]} \bar{ W}(t)e ^{-\delta t}< \bar{ W}(s)e^{-\delta s}, \quad \forall s\in [ 0,t_{0}). $$
(3.1)

It is easy to see that there exist \(l_{0}\in \{1,2 \}\) and a sequence \(\{(i_{k},j_{k})\}_{k=1}^{+\infty }\subseteq \mathbb{Z}^{2}\) such that

$$W ^{(l_{0})}_{i_{k},j_{k}}(t_{0})< 0,\quad \forall k\geq 1,\quad \mbox{and}\quad \lim_{k\rightarrow +\infty }W ^{(l_{0})}_{i_{k},j_{k}}(t_{0})= \bar{ W}(t_{0}). $$

Let \(\{t_{k}\}_{k=1}^{+\infty } \) be a sequence in \((0,t_{0}]\) such that

$$ W ^{(l_{0})}_{i_{k},j_{k}}(t_{k})e^{-\delta t_{k}}=\min _{t\in [0,t _{0}]}W ^{(l_{0})}_{i_{k},j_{k}}(t)e^{-\delta t}< 0. $$
(3.2)

Since \(W ^{(l_{0})}_{i,j}(t)\geq 0\) for any \((i,j,t)\in ( \mathbb{Z} ^{2}\times \mathbb{R}_{+}) \setminus \varOmega _{X_{0}}\), we must have \((i_{k},j_{k},t_{k})\in \varOmega _{X_{0}}\). It then follows from (3.1) that \(\lim_{k\rightarrow +\infty }t_{k}=t_{0}\). By the definition of \(\bar{ W}(t)\) we have

$$ \bar{ W}(t_{k}) \leq W ^{(l_{0})}_{i_{k},j_{k}}(t_{k}) \leq W ^{(l_{0})} _{i_{k},j_{k}}(t_{0})e^{-\delta (t_{0}-t_{k})}, $$

which implies that \(\lim_{k\rightarrow +\infty }W ^{(l_{0})} _{i_{k},j_{k}}(t_{k}) =\bar{ W}(t_{0})<0\) by taking \(k\rightarrow + \infty \).

We distinguish two cases.

Case 1. \(l_{0}=1\). By the definition of \(\bar{ W}(t)\) and the fact \(W ^{(l_{0})}_{i,j}(t)\geq 0\) for any \((i,j,t)\in ( \mathbb{Z} ^{2}\times \mathbb{R}_{+}) \setminus \varOmega _{X_{0}}\), for each \(k\geq 1\), we obtain

$$\begin{aligned} \bigl(\Delta _{2}W ^{(1)}\bigr)_{i_{k},j_{k}}(t_{k}) &=d\bigl[W_{i_{k}+1,j_{k}}^{(1)}(t _{k})+W_{i_{k}-1,j_{k}}^{(1)}(t_{k})+W_{i_{k},j_{k}+1}^{(1)}(t_{k})+W _{i_{k},j_{k}-1}^{(1)}(t_{k})-4W_{i_{k},j_{k}}^{(1)}(t_{k}) \bigr] \\ &\geq 4d\min \bigl\{ 0,\bar{ W}(t_{k})\bigr\} -4dW ^{(1)}_{i_{k},j_{k}}(t_{k}). \end{aligned}$$

Then by (3.2) we have

$$\begin{aligned} 0\geq{} & e^{\delta t_{k}}\frac{\mathrm{d}}{\mathrm{d}t} \bigl(W ^{(1)} _{i_{k},j_{k}}(t)e^{-\delta t} \bigr)\bigg|_{t=t_{k}^{-}} \\ ={} &\frac{\mathrm{d}}{\mathrm{d} t}W ^{(1)}_{i_{k},j_{k}}(t_{k})- \delta W ^{(1)}_{i_{k},j_{k}}(t_{k}) \\ \geq{} & \bigl(\Delta _{2}W ^{(1)}\bigr)_{i_{k},j_{k}}(t_{k})+ (\beta - \delta )W ^{(1)}_{i_{k},j_{k}}(t_{k})+\gamma _{2} W ^{(2)}_{i_{k},j_{k}}(t_{k}) \\ \geq{} & 4d\min \bigl\{ 0,\bar{ W}(t_{k})\bigr\} + (\beta - \delta -4d)W ^{(1)} _{i_{k},j_{k}}(t_{k})+\gamma _{2} \bar{ W}(t_{k}). \end{aligned}$$

Note that \(\bar{ W}(t_{0})<0\) and \(\delta >\beta +\gamma _{2}\). Sending \(k\rightarrow \infty \), we get \(0\geq \bar{ W}(t_{0}) [\beta + \gamma _{2}- \delta ]>0\), a contradiction.

Case 2. \(l_{0}=2\). Similarly, we have

$$\begin{aligned} 0\geq \frac{\mathrm{d}}{\mathrm{d}t}W ^{(2)}_{i_{k},j_{k}}(t_{k})- \delta W ^{(2)}_{i_{k},j_{k}}(t_{k}) \geq \gamma _{1} \bar{ W}(t_{k})+(- \gamma _{2}-\delta ) W ^{(2)}_{i_{k},j_{k}}(t_{k}). \end{aligned}$$

Letting \(k\rightarrow \infty \), we get \(0\geq \bar{ W}(t_{0}) [ \gamma _{1}-\gamma _{2}- \delta ]>0\), a contradiction.

These contradictions show that \(\bar{ W}(t)\geq 0\) for all \(t\geq 0 \), and hence \(W^{+}_{i,j}(t)\geq W^{-}_{i,j}(t)\) for all \((i,j,t)\in \mathbb{Z}^{2}\times \mathbb{R}^{+}\). This completes the proof. □

Let \(\lambda _{K}\) be the principle eigenvalue of the matrix

( f ′ ( 1 ) − γ 1 γ 2 γ 1 − γ 2 ) ,

and let \(\varrho =(\varrho _{1},\varrho _{2})\gg (0,0)\) be the associated eigenvector. Thus

$$ \textstyle\begin{cases} \lambda _{K}\varrho _{1}=(f'(1)-\gamma _{1}) \varrho _{1}+\gamma _{2} \varrho _{2}, \\ \lambda _{K}\varrho _{2}=\gamma _{1}\varrho _{1}-\gamma _{2}\varrho _{2}. \end{cases} $$
(3.3)

Since \(f'(1)<0\), it is easy to see that \(\lambda _{K}<0\). In fact, suppose on the contrary that \(\lambda _{K}\geq 0\). Then \(\lambda _{K}\varrho _{2} =\gamma _{1}\varrho _{1}-\gamma _{2}\varrho _{2}\geq 0\), and hence

$$ \lambda _{K}\varrho _{1} =\bigl(f'(1)- \gamma _{1}\bigr) \varrho _{1}+\gamma _{2} \varrho _{2}\leq f'(1) , $$

which yields that \(f'(1)\geq \lambda _{K}\geq 0\). This contradicts the assumption that \(f'(1)<0\).

Take \(\bar{\mu }\in (0,-\lambda _{K})\) and \(\epsilon \in (0,\min \{\bar{ \mu },\lambda _{*}(c)-\lambda _{1}(c)\})\). Since \(f\in C^{1}([0,\infty ))\) and \(f'(u)=f'(1)\) for all \(u\geq 1\), there exists \(\delta _{0} \in (0,1)\) such that

$$\begin{aligned} f'(u)< f'(1)+\epsilon\quad \text{for any }u\geq 1- \delta _{0}. \end{aligned}$$
(3.4)

Noting that \(\phi (+\infty )=1\), we can choose \(M_{0}>0\) sufficiently large such that

$$\begin{aligned} \phi (\xi )\in [1-\delta _{0},1)\quad \text{for all }\xi \geq M_{0}. \end{aligned}$$
(3.5)

We now define the weighted function

$$ \omega ^{\epsilon }(i,j):=\textstyle\begin{cases} {\mathrm{e}}^{-\lambda _{\epsilon }(i\cos \theta +j\sin \theta -M_{0})}, &i\cos \theta +j\sin \theta \leq M_{0}, \\ 1, &i\cos \theta +j\sin \theta >M_{0}, \end{cases} $$
(3.6)

where \(\lambda _{\epsilon }:=\lambda _{1}(c)+\epsilon \). Note that \(c\lambda _{\epsilon }>M(\lambda _{\epsilon })\). Take \(w^{0}=\{w^{0}_{i,j}\}_{(i,j)\in \mathbb{Z}^{2}}=\{(U^{0}_{i,j},V^{0}_{i,j})\}_{(i,j)\in \mathbb{Z}^{2}}\). We now state the main result.

Theorem 3.2

Assume that \(\mathbf{0}\leq w^{0}_{i,j}\leq {\mathbf{K}}\) and \(\|w^{0}_{i,j}-\varPhi (i\cos \theta +j\sin \theta )\|\omega ^{\epsilon }(i,j) \) is bounded on \(\mathbb{Z}^{2}\). Then there exists \(\mu _{0}>0\) such that the unique solution \(w_{i,j}(t; w^{0})=(u_{i,j}(t; w^{0}), v _{i,j}(t; w^{0}))\) of (1.1) satisfies \(\mathbf{0}\leq w_{i,j}(t; w^{0}) \leq {\mathbf{K}}\) for \((i,j)\in \mathbb{Z}^{2}\), \(t\geq 0 \), and for some \(C>0\),

$$ \sup_{(i,j)\in \mathbb{Z}^{2}} \bigl\Vert w_{i,j} \bigl(t; w^{0}\bigr)- \varPhi (i\cos \theta +j\sin \theta +ct) \bigr\Vert \leq Ce^{-\mu _{0}t},\quad \forall t\geq 0. $$
(3.7)

Proof

Take \(W^{\pm ,0}_{i,j}=(U^{\pm ,0}_{i,j},V^{\pm ,0}_{i,j})\), where

$$\begin{aligned} &U^{+,0}_{i,j}:=\max \bigl\{ u_{i,j}^{0}, \phi (i\cos \theta +j\sin \theta ) \bigr\} , \\ & V^{+,0}_{i,j} :=\max \bigl\{ v_{i,j}^{0}, \psi (i\cos \theta +j\sin \theta )\bigr\} ,\quad (i,j)\in \mathbb{Z}^{2}, \\ &U^{-,0}_{i,j}:=\min \bigl\{ u_{i,j}^{0}, \phi (i\cos \theta +j\sin \theta )\bigr\} , \\ &V^{-,0}_{i,j} :=\min \bigl\{ v_{i,j}^{0}, \psi (i\cos \theta +j\sin \theta )\bigr\} , \quad (i,j)\in \mathbb{Z}^{2}. \end{aligned}$$

Let \(W^{\pm }_{i,j}(t)=(U^{\pm }_{i,j}(t),V^{\pm }_{i,j}(t))\) be the unique solution of (1.1) with the initial value \(W^{\pm ,0}=\{W^{\pm ,0}_{i,j}\}_{(i,j)\in \mathbb{Z}^{2}}\). By comparison theorem we obtain

$$\begin{aligned} {\mathbf{0}}\leq W^{-}_{i,j}(t)\leq w_{i,j} \bigl(t;w^{0}\bigr), \qquad \varPhi (i\cos \theta +j\sin \theta +ct) \leq W_{i,j}^{+}(t)\leq {\mathbf{K}}, \quad (i,j) \in \mathbb{Z}^{2}, t>0, \end{aligned}$$

which implies that

$$\begin{aligned} & \bigl\Vert w_{i,j}\bigl(t;w^{0}\bigr)-\varPhi (i\cos \theta +j\sin \theta +ct) \bigr\Vert \\ &\quad \leq \max \bigl\{ \bigl\Vert W_{i,j}^{+}(t)- \varPhi (i\cos \theta +j\sin \theta +ct) \bigr\Vert , \bigl\Vert W_{i,j}^{-}(t)-\varPhi (i\cos \theta +j\sin \theta +ct) \bigr\Vert \bigr\} \end{aligned}$$

for any \((i,j)\in \mathbb{Z}^{2}\), \(t>0\). Thus, to prove the theorem, it suffices to prove the convergence of \(W^{\pm }_{i,j}(t)\) to \(\varPhi (i\cos \theta +j\sin \theta +ct)\). We only prove that \(W^{+}_{i,j}(t)\) converges to \(\varPhi (i\cos \theta +j\sin \theta +ct) \) in time, since the other case can be treated similarly.

Denote

$$ Z_{i,j}(t)=\bigl(Z_{i,j}^{(1)}(t),Z_{i,j}^{(2)}(t) \bigr):=W_{i,j}^{+}(t)-\varPhi (i \cos \theta +j\sin \theta +ct), \quad (i,j)\in \mathbb{Z}^{2}, t>0. $$

It is clear that, for all \((i,j)\in \mathbb{Z}^{2}\) and \(t>0 \),

$$ \begin{aligned}&Z_{i,j}(t)\geq {\mathbf{0}}, \\ &0\leq Z_{i,j}^{(1)}(0)\leq \bigl\vert u_{i,j}^{0}- \phi (i\cos \theta +j\sin \theta ) \bigr\vert , \\ &0\leq Z_{i,j}^{(2)}(0)\leq \bigl\vert v _{i,j}^{0}-\psi (i\cos \theta +j\sin \theta ) \bigr\vert . \end{aligned} $$

Hence \(\|Z_{i,j}(0)\|w^{\epsilon }(i,j)\) is bounded on \(\mathbb{Z} ^{2}\).

We divide the remainder of the proof into two steps.

Step 1. We show that there exists \(C_{1}>0\) such that

$$\begin{aligned} Z_{i,j}(t) \leq C_{1}\nu (\lambda _{\epsilon }){ \mathrm{e}}^{-(c \lambda _{\epsilon }-M(\lambda _{\epsilon }))t},\quad \forall (i,j,t)\in \bigl( \mathbb{Z}^{2} \times \mathbb{R}^{+} \bigr)\backslash \varOmega _{M_{0}}. \end{aligned}$$
(3.8)

By the assumption \(f'(u)\leq f'(0)\), \(u\in [0,1]\), direction computation shows that

$$\begin{aligned} \frac{{\mathrm{d}}Z_{i,j}^{(1)}(t)}{{\mathrm{d}}t} ={}&d\bigl[Z_{i+1,j}^{(1)}+Z _{i-1,j}^{(1)}+Z_{i,j+1}^{(1)}+Z_{i,j-1}^{(1)}-4Z_{i,j}^{(1)} \bigr]-\gamma _{1}Z_{i,j}^{(1)}(t)+\gamma _{2}Z_{i,j}^{(2)}(t) \\ & {} +f \bigl(U_{i,j}^{+}(t) \bigr)-f \bigl(\phi (i\cos \theta +j\sin \theta +ct) \bigr) \\ \leq{}& d\bigl[Z_{i+1,j}^{(1)}+Z_{i-1,j}^{(1)}+Z_{i,j+1}^{(1)}+Z_{i,j-1} ^{(1)}-4Z_{i,j}^{(1)}\bigr]+ \bigl(f'(0)-\gamma _{1}\bigr)Z_{i,j}^{(1)}(t)+ \gamma _{2}Z _{i,j}(t) \end{aligned}$$

and

$$ \frac{{\mathrm{d}}Z_{i,j}^{(2)}(t)}{{\mathrm{d}}t}=\gamma _{1}Z_{i,j} ^{(1)}(t)-\gamma _{2}Z_{i,j}^{(2)}(t). $$

Let \(\nu (\lambda _{\epsilon })=(\nu _{1}(\lambda _{\epsilon }),\nu _{2}( \lambda _{\epsilon }))\gg {\mathbf{0}}\) be the eigenvector of (2.1) corresponding to \(M(\lambda _{\epsilon })\). By the boundedness of \(\|Z_{i,j}(0)\|w^{\epsilon }(i,j)\) on \(\mathbb{Z}^{2}\) we can choose a sufficiently large \(C_{1}>0\) such that

$$ Z_{i,j}(0) \leq C_{1}\nu (\lambda _{\epsilon }){ \mathrm{e}}^{ \lambda _{\epsilon }(i\cos \theta +j\sin \theta -M_{0})}, \quad i,j\in \mathbb{Z}. $$

Define the function \(\overline{Z}_{i,j}(t)=(\overline{Z}_{i,j}^{(1)}(t), \overline{Z}_{i,j}^{(2)}(t))\) as follows:

$$ \overline{Z}_{i,j}(t)= C_{1}\nu (\lambda _{\epsilon }){\mathrm{e}}^{ \lambda _{\epsilon }(i\cos \theta +j\sin \theta -M_{0})+M( \lambda _{\epsilon })t} ,\quad i,j\in \mathbb{Z}, t\geq 0. $$

Using (2.1), we can easily verify that

$$\begin{aligned} \frac{\mathrm{d} \overline{Z}_{i,j}^{(1)}(t)}{\mathrm{d}t}= {}&C_{1}M( \lambda _{\epsilon })\nu _{1}(\lambda _{\epsilon }){\mathrm{e}}^{ \lambda _{\epsilon }(i\cos \theta +j\sin \theta -M_{0})+M( \lambda _{\epsilon })t} \\ ={} &C_{1}{\mathrm{e}}^{\lambda _{\epsilon }(i\cos \theta +j\sin \theta -M_{0})+M(\lambda _{\epsilon })t} \\ &{}\times \bigl\{ d\bigl[e^{\lambda _{\epsilon }\cos \theta }+e^{- \lambda _{\epsilon }\cos \theta }+e^{\lambda _{\epsilon }\sin \theta } +e ^{-\lambda _{\epsilon }\sin \theta }-4 \bigr]\nu _{1}(\lambda _{\epsilon }) \\ &{}+\bigl[f'(0) -\gamma _{1}\bigr]\nu _{1}(\lambda _{\epsilon })+\gamma _{2}\nu _{2}( \lambda _{\epsilon }) \bigr\} \\ ={} & d\bigl[\overline{Z}_{i+1,j}^{(1)}+ \overline{Z}_{i-1,j}^{(1)}+ \overline{Z}_{i,j+1}^{(1)}+ \overline{Z}_{i,j-1}^{(1)}-4\overline{Z} _{i,j}^{(1)} \bigr] +\bigl(f'(0)-\gamma _{1}\bigr) \overline{Z}_{i,j}^{(1)}(t)+\gamma _{2} \overline{Z}_{i,j}^{(2)}(t) \end{aligned}$$

for \(i,j\in \mathbb{Z}\), \(t>0\). Similarly, we have

$$ \frac{\mathrm{d} \overline{Z}_{i,j}^{(2)}(t)}{\mathrm{d}t} =\gamma _{1}\overline{Z}_{i,j}^{(1)}(t)- \gamma _{2}\overline{Z}_{i,j}^{(2)}(t),\quad i,j \in \mathbb{Z}, t>0. $$

Applying Lemma 3.1 with \(X_{0}=-\infty \) and \(\beta = f'(0) - \gamma _{1} \), we get

$$ Z_{i,j}(t)\leq \overline{Z}_{i,j}(t),\quad \forall i,j\in \mathbb{Z}, t>0. $$

Then, for any \(i,j\in \mathbb{Z}\), \(t\geq 0\) with \(i\cos \theta +j \sin \theta +ct\leq M_{0}\), we have

$$\begin{aligned} Z_{i,j}(t) &\leq C_{1} \nu (\lambda _{\epsilon }){ \mathrm{e}}^{ \lambda _{\epsilon }(i\cos \theta +j\sin \theta -M_{0})+M( \lambda _{\epsilon })t} \\ &= C_{1} \nu (\lambda _{\epsilon }){\mathrm{e}}^{\lambda _{\epsilon }(i \cos \theta +j\sin \theta +ct-M_{0})}{ \mathrm{e}}^{-(c\lambda _{\epsilon }-M(\lambda _{\epsilon }))t} \leq C_{1}\nu (\lambda _{\epsilon }){\mathrm{e}} ^{-(c\lambda _{\epsilon }-M(\lambda _{\epsilon }))t}. \end{aligned}$$

This proves (3.8).

Step 2. Let \(\mu _{0}=\min \{c\lambda _{\epsilon }-M( \lambda _{\epsilon }),-\lambda _{K}-\epsilon \}>0\). We prove that there exists \(L_{2}>0\) such that

$$ Z_{i,j}(t)\leq L_{2}\varrho {\mathrm{e}}^{-\mu _{0}t} , \quad \forall i,j\in \mathbb{Z}, t\geq 0 . $$
(3.9)

Note that \(\varPhi (i\cos \theta +j\sin \theta +ct) \leq W^{+}_{i,j}(t) \leq {\mathbf{K}}\) for all \(i,j\in \mathbb{Z}\) and \(t\geq 0 \) and \(\phi (\xi )\in [1-\delta _{0},1)\) for all \(\xi \geq M_{0} \). Thus it follows from (3.4) that for any \((i,j,t)\in \varOmega _{M_{0}} \),

$$\begin{aligned} \frac{{\mathrm{d}}Z_{i,j}^{(1)}(t)}{{\mathrm{d}}t} \leq{}& d\bigl[Z_{i+1,j} ^{(1)}+Z_{i-1,j}^{(1)}+Z_{i,j+1}^{(1)}+Z_{i,j-1}^{(1)}-4Z_{i,j}^{(1)} \bigr] \\ &{} +\bigl(f'(1)-\gamma _{1}+\epsilon \bigr)Z_{i,j}^{(1)}(t) +\gamma _{2}Z _{i,j}^{(2)}(t) \end{aligned}$$

and

$$ \frac{\mathrm{d} Z_{i,j}^{(2)}(t)}{\mathrm{d}t} =\gamma _{1}Z_{i,j} ^{(1)}(t)-\gamma _{2}Z_{i,j}^{(2)}(t). $$

We can choose \(L_{2}>0\) such that \(L_{2}\varrho \geq \max \{C_{1} \nu (\lambda _{\epsilon }),\mathbf{K}\}\). Define

$$ \hat{Z}_{i,j}(t)=\bigl(\hat{Z}_{i,j}^{(1)}(t), \hat{Z}_{i,j}^{(2)}(t)\bigr):=L _{2} \varrho e^{-\mu _{0}t},\quad \forall i,j\in \mathbb{Z}, t\geq 0 . $$

It is clear that \(\hat{Z}_{i,j}(0)\geq {\mathbf{K}} \geq Z_{i,j}(0)\). Since \(\mu _{0}\leq c\lambda _{\epsilon }-M(\lambda _{\epsilon })\), from (3.8) we see that \(Z_{i,j}(t)\leq \hat{Z}_{i,j}(t)\) for all \((i,j,t)\in (\mathbb{Z}^{2}\times \mathbb{R}^{+} )\backslash \varOmega _{M_{0}}\).

Due to \(\mu _{0}\leq - \lambda _{K}-\epsilon \leq -\lambda _{K}\), we get

$$\begin{aligned} \frac{{\mathrm{d}}\hat{Z}_{i,j}^{(1)}(t)}{{\mathrm{d}}t} ={}& {-}\mu _{0} \hat{Z}_{i,j}^{(1)}(t) \geq (\lambda _{K}+\epsilon )\hat{Z}_{i,j}^{(1)}(t) \\ ={} &f'(1)\hat{Z}_{i,j}^{(1)}(t)-\gamma _{1}\hat{Z}_{i,j}^{(1)}(t)+ \gamma _{2}\hat{Z}_{i,j}^{(2)}(t)+\epsilon \hat{Z}_{i,j}^{(1)}(t) \\ \geq{} & d\bigl[Z_{i+1,j}^{(1)}+Z_{i-1,j}^{(1)}+Z_{i,j+1}^{(1)}+Z_{i,j-1} ^{(1)}-4Z_{i,j}^{(1)}\bigr] \\ &{}+\bigl(f'(1)-\gamma _{1}+\epsilon \bigr) \hat{Z}_{i,j}^{(1)}(t) +\gamma _{2} \hat{Z}_{i,j}^{(2)}(t) \end{aligned}$$

and

$$ \frac{{\mathrm{d}}\hat{Z}_{i,j}^{(2)}(t)}{{\mathrm{d}}t}=-\mu _{0} \hat{Z}_{i,j}^{(2)}(t) \geq \lambda _{K} \hat{Z}_{i,j}^{(2)}(t) = \gamma _{1}\hat{Z}_{i,j}^{(1)}(t)-\gamma _{2}\hat{Z}_{i,j}^{(2)}(t). $$

Using Lemma 3.1 with \(X_{0}=M_{0} \) and \(\beta = f'(1) -\gamma _{1}+\epsilon \), we deduce that \(Z_{i,j}(t)\leq \hat{Z}_{i,j}(t)\) for all \(i,j\in \mathbb{Z}\) and \(t\geq 0 \), that is, (3.9) holds. Letting \(\tilde{M}:=L_{2}\|\varrho \|\), we conclude that

$$\begin{aligned} \bigl\Vert Z_{i,j}(t) \bigr\Vert = \bigl\Vert W_{i,j}^{+}(t)-\varPhi (i\cos \theta +j\sin \theta +ct) \bigr\Vert \leq \tilde{M} {\mathrm{e}}^{-\mu _{0}t} . \end{aligned}$$

This completes the proof. □

4 Discussions

In the previous sections, we have studied the stability of the traveling wave solutions of the two-dimensional lattice differential system (1.1). Since (1.1) is a monotone system, our method is mainly based on the comparison principle. We would like to mention that in the recent years, there are some results on the stability of the traveling wave solutions for nonmonotone diffusion systems by using the weighted energy method; see, for example, Wu, Zhao, and Liu [18] and Lin et al. [8].

On the other hand, by taking into account the effect of time delay, it is natural to consider the following delayed lattice differential system:

$$ \textstyle\begin{cases} \frac{\mathrm{d}u_{i,j}(t)}{\mathrm{d}t}=(\Delta _{2}u)_{i,j}+f(u_{i,j}(t),u_{i,j}(t-\tau ))-\gamma _{1}u_{i,j}(t)+\gamma _{2}v_{i,j}(t),\\ \frac{\mathrm{d}v_{i,j}(t)}{\mathrm{d}t}=\gamma _{1}u_{i,j}(t)-\gamma _{2}v_{i,j}(t). \end{cases} $$
(4.1)

For this delayed system, if the function \(f(u,v)\) is not monotone with respect to v, then the comparison principle cannot be applied to consider the stability of traveling waves. An interesting problem is to study the stability of the traveling waves for such nonmonotone lattice differential systems such as (4.1).