1 Introduction

In this paper, we consider the existence of traveling wave solutions of a higher-dimensional lattice competitive system with stage structure

$$\begin{aligned} \left\{ \begin{array}{ll} v'_{1\eta }(t)&{}=d_1 (\Delta _nv_1)_\eta +\alpha _1 u_{1\eta }(t)-\gamma _1 v_{1\eta }(t) -\alpha _1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )e^{-\gamma _1\tau _1}u_{1(\eta -\xi )}(t-\tau _1),\\ u'_{1\eta }(t)&{}=D_1(\Delta _nu_1)_\eta +\alpha _1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )e^{-\gamma _1\tau _1}u_{1(\eta -\xi )}(t-\tau _1) -a_1u^2_{1\eta }(t)-b_1u_{1\eta }(t)u_{2\eta }(t),\\ v'_{2\eta }(t)&{}=d_2 (\Delta _nv_2)_\eta +\alpha _2 u_{2\eta }(t)-\gamma _2 v_{2\eta }(t) -\alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi )e^{-\gamma _2\tau _2}u_{2(\eta -\xi )}(t-\tau _2),\\ u'_{2\eta }(t)&{}=D_2(\Delta _nu_2)_\eta +\alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi )e^{-\gamma _2\tau _2}u_{2(\eta -\xi )}(t-\tau _2) -b_2u_{1\eta }(t)u_{2\eta }(t)-a_2u^2_{2\eta }(t), \end{array} \right. \end{aligned}$$
(1.1)

where \(t>0,(\Delta _nw)_\eta =\sum _{|\xi -\eta |=1,\xi \in \mathbb {Z}^n} w_\xi -2nw_\eta ,\eta \in \mathbb {Z}^n,|\cdot |\) denotes the Euclidean norm in \({\mathbb {R}}^n,n\in \mathbb {Z}^+,\sum _{\xi \in \mathbb {Z}^n}J_i(\xi )=1,i=1,2\). System (1.1) is the spatially discrete version of stage-structured reaction–diffusion competitive system

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial v_1(x,t)}{\partial t}&{}=d_1\frac{\partial ^2v_1(x,t)}{\partial x^2} +\alpha _1 u_1(x,t)-\gamma _1 v_1(x,t)-\alpha _1\displaystyle \int _{{\mathbb {R}}^n} e^{-\gamma _1\tau _1}g_1(y)u_1(x-y,t-\tau _1)dy,\\ \frac{\partial u_1(x,t)}{\partial t}&{}=D_1\frac{\partial ^2u_1(x,t)}{\partial x^2}+\alpha _1\displaystyle \int _{{\mathbb {R}}^n} e^{-\gamma _1\tau _1}g_1(y)u_1(x-y,t-\tau _1)dy\\ &{}\quad -a_1u_1^2(x,t)-b_1u_1(x,t)u_2(x,t),\\ \frac{\partial v_2(x,t)}{\partial t}&{}=d_2\frac{\partial ^2v_2(x,t)}{\partial x^2}+\alpha _2 u_2(x,t)-\gamma _2 v_2(x,t) -\alpha _2\displaystyle \int _{{\mathbb {R}}^n} e^{-\gamma _2\tau _2}g_2(y)u_2(x-y,t-\tau _2)dy,\\ \frac{\partial u_2(x,t)}{\partial t}&{}=D_2\frac{\partial ^2u_2(x,t)}{\partial x^2}+\alpha _2\displaystyle \int _{{\mathbb {R}}^n} e^{-\gamma _2\tau _2}g_2(y)u_2(x-y,t-\tau _2)dy\\ &{}\quad -b_2u_1(x,t)u_2(x,t)-a_2u_2^2(x,t), \end{array} \right. \end{aligned}$$
(1.2)

where all the parameters are positive constants, \(t>0,x\in {\mathbb {R}}^n,\int _{{\mathbb {R}}^n}g_i(y)dy=1,v_i,u_i,d_i,D_i,\alpha _i, a_i,b_i,\gamma _i,\tau _i,\alpha _i\int _{{\mathbb {R}}^n}e^{-\gamma _i\tau _i} g_i(y)u_i(x-y,t-\tau _i)dy,i=1,2,\) denote the densities of the immature population, the densities of the mature population, the diffusive rate of the immature population, the diffusive rate of the mature population, the birth rate, the mature death and overcrowding rate, the rate of competition, the death rate of the immature population, the mature period, the number born at the location y and at the time \(t-\tau _i\) and still alive now at the location x and at the time t of two species, respectively. Xu et al. [30] considered the existence of traveling wave solutions of (1.2) with a special case \(g_i(x)=\frac{1}{\sqrt{4\pi d_i\tau _i}}e^{-\frac{x^2}{4d_i\tau _i}}\) and \(n=1\). For more results about traveling wave solutions for reaction–diffusion equations with stage structure, one can refer to [1,2,3,4, 10, 11, 19, 23, 25].

Note that the properties of solutions of system (1.1) are determined by the following system

$$\begin{aligned} \left\{ \begin{array}{ll} u'_{1\eta }(t)=D_1(\Delta _nu_1)_\eta +\alpha _1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )e^{-\gamma _1\tau _1}u_{1(\eta -\xi )}(t-\tau _1) -a_1u^2_{1\eta }(t)-b_1u_{1\eta }(t)u_{2\eta }(t),\\ u'_{2\eta }(t)=D_2(\Delta _nu_2)_\eta +\alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi )e^{-\gamma _2\tau _2}u_{2(\eta -\xi )}(t-\tau _2) -b_2u_{1\eta }(t)u_{2\eta }(t)-a_2u^2_{2\eta }(t) \end{array} \right. \end{aligned}$$
(1.3)

for \(t>0\), so we only consider this system. System (1.3) has four equilibria \((0,0),(\frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1},0), (0,\frac{\alpha _2e^{-\gamma _2\tau _2}}{a_2}),(\frac{a_2\alpha _ 1e^{-\gamma _1\tau _1}-b_1\alpha _2e^{-\gamma _2\tau _2}}{a_1a_2-b_1b_2}, \frac{a_1\alpha _2e^{-\gamma _2\tau _2}-b_2\alpha _1e^ {-\gamma _1\tau _1}}{a_1a_2-b_1b_2})\). Two semi-trivial equilibria have an important significance in biology, which represent the competitive ability of the populations in a community. The traveling wave solutions connecting two semi-trivial equilibria are usually called invasion waves, which means that one species will eventually exterminate the other species if one species invades at a certain speed. Based on this idea, many authors are devoted to studying invasion waves, one can refer to [17, 18, 20, 27] for reaction–diffusion systems with delays or without delays, [21, 24] for nonlocal dispersal systems and [13] for lattice systems. However, there are no results for higher-dimensional lattice system (1.3). We notice that there are many results about the traveling wave solutions for different dimensional lattice equations. For a single equation, one can refer to [5,6,7,8,9, 12, 14, 26, 29, 31,32,33] for one or two-dimensional lattices and [28, 34, 35] for higher-dimensional lattices. For systems with two equations and different monotonicity, one can refer to [13, 15, 22] for one-dimensional lattice and [16] for higher-dimensional lattices.

Motivated by the techniques in [15, 16, 22], we will use Schauder’s fixed point theorem and upper and lower solution method to establish the existence of traveling wave solutions of (1.3) connecting \((\frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1},0)\) with \((0,\frac{\alpha _2e^{-\gamma _2\tau _2}}{a_2})\) when

$$\begin{aligned} a_2\alpha _1e^{-\gamma _1\tau _1}<b_1\alpha _2e^{-\gamma _2\tau _2} \ \ \text{ and } \ \ a_1\alpha _2e^{-\gamma _2\tau _2}>b_2\alpha _1e^{-\gamma _1\tau _1}. \end{aligned}$$
(1.4)

Let \(u_n^*=\frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1}-u_n\) and drop the star. Then, the properties of traveling wave solutions of system (1.3) connecting \((\frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1},0)\) with \((0,\frac{\alpha _2e^{-\gamma _2\tau _2}}{a_2})\) are equivalent to the properties of traveling wave solutions of the following system

$$\begin{aligned} \left\{ \begin{array}{ll} u'_{1\eta }(t)=&{}D_1(\Delta _nu_1)_\eta +\alpha _1\displaystyle \sum _{\xi \in \mathbb {Z} ^n}J_1(\xi )e^{-\gamma _1\tau _1}u_{1(\eta -\xi )}(t-\tau _1)\\ &{}-2\alpha _1e^{-\gamma _1\tau _1}u_{1\eta }(t)+a_1u^2_{1\eta }(t)+b_1 \left( \frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1}-u_{1\eta }(t)\right) u_{2\eta }(t),\\ u'_{2\eta }(t)=&{}D_2(\Delta _nu_2)_\eta +\alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi )e^{-\gamma _2\tau _2}u_{2(\eta -\xi )}(t-\tau _2)\\ &{}-b_2\left( \frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1}-u_{1\eta }(t)\right) u_{2\eta } (t)-a_2u^2_{2\eta }(t) \end{array} \right. \end{aligned}$$
(1.5)

connecting \(\mathbf{0 }=(0,0)\) with \(\mathbf{K }=(k_1,k_2)=(\frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1}, \frac{\alpha _2e^{-\gamma _2\tau _2}}{a_2})\).

This paper is organized as follows: In Sect. 2, we establish the existence of traveling wave solutions of (1.5) by Schauder’s fixed point theorem. In Sect. 3, we are devoted to constructing a pair of upper and lower solutions.

2 The Existence of Traveling Wave Solutions

In this paper, we use the usual notations for the standard ordering in \({\mathbb {R}}^2\), that is, for \(u=(u_1,u_2)\) and \(v=(v_1,v_2)\), we denote \(u\le v\) if \(u_i\le v_i,i=1,2\), and \(u<v\) if \(u\le v\) but \(u\ne v\). In particular, we denote \(u\ll v\) if \(u\le v\) but \(u_i\ne v_i,i=1,2\). If \(u\le v\), we also denote \((u,v]=\{\omega \in {\mathbb {R}}^2,u<\omega \le v\}, [u,v)=\{\omega \in {\mathbb {R}}^2,u\le \omega < v\}\), \([u,v]=\{\omega \in {\mathbb {R}}^2,u\le \omega \le v\}\). In the following, \(|\cdot |\) denotes the Euclidean norm in \({\mathbb {R}}^2\) or \({\mathbb {R}}^n\) and \(\parallel \cdot \parallel \) denotes the supremum norm in \(C([-\tau ,0],{\mathbb {R}}^2)\).

Let

$$\begin{aligned} C_{[\mathbf{0 },\mathbf{K }]}({\mathbb {R}},{\mathbb {R}}^2)=\{(\phi ,\psi )\in C({\mathbb {R}},{\mathbb {R}}^2):\mathbf{0 }\le (\phi (s),\psi (s))\le \mathbf{K },s\in {\mathbb {R}}\}. \end{aligned}$$

Denote

$$\begin{aligned} \left\{ \begin{array}{ll} f_{1}(u_{\eta }(s),v_{\eta }(s))=&{}\alpha _1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )e^{-\gamma _1\tau _1}u_{(\eta -\xi )}(-\tau _1)\\ &{}-2\alpha _1e^{-\gamma _1\tau _1}u_{\eta }(0)+a_1u^2_{\eta } (0)+b_1\left( \frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1}-u_{\eta } (0)\right) v_{\eta }(0),\\ f_{2}(u_{\eta }(s),v_{\eta }(s))=&{}\alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n} J_2(\xi )e^{-\gamma _2\tau _2}v_{(\eta -\xi )}(-\tau _2)\\ &{}-b_2\left( \frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1}-u_{\eta }(0)\right) v_{\eta }(0)-a_2v^2_{\eta }(0). \end{array} \right. \end{aligned}$$

It is easy to see that there exist \(L_i>0\) such that

$$\begin{aligned}&|f_i(u_{1\eta },v_{1\eta })-f_i(u_{2\eta },v_{2\eta })|\le L_i\parallel U-V \parallel \end{aligned}$$

for \(U=(u_{1\eta },v_{1\eta }),V=(u_{2\eta },v_{2\eta })\in C([-\tau ,0],{\mathbb {R}}^2)\) with \(\mathbf{0 }\le (u_{i\eta }(s),v_{i\eta }(s))\le \mathbf{K },s\in [-\tau ,0],i=1,2,\) where \(\tau =\max \{\tau _1,\tau _2\}\).

Lemma 2.1

\(f(\phi ,\psi )=(f_{1}(\phi ,\psi ),f_{2}(\phi ,\psi ))\) satisfies quasimonotone condition:

  1. (QM)

    there exist \(\beta _1>0\) and \(\beta _2>0\) such that

    $$\begin{aligned}&f_1(u_{1\eta }(s),v_{1\eta }(s))-f_1(u_{2\eta }(s),v_{2\eta }(s)) +\beta _1[u_{1\eta }(0)-u_{2\eta }(0)]\\&\quad \ge 2nD_1[u_{1\eta }(0)-u_{2\eta }(0)],\\&f_2(u_{1\eta }(s),v_{1\eta }(s))-f_2(u_{2\eta }(s),v_{2\eta }(s)) +\beta _2[v_{1\eta }(0)-v_{2\eta }(0)]\\&\quad \ge 2nD_2[v_{1\eta }(0)-v_{2\eta }(0)] \end{aligned}$$

    for \(u_{i\eta }(s),v_{i\eta }(s)\in C([-\tau ,0],{\mathbb {R}}),i=1,2,\) with \(\mathbf{0 }\le (u_{2\eta }(s),v_{2\eta }(s))\le (u_{1\eta }(s),v_{1\eta }(s)) \le \mathbf{K }\) for \(s\in [-\tau ,0]\).

Proof

For any \((u_{1\eta },v_{1\eta }),(u_{2\eta },v_{2\eta })\in C([-\tau ,0],{\mathbb {R}}^2)\) with \(\mathbf{0 }\le (u_{2\eta }(s),v_{2\eta }(s))\le (u_{1\eta } (s),v_{1\eta }(s))\le \mathbf{K }\), taking \(\beta _1>2\alpha _1e^{-\gamma _1\tau _1}+b_1k_2+2nD_1\), we have

$$\begin{aligned}&f_1(u_{1\eta },v_{1\eta })-f_1(u_{2\eta },v_{2\eta }) -2nD_1[u_{1\eta }(0)-u_{2\eta }(0)]\\&\quad \ge [-2\alpha _1e^{-\gamma _1\tau _1}-b_1v_{1\eta }(0)] [u_{1\eta }(0)-u_{2\eta }(0)]-2nD_1[u_{1\eta }(0)-u_{2\eta }(0)]\\&\quad \ge -(2\alpha _1e^{-\gamma _1\tau _1}+b_1k_2+2nD_1) [u_{1\eta }(0)-u_{2\eta }(0)]\\&\quad \ge -\beta _1[u_{1\eta }(0)-u_{2\eta }(0)]. \end{aligned}$$

Similarly, one can establish the inequality for \(f_2\). The proof is completed. \(\square \)

Definition 2.1

A traveling wave solution of (1.5) has the special form \(u_{1\eta }(t)=\phi (\sigma \cdot \eta +ct),u_{2\eta }(t)=\psi (\sigma \cdot \eta +ct)\), where \(\phi (\pm \infty )\) and \(\psi (\pm \infty )\) both exist, \(\sigma =(\sigma _1,\sigma _2,\ldots ,\sigma _n)\in {\mathbb {R}}^n\) is a unit vector, \(c>0\) is the wave speed, \((\phi ,\psi )\) is the wave profile.

Denoting \(\sigma \cdot \eta +ct\) by t, we find the solutions of the wave system

$$\begin{aligned} \left\{ \begin{array}{ll} c\phi '(t)=D_1\displaystyle \sum ^n_{k=1}[\phi (t+\sigma _k)-2\phi (t)+\phi (t-\sigma _k)] +f_1^c(\phi _t,\psi _t),\\ c\psi '(t)=D_2\displaystyle \sum ^n_{k=1}[\psi (t+\sigma _k)-2\psi (t)+\psi (t-\sigma _k)] +f_2^c(\phi _t,\psi _t) \end{array} \right. \end{aligned}$$
(2.1)

satisfying

$$\begin{aligned} \displaystyle \lim _{t\rightarrow -\infty }(\phi (t),\psi (t))=\mathbf{0 },\ \ \ \displaystyle \lim _{t\rightarrow +\infty }(\phi (t),\psi (t))=\mathbf{K }, \end{aligned}$$
(2.2)

where \(\phi _t(s)=\phi (t+s),\psi _t(s)=\psi (t+s),s\in [- c \tau ,0] ,f_i^c(\phi ,\psi ),i=1,2\), is defined by

$$\begin{aligned} \left\{ \begin{array}{ll} f^c_{1}(\phi _t,\psi _t)=&{}\alpha _1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1( \xi )e^{-\gamma _1\tau _1}\phi (t-\sigma \cdot \xi -c\tau _1)\\ &{}-2\alpha _1e^{-\gamma _1\tau _1}\phi (t)+a_1\phi ^2(t)+b_1\left( \frac{\alpha _1e^{-\gamma _1 \tau _1}}{a_1}-\phi (t)\right) \psi (t),\nonumber \\ f^c_{2}(\phi _t,\psi _t)=&{}\alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi ) e^{-\gamma _2\tau _2}\psi (t-\sigma \cdot \xi -c\tau _2)\\ &{}-b_2\left( \frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1}-\phi (t)\right) \psi (t)-a_2\psi ^2(t). \end{array} \right. \end{aligned}$$

Define \(H=(H_1,H_2):C_{[\mathbf{0 },\mathbf{K }]}({\mathbb {R}},{\mathbb {R}}^2)\rightarrow C({\mathbb {R}},{\mathbb {R}}^2)\) by

$$\begin{aligned} \left\{ \begin{array}{ll} H_1(\phi ,\psi )(t)=f_1^c(\phi _t,\psi _t)+\beta _1\phi (t)+D_1\displaystyle \sum ^n_{k=1} [\phi (t+\sigma _k)-2\phi (t)+\phi (t-\sigma _k)],\\ H_2(\phi ,\psi )(t)=f_2^c(\phi _t,\psi _t)+\beta _2\psi (t)+D_2\displaystyle \sum ^n_{k=1} [\psi (t+\sigma _k)-2\psi (t)+\psi (t-\sigma _k)]. \end{array} \right. \end{aligned}$$

Then, (2.1) can be rewritten as

$$\begin{aligned} \left\{ \begin{array}{ll} c\phi '=-\beta _1\phi +H_1(\phi ,\psi ),\\ c\psi '=-\beta _2\psi +H_2(\phi ,\psi ). \end{array} \right. \end{aligned}$$
(2.3)

Define \(F=(F_1,F_2):C_{[\mathbf{0 },\mathbf{K }]}({\mathbb {R}},{\mathbb {R}}^2)\rightarrow C({\mathbb {R}},{\mathbb {R}}^2)\) by

$$\begin{aligned} F_i(\phi ,\psi )(t)= & {} \frac{1}{c}e^{-\frac{\beta _i}{c}t}\displaystyle \int ^t_{-\infty } e^{\frac{\beta _i}{c}s}H_i(\phi ,\psi )(s)ds,\ \ i=1,2. \end{aligned}$$

Then, F is well defined, and for any \((\phi ,\psi )\in C_{[\mathbf{0 },\mathbf{K }]}({\mathbb {R}},{\mathbb {R}}^2)\), we have

$$\begin{aligned} c(F_i(\phi ,\psi ))'(t)=-\beta _i F_i(\phi ,\psi )(t)+H_i(\phi ,\psi )(t),\ \ \ i=1,2. \end{aligned}$$
(2.4)

We only need to find a fixed point of (2.4) satisfying (2.2).

Let \(\mu \in (0,\displaystyle \min \{\beta _1/c,\beta _2/c\})\) and equip \(C({\mathbb {R}},{\mathbb {R}}^2)\) with the norm \(|\cdot |_\mu \) defined by

$$\begin{aligned} |\Phi |_\mu =\displaystyle \sup _{t\in {\mathbb {R}}}|\Phi (t)|e^{-\mu |t|} \text{ and } B_\mu ({\mathbb {R}},{\mathbb {R}}^2)=\left\{ \Phi \in C({\mathbb {R}},{\mathbb {R}}^2):\displaystyle \sup _{t\in {\mathbb {R}}}|\Phi (t)|e^{-\mu |t|}<\infty \right\} . \end{aligned}$$

Then, we can prove that \((B_\mu ({\mathbb {R}},{\mathbb {R}}^2),|\cdot |_\mu )\) is a Banach space.

Now we give the definition of upper and lower solutions of (2.1), which is similar to that in [16].

Definition 2.2

A pair of functions \({\bar{\Phi }}=({\bar{\phi }},{\bar{\psi }})\in C({\mathbb {R}},{\mathbb {R}}^2)\) is called an upper solution of (2.1) if there exist finitely many constants \(T_i,i=1,\ldots ,p,\) such that \({\bar{\Phi }}\) is differentiable in \({\mathbb {R}}{\setminus } T,T:=\{T_i:i=1,\ldots ,p\}\), and satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} c{\bar{\phi }}'(t)\ge &{}D_1\displaystyle \sum ^n_{k=1}[{\bar{\phi }}(t+\sigma _k) -2{\bar{\phi }}(t)+{\bar{\phi }}(t-\sigma _k)]+f_1^c({\bar{\phi }}_t, {\bar{\psi }}_t)\ \ \text{ for } t\in {\mathbb {R}}{\setminus } T,\\ c{\bar{\psi }}'(t)\ge &{}D_2\displaystyle \sum ^n_{k=1}[{\bar{\psi }}(t+\sigma _k) -2{\bar{\psi }}(t)+{\bar{\psi }}(t-\sigma _k)]+f_2^c({\bar{\phi }}_t, {\bar{\psi }}_t)\ \ \text{ for } t\in {\mathbb {R}}{\setminus } T. \end{array} \right. \end{aligned}$$
(2.5)

A lower solution \(\underline{\Phi }=(\underline{\phi },\underline{\psi })\in C({\mathbb {R}},{\mathbb {R}}^2)\) of (2.1) can only be defined by reversing inequalities of (2.5).

In what follows, we assume that an upper solution \({\bar{\Phi }}=({\bar{\phi }},{\bar{\psi }})\) and a lower solution \(\underline{\Phi }=(\underline{\phi },\underline{\psi })\) of (2.1) satisfy

  1. (A1)

    \(\mathbf{0 }\le (\underline{\phi }(t),\underline{\psi }(t))\le ({\bar{\phi }}(t), {\bar{\psi }}(t))\le \mathbf{K };\)

  2. (A2)

    \(\displaystyle \lim _{t\rightarrow -\infty }(\underline{\phi }(t),\underline{\psi }(t)) =\mathbf{0 },\displaystyle \lim _{t\rightarrow +\infty }({\bar{\phi }}(t),{\bar{\psi }}(t)) =\mathbf{K };\)

  3. (A3)

    \(\sup _{s\le t}\underline{\phi }(s)\le {\bar{\phi }}(t)\) and \(\sup _{s\le t}\underline{\psi }(s)\le {\bar{\psi }}(t)\) for all \(t\in {\mathbb {R}};\)

  4. (A4)

    \(\sup _{t\in {\mathbb {R}}}\underline{\phi }(t)>0,\sup _{t\in {\mathbb {R}}}\underline{\psi }(t)>0.\)

We make the assumption on \(J_i,i=1,2.\)

(J) For any \(\xi \in \mathbb {Z}^n,J_i(\xi )\ge 0\) is symmetric for each component, \(\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_i(\xi )e^{\lambda |\xi |}<\infty \) for any \(\lambda \in \Bigl (0,\displaystyle \min \Bigl \{\frac{\beta _1}{c},\frac{\beta _2}{c}\Bigr \}\Bigr ),i=1,2.\)

From the definitions of H and F, we have the following lemma.

Lemma 2.2

Assume that (J) holds. Then,

  1. (i)

    If \((\phi ,\psi )\in C_{[\mathbf{0 },\mathbf{K }]}({\mathbb {R}},{\mathbb {R}}^2)\) is nondecreasing in \(t\in {\mathbb {R}},\) then \(H_i(\phi ,\psi )(t)\) is nondecreasing in \(t\in {\mathbb {R}}\); furthermore, \(F_i(\phi ,\psi )(t)\) is nondecreasing in \(t\in {\mathbb {R}},i=1,2;\)

  2. (ii)

    If \((\phi _i,\psi _i)\in C_{[\mathbf{0 },\mathbf{K }]}({\mathbb {R}},{\mathbb {R}}^2)\) satisfy \((\phi _2(t),\psi _2(t))\le (\phi _1(t),\psi _1(t))\) for \(t\in {\mathbb {R}},\) then \(H_i(\phi _2,\psi _2)(t)\le H_i(\phi _1,\psi _1)(t)\) for \(t\in {\mathbb {R}}\); furthermore, \( F_i(\phi _2,\psi _2)(t)\le F_i(\phi _1,\psi _1)(t)\) for \(t\in {\mathbb {R}},i=1,2.\)

Define

$$\begin{aligned}&\Gamma ([\underline{\phi },\underline{\psi }],[{\bar{\phi }},{\bar{\psi }}])\\&\quad = \left\{ (\phi ,\psi )\in C_{[\mathbf{0 },\mathbf{K }]}({\mathbb {R}},{\mathbb {R}}^2): \begin{array}{ll} &{}(i)\ (\phi (t),\psi (t)) \text{ is } \text{ nondecreasing } \text{ in } t\in {\mathbb {R}},\\ &{}(ii)\ (\underline{\phi }(t),\underline{\psi }(t))\le (\phi (t),\psi (t)) \le ({\bar{\phi }}(t),{\bar{\psi }}(t)),\ \ t\in {\mathbb {R}}\end{array} \right\} . \end{aligned}$$

(A3) implies that \(\Gamma ([\underline{\phi },\underline{\psi }],[{\bar{\phi }},{\bar{\psi }}])\) is nonempty since \((\displaystyle \sup _{s\le t}\underline{\phi }(s),\displaystyle \sup _{s\le t}\underline{\psi }(s))\in \Gamma ([\underline{\phi },\underline{\psi }],[{\bar{\phi }},{\bar{\psi }}])\).

Lemma 2.3

Assume that (J) holds. Then, \(F=(F_1,F_2):C_{[\mathbf{0 },\mathbf{M }]}({\mathbb {R}},{\mathbb {R}}^2)\rightarrow C({\mathbb {R}},{\mathbb {R}}^2)\) is continuous with respect to the norm \(|\cdot |_\mu \) in \(B_\mu ({\mathbb {R}},{\mathbb {R}}^2)\).

Proof

We only show that \(F_1:C_\mathbf{[0,K] }({\mathbb {R}},{\mathbb {R}}^2)\rightarrow C({\mathbb {R}},{\mathbb {R}}^2)\) is continuous with respect to the norm \(|\cdot |_\mu \) since \(F_2\) is similar. Without loss of generality, assume \(\sigma _k>0,i=1,2,\ldots ,n\). For \(\Phi =(\phi _1,\psi _1),\Psi =(\phi _2,\psi _2) \in C_\mathbf{[0,K] }({\mathbb {R}},{\mathbb {R}}^2)\), we have

$$\begin{aligned}&|H_1(\phi _1,\psi _1)(t)-H_1(\phi _2,\psi _2)(t)|e^{-\mu |t|}\\&\quad \le |f^c_{1}(\phi _{1t},\psi _{1t})-f^c_{1}(\phi _{2t}, \psi _{2t})|e^{-\mu |t|}+(\beta _1+2nD_1)|\phi _1(t)-\phi _2(t)|e^{-\mu |t|}\\&\qquad +D_1\displaystyle \sum ^n_{k=1}|\phi _1(t+\sigma _k)-\phi _2(t+\sigma _k)| +D_1\displaystyle \sum ^n_{k=1}|\phi _1(t-\sigma _k)-\phi _2(t-\sigma _k)|e^{-\mu |t|}\\&\quad \le L_1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )|\phi _1(t-\sigma \cdot \xi -c\tau _1)-\phi _2(t-\sigma \cdot \xi -c\tau _1)|e^{-\mu |t|}\\&\qquad +\Bigl (\beta _1+2nD_1+2D_1\displaystyle \sum ^n_{k=1}e^{\mu \sigma _k }\Bigr )| \phi _1-\phi _2|_\mu \\&\quad \le L_1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )e^{\mu |\sigma \cdot \xi +c\tau _1|} |\phi _1-\phi _2|_\mu +\Bigl (\beta _1 +2nD_1+2D_1\displaystyle \sum ^n_{k=1}e^{\mu \sigma _k }\Bigr )|\phi _1-\phi _2|_\mu \\&\quad \le \Bigg (L_1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )e^{\mu |\xi |}e^{\mu c\tau _1}+\beta _1+2nD_1+2D_1\displaystyle \sum ^n_{k=1}e^{\mu \sigma _k }\Bigg )|\Phi -\Phi |_\mu . \end{aligned}$$

Next we estimate \(F_1\).

$$\begin{aligned}&|F_1(\phi _1,\psi _1)(t)-F_1(\phi _2,\psi _2)(t)|e^{-\mu |t|}\\&\quad \le \frac{1}{c}e^{-\frac{\beta _1}{c}t}e^{-\mu |t|} \int ^t_{-\infty }e^{\frac{\beta _1}{c}s}|H_1(\phi _1,\psi _1)(s)-H_1 (\phi _2,\psi _2)(s)|ds\\&\quad \le |H_1(\phi _1,\psi _1)-H_1(\phi _2,\psi _2)|_\mu \frac{1}{c}e^{-\frac{\beta _1}{c}t}e^{-\mu |t|}\int ^t_{-\infty }e^{\frac{\beta _1}{c}s+\mu |s|}ds\\&\quad \le \frac{1}{\beta _1-c\mu }|H_1(\phi _1,\psi _1)-H_1(\phi _2,\psi _2)|_\mu . \end{aligned}$$

Therefore,

$$\begin{aligned}&|F_1(\phi _1,\psi _1)-F_1(\phi _2,\psi _2)|_\mu \\&\quad \le \frac{L_1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )e^{\mu |\xi |}e^{\mu c\tau _1}+\beta _1+2nD_1+2D_1\displaystyle \sum ^n_{k=1}e^{\mu \sigma _k }}{\beta _1-c\mu }|\Phi -\Psi |_\mu , \end{aligned}$$

which implies that \(F_1:C_\mathbf{[0,K] }({\mathbb {R}},{\mathbb {R}}^2)\rightarrow C({\mathbb {R}},{\mathbb {R}}^2)\) is continuous with respect to the norm \(|\cdot |_\mu \) in \(B_\mu ({\mathbb {R}},{\mathbb {R}}^2)\). This completes the proof. \(\square \)

By an analogous argument in [14,15,16, 22], we have the following results.

Lemma 2.4

Assume that (J) holds. Then, \(F(\Gamma ([\underline{\phi },\underline{\psi }],[{\bar{\phi }},{\bar{\psi }}])) \subset \Gamma ([\underline{\phi },\underline{\psi }],[{\bar{\phi }},{\bar{\psi }}])\) and \(F:\Gamma ([\underline{\phi },\underline{\psi }],[{\bar{\phi }},{\bar{\psi }}]) \rightarrow \Gamma ([\underline{\phi },\underline{\psi }],[{\bar{\phi }},{\bar{\psi }}])\) is compact with respect to the norm \(|\cdot |_\mu \).

Theorem 2.1

Assume that (J) and (1.4) hold. If (2.1) has an upper solution \(({\bar{\phi }},{\bar{\psi }})\in C_{[\mathbf{0 },\mathbf{K }]}({\mathbb {R}},{\mathbb {R}}^2)\) and a lower solution \((\underline{\phi },\underline{\psi })\in C_{[\mathbf{0 },\mathbf{K }]}({\mathbb {R}},{\mathbb {R}}^2)\) such that (A1)–(A4) are satisfied, then (2.1) has a monotone solution satisfying (2.2).

Proof

Combining with Lemmas 2.3, 2.4 and Schauder’s fixed point theorem, F has a fixed point \((\phi ^*,\psi ^*)\in \Gamma ([\underline{\phi },\underline{\psi }],[{\bar{\phi }},{\bar{\psi }}])\) satisfying (2.2). The proof is completed. \(\square \)

3 Construct Upper and Lower Solutions

In this section, we construct a pair of upper and lower solutions of (2.1). Let

$$\begin{aligned} \Delta (\lambda ,c):= & {} D_2\displaystyle \sum ^n_{k=1}(e^{\lambda \sigma _k} +e^{-\lambda \sigma _k}-2)-c\lambda \\&+\,\alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi )e^{-\gamma _2\tau _2} e^{-\lambda (\sigma \cdot \xi +c\tau _2)} -\frac{b_2\alpha _1e^{-\gamma _1\tau _1}}{a_1}. \end{aligned}$$

The following lemma is easy to check.

Lemma 3.1

Assume that (1.4) holds. Then, there exist \(c^*>0\) and \(\lambda ^*>0\) such that \(\Delta (\lambda ^*,c^{*})=0\) and \(\frac{\partial \Delta (\lambda ,c^*)}{\partial \lambda }|_{\lambda =\lambda ^*}=0\). Moreover, for \(c>c^*\), the equation \(\Delta (\lambda ,c)=0\) has two positive roots \(\lambda _1,\lambda _2\) with \(0<\lambda _1<\lambda ^*_1<\lambda _2\), and

$$\begin{aligned} \Delta _1(\lambda ,c) \left\{ \begin{array}{ll}>0,\ \ \ \ \ &{}\lambda<\lambda _1,\\<0,\ \ &{}\lambda _1<\lambda <\lambda _2, \\>0,\ \ \ \ \ &{}\lambda >\lambda _2, \end{array} \right. \end{aligned}$$

for \(0<c<c^*\), \(\Delta (\lambda ,c)>0 \) for \(\lambda \in {\mathbb {R}}.\)

For \(c>c^*\), define the continuous functions as follows:

$$\begin{aligned} {\bar{\phi }}(t)= \left\{ \begin{array}{ll} k_1e^{\lambda _1t},&{}t\le 0,\\ k_1,\ \ &{}t> 0, \end{array} \right. \ \ \ \ \ \ \ {\bar{\psi }}(t)= \left\{ \begin{array}{ll} k_2e^{\lambda _1t},&{}t\le 0,\\ k_2,\ \ &{}t> 0, \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \underline{\phi }(t)\equiv 0,\ \ \ \ \ \ t\in {\mathbb {R}},\ \ \ \ \ \underline{\psi }(t)= \left\{ \begin{array}{ll} k_2(e^{\lambda _1t}-qe^{\eta \lambda _1t}),&{}t\le t_0,\\ 0,\ \ &{}t> t_0, \end{array} \right. \end{aligned}$$

where \(q>0\) is large enough and will be determined later, \(\eta \in (1,\min \{2,\frac{\lambda _2}{\lambda _1}\}), t_0(q)=\frac{1}{(\eta -1)\lambda _1}\ln \frac{1}{q}<0\). \({\bar{\phi }}(t),{\bar{\psi }}(t),\underline{\phi }(t)\) and \(\underline{\psi }(t)\) satisfy (A1)-(A4) for large enough q. By the choice of \(\eta \), \(\Delta (\eta \lambda _1,c)<0\).

Lemma 3.2

Assume that (1.4) holds and \(D_1\le D_2\),

$$\begin{aligned}&\alpha _1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )e^{-\gamma _1\tau _1} e^{-\lambda _1(\sigma \cdot \xi +c\tau _1)} +\Bigl (-2\alpha _1e^{-\gamma _1\tau _1}+\frac{b_1\alpha _2e^{-\gamma _2\tau _2}}{a_2}\Bigr )\nonumber \\&\quad \le \alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi )e^{-\gamma _2 \tau _2}e^{-(\sigma \cdot \xi +\lambda c\tau _2)} -b_2\frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1}. \end{aligned}$$
(3.1)

Then, \(({\bar{\phi }}(t),{\bar{\psi }}(t))\) is an upper solution and \((\underline{\phi }(t),\underline{\psi }(t))\) is a lower solution of (2.1), respectively.

Proof

Without loss of generality, assume \(\sigma _k>0\). Otherwise, we only need to distinguish them from positive, negative or zero. We only verify \({\bar{\phi }}(t)\) and \(\underline{\phi }(t)\); the others are similar. For \({\bar{\phi }}(t)\), we have two cases to verify.

  1. (i)

    For \(t<0\), since \({\bar{\phi }}(t\pm \sigma _k)\le k_1e^{\lambda _1(t\pm \sigma _k)}\) and \({\bar{\phi }}(t-\sigma \cdot \xi -c\tau _1)\le k_1e^{\lambda _1(t-\sigma \cdot \xi -c\tau _1)}\), by (3.1), we have

    $$\begin{aligned}&D_1\displaystyle \sum ^n_{k=1}[{\bar{\phi }}(t+\sigma _k)-2{\bar{\phi }}(t)+{\bar{\phi }} (t-\sigma _k)]-c{\bar{\phi }}'(t)\\&\qquad +\,\alpha _1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )e^{-\gamma _1\tau _1} {\bar{\phi }}(t-\sigma \cdot \xi -c\tau _1)\\&\qquad -\,2\alpha _1e^{-\gamma _1\tau _1}{\bar{\phi }}(t)+a_1 {\bar{\phi }}^2(t)+b_1\Bigl (\frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1} -{\bar{\phi }}(t)\Bigr ){\bar{\psi }}(t)\\&\quad =D_1\displaystyle \sum ^n_{k=1}[{\bar{\phi }}(t+\sigma _k)-2{\bar{\phi }}(t) +{\bar{\phi }}(t-\sigma _k)]-c{\bar{\phi }}'(t)\\&\qquad +\,\alpha _1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )e^{-\gamma _1\tau _1} {\bar{\phi }}(t-\sigma \cdot \xi -c\tau _1)\\&\qquad -\,\alpha _1e^{-\gamma _1\tau _1}{\bar{\phi }}(t) +\Bigl (\frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1}-{\bar{\phi }}(t)\Bigr ) (-a_1{\bar{\phi }}(t)+b_1{\bar{\psi }}(t))\\&\quad =D_1\displaystyle \sum ^n_{k=1}[{\bar{\phi }}(t+\sigma _k)-2{\bar{\phi }}(t) +{\bar{\phi }}(t-\sigma _k)]-c{\bar{\phi }}'(t)\\&\qquad +\,\alpha _1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )e^{-\gamma _1\tau _1} {\bar{\phi }}(t-\sigma \cdot \xi -c\tau _1)\\&\qquad -\,\alpha _1e^{-\gamma _1\tau _1}{\bar{\phi }}(t)+e^{\lambda _1t} \Bigl (-\alpha _1e^{-\gamma _1\tau _1} +\frac{b_1\alpha _2e^{-\gamma _2\tau _2}}{a_2}\Bigr )\Bigl (\frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1}-{\bar{\phi }}(t)\Bigr )\\&\quad \le k_1e^{\lambda _1t}\Bigl [D_1\displaystyle \sum ^n_{k=1}(e^{\lambda _1\sigma _k}+e^ {-\lambda _1\sigma _k}-2)-c\lambda _1\\&\qquad +\,\alpha _1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )e^{-\gamma _1\tau _1}e^ {-\lambda _1(\sigma \cdot \xi +c\tau _1)} +\,\Bigl (-2\alpha _1e^{-\gamma _1\tau _1}+\frac{b_1\alpha _2e^{-\gamma _2 \tau _2}}{a_2}\Bigr )\Bigr ]\\&\quad \le k_1e^{\lambda _1t}\Delta (\lambda _1,c)=0. \end{aligned}$$
  2. (ii)

    For \(t>0\), in view of \({\bar{\phi }}(t\pm \sigma _k)\le k_1,{\bar{\phi }}(t-\sigma \cdot \xi -c\tau _1)\le k_1\), we have

    $$\begin{aligned}&D_1\displaystyle \sum ^n_{k=1}[{\bar{\phi }}(t+\sigma _k)-2{\bar{\phi }}(t) +{\bar{\phi }}(t-\sigma _k)]-c{\bar{\phi }}'(t)\\&\qquad +\,\alpha _1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )e^{-\gamma _1\tau _1} {\bar{\phi }}(t-\sigma \cdot \xi -c\tau _1)\\&\qquad -\,2\alpha _1e^{-\gamma _1\tau _1}{\bar{\phi }}(t)+a_1 {\bar{\phi }}^2(t)+b_1\Bigl (\frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1} -{\bar{\phi }}(t)\Bigr ){\bar{\psi }}(t)\\&\quad = D_1\displaystyle \sum ^n_{k=1}[{\bar{\phi }}(t+\sigma _k)-2k_1+{\bar{\phi }} (t-\sigma _k)]\nonumber \\&\qquad +\,\alpha _1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )e^{-\gamma _1\tau _1} {\bar{\phi }}(t-\sigma \cdot \xi -c\tau _1)-k_1\alpha _1e^{-\gamma _1\tau _1}\\&\quad \le \alpha _1\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_1(\xi )e^{-\gamma _1\tau _1} k_1-k_1\alpha _1e^{-\gamma _1\tau _1}=0. \end{aligned}$$

We have two cases to verify for \({\bar{\psi }}(t)\).

  1. (i)

    For \(t<0\), since \({\bar{\psi }}(t\pm \sigma _k)\le k_2e^{\lambda _1(t\pm \sigma _k)},b_2{\bar{\phi }}(t)\le a_2{\bar{\psi }}(t)\), by (1.4), we have

    $$\begin{aligned}&D_2\displaystyle \sum ^n_{k=1}[{\bar{\psi }}(t+\sigma _k)-2{\bar{\psi }}(t) +{\bar{\psi }}(t-\sigma _{k})]-c{\bar{\psi }}'(t)\\&\qquad +\,\alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi )e^{-\gamma _2\tau _2} {\bar{\psi }}(t-\sigma \cdot \xi -c\tau _2)\\&\qquad -\,b_2\Bigl (\frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1}-{\bar{\phi }}(t)\Bigr ) {\bar{\psi }}(t)-a_2{\bar{\psi }}^2(t)\\&\quad \le D_2\displaystyle \sum ^n_{k=1}[{\bar{\psi }}(t+\sigma _k) -2{\bar{\psi }}(t)+{\bar{\psi }}(t-\sigma _k)]-c{\bar{\psi }}'(t)\\&\qquad +\,\alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi )e^{-\gamma _2\tau _2} {\bar{\psi }}(t-\sigma \cdot \xi -c\tau _2)-\frac{b_2\alpha _1e^{-\gamma _1\tau _1}}{a_1}{\bar{\psi }}(t)\\&\quad \le k_2e^{\lambda _1t}\Delta (\lambda _1,c)=0. \end{aligned}$$
  2. (ii)

    For \(t>0\), since \({\bar{\psi }}(t\pm \sigma _k)\le k_2\) and \({\bar{\psi }}(t-\sigma \cdot \xi -c\tau _2)\le k_2\),

    $$\begin{aligned}&D_2\displaystyle \sum ^n_{k=1}[{\bar{\psi }}(t+\sigma _k)-2{\bar{\psi }}(t) +{\bar{\psi }}(t-\sigma _k)]-c{\bar{\psi }}'(t)\\&\qquad +\,\alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi ) e^{-\gamma _2\tau _2}{\bar{\psi }}(t-\sigma \cdot \xi -c\tau _2)\\&\qquad -\,b_2\Bigl (\frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1} -{\bar{\phi }}(t)\Bigr ){\bar{\psi }}(t)-a_2{\bar{\psi }}^2(t)\\&\quad = D_2\displaystyle \sum ^n_{k=1}[{\bar{\psi }}(t+\sigma _k)-2k_2 +{\bar{\psi }}(t-\sigma _k)]\\&\qquad +\,\alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi )e^{-\gamma _2\tau _2} {\bar{\psi }}(t-\sigma \cdot \xi -c\tau _2)-a_2{\bar{\psi }}^2(t)\\&\quad \le \alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi )e^{-\gamma _2\tau _2} k_2-a_2k_2^2=0. \end{aligned}$$

For \(\underline{\phi }(t)\equiv 0\), the proof is trivial. For \(\underline{\psi }(t)\), we have two cases to verify.

  1. (i)

    For \(t\le t_0\), since \(\underline{\psi }(t\pm \sigma _k)\ge k_2(e^{\lambda _1(t\pm \sigma _k)}-qe^{\eta \lambda _1(t\pm \sigma _k)}), \underline{\psi }(t-\sigma \cdot \xi -c\tau _2)\ge k_2(e^{\lambda _1(t-\sigma \cdot \xi -c\tau _2)}-qe^{\eta \lambda _1(t- \sigma \cdot \xi -c\tau _2)})\), we have

    $$\begin{aligned}&D_2\displaystyle \sum ^n_{k=1}[\underline{\psi }(t+\sigma _k)-2\underline{\psi }(t) +\underline{\psi }(t-\sigma _k)]-c\underline{\psi }'(t)\\&\qquad +\,\alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi )e^{-\gamma _2\tau _2} \underline{\psi }(t-\sigma \cdot \xi -c\tau _2)\\&\qquad -\,b_2\Bigl (\frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1} -\underline{\phi }(t)\Bigr )\underline{\psi }(t)-a_2\underline{\psi }^2(t)\\&\quad \ge D_2\displaystyle \sum ^n_{k=1}[\underline{\psi }(t+\sigma _k)-2\underline{\psi }(t) +\underline{\psi }(t-\sigma _k)]-c\underline{\psi }'(t)\\&\qquad +\,\alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi )e^{-\gamma _2 \tau _2}\underline{\psi }(t-\sigma \cdot \xi -c\tau _2)\\&\qquad -\,\frac{b_2\alpha _1e^{-\gamma _1\tau _1}}{a_1}\underline{\psi }(t)-a_2\underline{\psi }^2(t)\\&\quad \ge -k_2qe^{\eta \lambda _1t}\Delta _1(\eta \lambda _1,c) -a_2k_2^2(e^{\lambda _1t}-qe^{\eta \lambda _1t})^2\\&\quad \ge -\,k_2qe^{\eta \lambda _1t}\Delta _1(\eta \lambda _1,c) -a_2k_2^2(e^{\lambda _1t}-qe^{\eta \lambda _1t})e^{\lambda _1t}\\&\quad \ge -\,k_2e^{\eta \lambda _1t}[q\Delta _1(\eta \lambda _1,c) +a_2k_2e^{(2-\eta )\lambda _1 t}]\ge 0. \end{aligned}$$

    The last inequality holds since \(-t_0(q)\gg 0\) is sufficiently small for sufficiently large \(q>0\).

  2. (ii)

    For \(t> t_0\), it follows that

    $$\begin{aligned}&D_2\displaystyle \sum ^n_{k=1}[\underline{\psi }(t+\sigma _k)-2\underline{\psi }(t)+\underline{\psi }(t-\sigma _k)] -c\underline{\psi }'(t)\\&\qquad +\,\alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi )e^{-\gamma _2\tau _2} \underline{\psi }(t-\sigma \cdot \xi -c\tau _2)\\&\qquad -\,b_2\Bigl (\frac{\alpha _1e^{-\gamma _1\tau _1}}{a_1}-\underline{\phi }(t)\Bigr )\underline{\psi }(t) -a_2\underline{\psi }^2(t)\\&\quad = D_2\displaystyle \sum ^n_{k=1}\underline{\psi }(t-\sigma _k)+\alpha _2\displaystyle \sum _{\xi \in \mathbb {Z}^n}J_2(\xi )e^{-\gamma _2\tau _2} \underline{\psi }(t-\sigma \cdot \xi -c\tau _2)\ge 0. \end{aligned}$$

The proof is completed. \(\square \)

By Theorem 2.1, we have the following existence result.

Theorem 3.1

Assume that \(D_1\le D_2\), (1.4) and (3.1) hold. Then, for every \(c> c^*\), (1.5) has a monotone traveling wave solution \((\phi (\sigma \cdot \eta +ct),\psi (\sigma \cdot \eta +ct))\) with the wave speed c which connects 0 with K.