Traveling Wave Solutions in Temporally Discrete Lotka-Volterra Competitive Systems with Delays

Abstract

In this paper, we investigate the existence of traveling wave solution for temporally discrete Lotka Volterra competitive system with delays. By using the cross iteration method and Schauder’s fixed point theorem, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The obtained results makes up and improves the results of the existence of traveling wave solutions for this systems.

1 Introduction

Traveling wave solutions have been widely investigated for reaction diffusion systems, such as Britton [1], Hosono [2], Guo and Liang [3], Huang and Han [4]and the references cited therein. For time delayed reaction diffusion equations such as [5,6,7,8,9,10,11,12]and the references cited therein. For discrete reaction diffusion equations [13,14,15] and the references cited therein. For delayed lattice differential equations, Wu and Zou [16]use iterative scheme and the upper-lower solution method to prove the existence of traveling wave fronts of lattice differential equation. The problems on traveling wave solutions for other types of spatio-temporal delays see [15, 17,18,19,20], and references therein.

For some organisms of non-overlapping generations,temporally discrete and spatially continuous diffusion model will be more suitable than its corresponding time continuous diffusion model to study the dynamic behavior of a single species that living in a spatially continuous habitat in population ecology. Thus the study of the time discontinuous model is necessary, Lin and Li [21] used the same approach as [16] presented a temporally discrete reaction diffusion equation with delay

$$\begin{aligned} u_{n}(x)-u_{n-1}(x)=d\Delta u_{n}(x)+f(u_{n}(x),u_{n-\tau }(x)),~~~n\in \mathbb {N}, x\in \mathbb {R}, \end{aligned}$$

(1)

which can be considered as a temporal discretization of the following differential equation

$$\begin{aligned} \frac{\partial u(t,x)}{\partial t}=d\frac{\partial ^{2} u(t,x)}{\partial x^{2}}+f(u(t,x),u(t-\tau ,x))~~t\ge 0,~~ x\in \mathbb {R}. \end{aligned}$$

(2)

Wang and Zhang [22] by using Schauder’s fixed point theorem for (1) and establish the existence of traveling wave fronts. For more researches on this topic see [23,24,25,26,27].

In nature, the populations of both species are affected by their respective the effect of inherent growth rate, at the same time, in time of t and $t-\tau _{i}(i=1,2,3,4)$, as the population density of both its own and competing species increases, the growth rate of population density declined for both species.This suggests that the population density of both competing species is affected separately by their own time t,$t-\tau _{i}(i=1,3)$, and competing species density at timet,$t-\tau _{i}(i=2,4)$ is affected. the model is as follows:

$$\begin{aligned} \left\{ \begin{array}{lll} \frac{\partial u(x,t)}{\partial t}& =D_{1}\frac{\partial ^{2}u(x,t)}{\partial x^{2}}+r_{1}u(x,t)[1-a_{1}u(x,t)-b_{1}u(x,t-\tau _{1})\\ & ~~~-c_{1}v(x,t)-d_{1}v(x,t-\tau _{2})],\\ \frac{\partial v(x,t)}{\partial t}& =D_{2}\frac{\partial ^{2}v(x,t)}{\partial x^{2}}+r_{2}v(x,t)[1-a_{2}v(x,t)-b_{2}v(x,t-\tau _{3})\\ & ~~~-c_{2}u(x,t)-d_{2}u(x,t-\tau _{4})]. \end{array} \right. \end{aligned}$$

(3)

In system (3), if $a_{1}=0$,$c_{1}=0$,$a_{2}=0$,$c_{2}=0$, the equations was considered by Li [11], they use a cross iteration scheme and Schauder’s fixed point theorem established the existence of traveling wave solutions. Xia and Yu [23] apply nonstandard finite difference schemes and Euler’s method to the models of [11] and obtain the existence of traveling wave solutions for a class of temporally discrete reaction-diffusion systems with delays.

Motivated by the above works, we apply nonstandard finite difference schemes and Euler’s method to the models (3) and can obtain the discrete-time models

$$\begin{aligned} \left\{ \begin{array}{lll} u_{n}(x)-u_{n-1}(x)& =D_{1}\Delta u_{n}(x)+r_{1}u_{n}(x)[1-a_{1}u_{n}(x)-b_{1}u_{n-\tau _{1}} (x)\\ & ~~~-c_{1}v_{n}(x)-d_{1}v_{n-\tau _{2}}(x)],\\ v_{n}(x)-v_{n-1}(x)& =D_{2}\Delta v_{n}(x)+r_{2}v_{n}(x)[1-a_{2}v_{n}(x)-b_{2}v_{n-\tau _{3}}(x)\\ & ~~~-c_{2}u_{n}(x)-d_{2}u_{n-\tau _{4}}(x)], \end{array} \right. \end{aligned}$$

(4)

where $u_{n}(x)$, $v_{n}(x)$ are the densities of populations of two species at time n and location x respectively, $x\in \mathbb {R}$, $n\in \mathbb {Z}$. An interesting problem is that whether (3) and (4) can have similar dynamical behavior. In this paper,we will consider the existence of traveling wave solution of (4) by using the cross iteration method and Schauder’s fixed point theorem,which was used by [9, 10].

The organization of this paper is as follows. In section 2, we introduce abstract results and obtain the existence of traveling wave solutions for more general equations with discrete delays under some conditions. Section 3 is invoked to derive the existence of travelling waves by constructing a pair of upper and lower solutions for temporally discrete diffusion-competition systems (4).

2 Preliminaries

In this section, we will consider the existence of traveling wave solution for more general equations, its generalization with delays can be written as

$$\begin{aligned} \left\{ \begin{array}{l} u_{n}(x)-u_{n-1}(x)=D_{1}\Delta u_{n}(x)+f_{1}(u_{n-\tau _{1}}(x),v_{n-\tau _{2}}(x)),\\ v_{n}(x)-v_{n-1}(x)=D_{2}\Delta v_{n}(x)+f_{2}(u_{n-\tau _{3}}(x),v_{n-\tau _{4}}(x)), \end{array} \right. \end{aligned}$$

(5)

where $D_{1}, D_{2}>0$, $f_{i}: C([-c\tau ,0],\mathbb {R}^{2} )\rightarrow \mathbb {R}$ is a continuous function, $\tau =\max \limits _{1\le i\le 4}\{\tau _{i}\}$.

A traveling wave solution of (5) is a special solution with the form $u_{n}(x)=\phi (x+cn)$, $v_{n}(x)=\psi (x+cn)$, where $\phi , \psi \in C^{2}(\mathbb {R}, \mathbb {R})$ and c is a positive constant corresponding to the wave speed. Substituting $u_{n}(x)=\phi (x+cn)$, $v_{n}(x)=\psi (x+cn)$ into (5) and denoting $\phi _{s}(t)=\phi (t+s)$,$\psi _{s}(t)=\psi (t+s)$ and $x+cn$ by t, we obtain the following system

$$\begin{aligned} \left\{ \begin{array}{l} D_{1}\phi ^{''}(t)-\phi (t)+\phi (t-c)+f_{1}(\phi _{t}(-c\tau _{1}), \psi _{t}(-c\tau _{2}))=0,\\ D_{2}\psi ^{''}(t)-\psi (t)+\psi (t-c)+f_{2}(\phi _{t}(-c\tau _{3}), \psi _{t}(-c\tau _{4}))=0, \end{array} \right. \end{aligned}$$

(6)

here, $(\phi ,\psi )$ is called a profile of the traveling wave solution. Motivated by the background of traveling wave solution, we also require that $(\phi (t),\psi (t))$ satisfy the following asymptotic boundary conditions

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }\phi (t)=\phi _{-},~~\lim \limits _{t\rightarrow +\infty }\phi (t)=\phi _{+},~~ \lim \limits _{t\rightarrow -\infty }\psi (t)=\psi _{-},~~\lim \limits _{t\rightarrow +\infty }\psi (t)=\psi _{+},\nonumber \\ \end{aligned}$$

(7)

where $(\phi _{-},\psi _{-})$ and $(\phi _{+},\psi _{+})$ are two equilibria of (5). Without loss of generality, we may assume that $\phi _{-}=0$, $\psi _{-}=0$ and $\phi _{+}=k_{1}>0$, $\psi _{+}=k_{2}>0$. Thus (7) reads as

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }\phi (t)=0,~~\lim \limits _{t\rightarrow +\infty }\phi (t)=k_{1},~~ \lim \limits _{t\rightarrow -\infty }\psi (t)=0,~~\lim \limits _{t\rightarrow +\infty }\psi (t)=k_{2}. \end{aligned}$$

(8)

For convenience of statements, we make the following hypothesis:

(P1) There exists $k=(k_{1},k_{2})$ with $k_{i}>0$ such that

$$f_{i}(0,0)=f_{i}(k_{1},k_{2})=0,~ \text {for}~ i=1,2.$$

(P2) There exist two positive constants $L_{1}>0$ and $L_{2}>0$ such that

$$|f_{1}(\phi _{1}, \psi _{1})-f_{1}(\phi _{2}, \psi _{2})|\le L_{1}\Vert \Phi -\Psi \Vert ,$$

$$|f_{2}(\phi _{1}, \psi _{1})-f_{2}(\phi _{2}, \psi _{2})|\le L_{2}\Vert \Phi -\Psi \Vert ,$$

for $\Phi =(\phi _{1}, \psi _{1})$, $\Psi =(\phi _{2}, \psi _{2})\in C([-c\tau , 0], \mathbb {R}^{2})$ with $0\le \phi _{i}(s), \psi _{i}(s)\le M_{i},~s\in [-c\tau , 0]$ for $M=(M_{1},M_{2}),~~M_{j}\ge k_{j}$, $\tau = \max \limits _{1\le i\le 4}\{\tau _{i}\}$.

Denote

$$C_{[0,M]}(\mathbb {R}, \mathbb {R}^{2})=\{(\phi ,\psi )\in C(\mathbb {R}, \mathbb {R}^{2}): 0\le \phi (s)\le M_{1}, 0\le \psi (s)\le M_{2}, s\in \mathbb {R}\}.$$

For any given positive constants $\beta _1>0, \beta _2>0$, define the operator $H=(H_{1},H_{2}): C_{[0,M]}(\mathbb {R},\mathbb {R}^{2})\rightarrow C(\mathbb {R},\mathbb {R}^{2})$ by

$$\begin{aligned} H_{1}(\phi ,\psi )(t)= & f_{1}(\phi _{t}(-c\tau _{1}), \psi _{t}(-c\tau _{2})) +(\beta _{1}-1)\phi (t)+\phi (t-c), \end{aligned}$$

(9)

$$\begin{aligned} H_{2}(\phi ,\psi )(t)= & f_{2}(\phi _{t}(-c\tau _{3}), \psi _{t}(-c\tau _{4})) +(\beta _{2}-1)\psi (t)+\psi (t-c). \end{aligned}$$

(10)

In terms of the expressions of $H_{1}$ and $H_{2}$, system (5) can be rewritten as

$$\begin{aligned} \left\{ \begin{array}{l} d_{1}\phi ^{''}(t)-\beta _{1}\phi (t)+H_{1}(\phi ,\psi )(t)=0,\\ d_{2}\psi ^{''}(t)-\beta _{2}\psi (t)+H_{2}(\phi ,\psi )(t)=0. \end{array} \right. \end{aligned}$$

(11)

For $(\phi ,\psi )\in C_{[0,M]}(\mathbb {R},\mathbb {R}^{2})$, define $F=(F_{1},F_{2}): C_{[0,M]}(\mathbb {R},\mathbb {R}^{2}) \rightarrow C(\mathbb {R},\mathbb {R}^{2})$ by

$$\begin{aligned} F_{1}(\phi ,\psi )(t)= & \frac{1}{d_{1}(\lambda _{2}-\lambda _{1})} [\int _{-\infty }^{t}e^{\lambda _{1}(t-s)} H_{1}(\phi ,\psi )(s)ds\\ & +\int _{t}^{\infty }e^{\lambda _{2}(t-s)} H_{1}(\phi ,\psi )(s)ds],\\ F_{2}(\phi ,\psi )(t)= & \frac{1}{d_{2}(\lambda _{4}-\lambda _{3})} [\int _{-\infty }^{t}e^{\lambda _{3}(t-s)} H_{2}(\phi ,\psi )(s)ds\\ & +\int _{t}^{\infty }e^{\lambda _{4}(t-s)} H_{2}(\phi ,\psi )(s)ds], \end{aligned}$$

where

$$\lambda _{1}=-\sqrt{\frac{\beta _{1}}{d_{1}}},~~\lambda _{2}= \sqrt{\frac{\beta _{1}}{d_{1}}}, ~~\lambda _{3}=-\sqrt{\frac{\beta _{2}}{d_{2}}},~~\lambda _{4}= \sqrt{\frac{\beta _{2}}{d_{2}}}.$$

It is easy to show that $F=(F_{1},F_{2})$ is well defined and for any $(\phi ,\psi )\in C_{[0, M]}(\mathbb {R},\mathbb {R}^{2})$, $F_{1}(\phi ,\psi )$ and $F_{2}(\phi ,\psi )$ satisfy

$$\begin{aligned} \left\{ \begin{array}{l} d_{1}F_{1}(\phi ,\psi )^{''}(t)-\beta _{1}F_{1}(\phi ,\psi )(t) +H_{1}(\phi ,\psi )(t)=0,\\ d_{2}F_{2}(\phi ,\psi )^{''}(t)-\beta _{2}F_{2}(\phi ,\psi )(t) +H_{2}(\phi ,\psi )(t)=0. \end{array} \right. \end{aligned}$$

(12)

Thus, if $F(\phi ,\psi )=(F_{1}(\phi ,\psi ),F_{2}(\phi ,\psi ))=(\phi ,\psi )$, i.e., $(\phi ,\psi )$ is a fixed point of F, then (11) has a solution $(\phi ,\psi )$. If this solution further satisfies the boundary condition (8), then it is a traveling wave solution of (5).

In order to obtain a fixed point of F,we propose a condition on the reaction terms:

(P3)   There exist two positive constants $\beta _{1}$ and $\beta _{2}$ such that

$$f_{1}(\phi _{1}(-c\tau _{1}), \psi _{1}(-c\tau _{2}))-f_{1}(\phi _{2}(-c\tau _{1}), \psi _{1}(-c\tau _{2}))+(\beta _{1}-1)[\phi _{1}(0)-\phi _{2}(0)]\ge 0,$$

$$f_{1}(\phi _{1}(-c\tau _{1}), \psi _{1}(-c\tau _{2}))-f_{1}(\phi _{1}(-c\tau _{1}), \psi _{2}(-c\tau _{2}))\le 0,$$

$$f_{2}(\phi _{1}(-c\tau _{3}), \psi _{1}(-c\tau _{4}))-f_{2}(\phi _{1}(-c\tau _{3}), \psi _{2}(-c\tau _{4}))+(\beta _{2}-1)[\psi _{1}(0)-\psi _{2}(0)]\ge 0,$$

$$f_{2}(\phi _{1}(-c\tau _{3}), \psi _{1}(-c\tau _{4}))-f_{2}(\phi _{2}(-c\tau _{3}), \psi _{1}(-c\tau _{4}))\le 0,$$

for any $\phi _{1}(s),~\phi _{2}(s),~\psi _{1}(s),~\psi _{2}(s) \in C([-c\tau , 0], \mathbb {R})$ with

$(i)~~ 0\le \phi _{2}(s)\le \phi _{1} (s)\le M_{1}, 0\le \psi _{2}(s)\le \psi _{1} (s)\le M_{2}, s \in [-c\tau , 0],$

$(ii)~~ e^{\beta _{1}s}[\phi _{1}(s)-\phi _{2}(s)] ~\text {and} ~e^{\beta _{2}s}[\psi _{1}(s)-\psi _{2}(s)]~\text {are nondecreasing in }~s \in [-c\tau , 0].$

In the following, we give the definition of weak upper and lower solutions of system (5).

Definition 1

  A pair of continuous functions $\overline{\Phi }(t)=(\overline{\phi }(t),\overline{\psi }(t))$ and $\underline{\Phi }(t)= (\underline{\phi }(t),\underline{\psi }(t))$ are called a pair of weak upper and lower solutions of (5), respectively, if there exist constants $T_{i}$ $(i=1, 2, \cdots , m)$ such that $\overline{\Phi }$ and $\underline{\Phi }$ are two continuously differentiable functions in $\mathbb {R}\setminus \{T_{i}, i=1, 2, \cdots , m\}$ and satisfy

$$\begin{aligned} \left\{ \begin{array}{l} d_{1}\overline{\phi }^{''}(t)-\overline{\phi }(t) +\overline{\phi }(t-c) +f_{1}(\overline{\phi }_{t}(-c\tau _{1}), \underline{\psi }_{t}(-c\tau _{2}))\\ ~~~~\le 0, t\in \mathbb {R}\setminus \{T_{i},i=1,2,\cdots , m\},\\ d_{2}\overline{\psi }^{''}(t)-\overline{\psi }(t) +\overline{\psi }(t-c) +f_{2}(\overline{\phi }_{t}(-c\tau _{3}), \overline{\psi }_{t}(-c\tau _{4}))\\ ~~~~\le 0, t\in \mathbb {R}\setminus \{T_{i},i=1,2,\cdots , m\}, \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{l} d_{1}\underline{\phi }^{''}(t)-\underline{\phi }(t) +\underline{\phi }(t-c) +f_{1}(\underline{\phi }_{t}(-c\tau _{1}), \overline{\psi }_{t}(-c\tau _{2}))\\ ~~~~\ge 0, t\in \mathbb {R}\setminus \{T_{i},i=1,2,\cdots , m\},\\ d_{2}\underline{\psi }^{''}(t)-\underline{\psi }(t) +\underline{\psi }(t-c) +f_{2}(\underline{\phi }_{t}(-c\tau _{3}), \underline{\psi }_{t}(-c\tau _{4}))\\ ~~~~\ge 0, t\in \mathbb {R}\setminus \{T_{i},i=1,2,\cdots , m\}. \end{array} \right. \end{aligned}$$

For $\mu \in (0,\min \{\lambda _{2},\lambda _{4}\})$, $C(\mathbb {R}, \mathbb {R}^{2})$ can be equipped with the exponential decay norm defined by $|\Phi |_{\mu }=\sup \limits _{t\in \mathbb {R}}|\Phi (t)|e^{-\mu |t|}.$ Let

$$B_{\mu }(\mathbb {R}, \mathbb {R}^{2})=\{\Phi \in C(\mathbb {R}, \mathbb {R}^{2}):~|\Phi |_{\mu }<\infty \}.$$

Then it is easy to check that $(B_{\mu }(\mathbb {R}, \mathbb {R}^{2}),|\cdot |_{\mu } )$ is a Banach space.

In what follows, we assume that there exists a pair of weak upper and lower solutions $(\overline{\phi }(t), \overline{\psi }(t))$ and $(\underline{\phi }(t), \underline{\psi }(t))$ to (5) satisfying

(A1)  $(0,0)\le (\underline{\phi }(t),\underline{\psi }(t))\le (\overline{\phi }(t),\overline{\psi }(t))\le (M_{1},M_{2})$,

(A2)  $\lim \limits _{t\rightarrow -\infty }(\overline{\phi }(t),\overline{\psi }(t))=(0,0)$, $\lim \limits _{t\rightarrow \infty }(\underline{\phi }(t),\underline{\psi }(t))= \lim \limits _{t\rightarrow \infty }(\overline{\phi }(t),\overline{\psi }(t))=(k_{1},k_{2})$,

(A3)  $\overline{\phi }'(t^{+}) \le \overline{\phi }'(t^{-})$, $\overline{\psi }'(t^{+}) \le \overline{\psi }'(t^{-})$,     $\underline{\phi }'(t^{+}) \ge \underline{\phi }'(t^{-})$, $\underline{\psi }'(t^{+}) \ge \underline{\psi }'(t^{-})$,

Define the set of profiles by

$$\Gamma := \{(\phi ,\psi )\in C_{[0,M]}(\mathbb {R},\mathbb {R}^{2}):~\underline{\phi }(t)\le \phi (t)\le \overline{\phi }(t)~{\text{ and }} ~\underline{\psi }(t)\le \psi (t)\le \overline{\psi }(t) \}.$$

It is easy to see that $\Gamma$ is nonempty. In fact, by (A1), we know that $(\underline{\phi }(t), \underline{\psi }(t))$ and $(\overline{\phi }(t), \overline{\psi }(t))$ satisfy. Moreover, it is obvious that $\Gamma$ is convex, closed and bounded in $B_{\mu }(\mathbb {R}, \mathbb {R}^{2})$.

Now we explore some basic properties of the operator H and F.

Lemma 1

Assume that (P1),(P2) and (P3) hold. Then we have

$$\begin{aligned} \left\{ \begin{array}{l} H_{1}(\phi _{2},\psi _{1})(t)\le H_{1}(\phi _{1},\psi _{1})(t),\\ H_{1}(\phi _{1},\psi _{1})(t)\le H_{1}(\phi _{1},\psi _{2})(t),\\ H_{2}(\phi _{2},\psi _{2})(t)\le H_{2}(\phi _{1},\psi _{1})(t), \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{l} F_{1}(\phi _{2},\psi _{1})(t)\le F_{1}(\phi _{1},\psi _{1})(t),\\ F_{1}(\phi _{1},\psi _{1})(t)\le F_{1}(\phi _{1},\psi _{2})(t),\\ F_{2}(\phi _{2},\psi _{2})(t)\le F_{2}(\phi _{1},\psi _{1})(t), \end{array} \right. \end{aligned}$$

for any $\phi _{i},\psi _{i} \in C([-c\tau ,0],\mathbb {R}), ~i=1,2$, with $0\le \phi _{2}(s)\le \phi _{1}(s)\le M_{1},~~0\le \psi _{2}(s)\le \psi _{1}(s)\le M_{2}$.

Next, we further explore the profile of the operator F.

Lemma 2

Assume (P1),(P2) holds, then $F=(F_{1},F_{2})$ is continuous with respect to the norm $|\cdot |_{\mu }$ in $B_{\mu }( \mathbb {R},\mathbb {R}^{2})$.

In order to apply Schauder’s fixed point theorem, we must prove

Lemma 3

If (P3) hold, then $F(\Gamma )\subset \Gamma$.

Lemma 4

If (P3) hold, then the operator $F: \Gamma \rightarrow \Gamma$ is compact with respect to the delay norm $|\cdot |_{\mu }$.

Since the proofs of lemma1–4 are similar to [11, 20, 23], so we omit it here.

Now, we are in the position to state and prove the following existence theorem.

Theorem 5

Auppose that (P1),(P2) and (P3) hold. If (5) has a pair of weak upper $(\overline{\phi },\overline{\psi })$ and weak lower solutions $(\underline{\phi },\underline{\psi })$ satisfying (A1)-(A3), then system (5) has a traveling wave solution.

Proof

It is easy to verify that $\Gamma$ is a nonempty, closed and convex subset of $B_{\mu }(\mathbb {R},\mathbb {R}^{2})$, combining lemma 1–4 with schauder’s fixed theorem, we know that there exists a fixed point $(\phi ^{*}(t),\psi ^{*}(t))$ of F in $\Gamma$. In order to show this fixed point is traveling wave solution, we need to verify the boundary condition (8).

By (A2) and the fact that $0\le (\underline{\phi }(t),\underline{\psi }(t))\le (\phi ^{*}(t),\psi ^{*}(t))\le (\overline{\phi }(t),\overline{\psi }(t)) \le (M_{1},M_{2})$, we know that $\lim \limits _{t\rightarrow -\infty }(\phi ^{*}(t),\psi ^{*}(t))=(0,0)$, and $\lim \limits _{t\rightarrow +\infty }(\phi ^{*}(t),\psi ^{*}(t))=(k_{1},k_{2})$. Therefore, the fixed points $(\phi ^{*}(t),\psi ^{*}(t))$ satisfies the boundary conditions. This completes the proof.  $\square$

3 Traveling Wave Solutions of (4)

In the section, we shall apply the result of section 2 to temporally discrete for competitive system (4). Let $a_{1}+b_{1}>c_{2}+d_{2}$, $a_{2}+b_{2}>c_{1}+d_{1}$, system (4) have four steady states namely $E_{1}(0,0)$,$E_{2}(0,\frac{1}{a_{2}+b_{2}})$, $E_{3}(\frac{1}{a_{1}+b_{1}},0)$, $E_{4}(k_{1},k_{2})$, where $k_{1}=\frac{(a_{2}+b_{2})-(c_{1}+d_{1})}{(a_{1}+b_{1})(a_{2}+b_{2})- (c_{1}+d_{1})(c_{2}+d_{2})}>0$, $k_{2}=\frac{(a_{1}+b_{1})-(c_{2}+d_{2})}{(a_{1}+b_{1})(a_{2}+b_{2})- (c_{1}+d_{1})(c_{2}+d_{2})}>0$. Assume that $c>0$, and substituting

$$u_{n}(x)=\phi (x+cn)=\phi (t), ~~v_{n}(x)=\psi (x+cn)=\psi (t), ~~t=x+cn,$$

into equation (4) then the corresponding wave system is

$$\begin{aligned} \left\{ \begin{array}{lll} D_{1}\phi ^{''}(t)-\phi (t)+\phi (t-c)+r_{1}\phi (t)[1-a_{1}\phi (t)-b_{1} \phi (t-c\tau _{1})\\ ~~~-c_{1}\psi (t)-d_{1}\psi (t-c\tau _{2})]=0,\\ D_{2}\psi ^{''}(t)-\psi (t)+\psi (t-c)+r_{2}\psi (t)[1-a_{2}\psi (t)-b_{2} \psi (t-c\tau _{3})\\ ~~~-c_{2}\phi (t)-d_{2}\phi (t-c\tau _{4})]=0, \end{array} \right. \end{aligned}$$

(13)

where, $\phi ,\psi \in c([-c\tau ,0],\mathbb {R})$,$\tau =\max \{\tau _{1},\tau _{2}, \tau _{3},\tau _{4}\}$. We are interested in solution of (13) satisfy the following asymptotic boundary condition

$$\begin{aligned} \lim \limits _{t\rightarrow -\infty }\phi (t)=0,~~\lim \limits _{t\rightarrow +\infty }\phi (t)=k_{1},~~ \lim \limits _{t\rightarrow -\infty }\psi (t)=0,~~\lim \limits _{t\rightarrow +\infty }\psi (t)=k_{2}. \end{aligned}$$

(14)

Define $f(\phi ,\psi )=(f_{1}(\phi ,\psi ),f_{2}(\phi ,\psi ))$ by

$$f_{1}(\phi ,\psi )=r_{1}\phi (0)[1-a_{1}\phi (0)-b_{1}\phi (-c\tau _{1}) -c_{1}\psi (0)-d_{1}\psi (-c\tau _{2})],$$

$$f_{2}(\phi ,\psi )=r_{2}\psi (0)[1-a_{2}\psi (0)-b_{2}\psi (-c\tau _{3}) -c_{2}\phi (0)-d_{2}\phi (-c\tau _{4})].$$

Obviously, $f_{1}$ and $f_{2}$ satisfies (P1) and (P2). Now, let us prove that $f(\phi ,\psi )$ satisfies (P3).

Lemma 6

When $\tau _{1},\tau _{3}$ are small enough, the function $f(\phi ,\psi )$ satisfies (P3).

Proof

For any $0\le \phi _{2}(s)\le \phi _{1}(s)\le M_{1}, 0\le \psi _{2}(s)\le \psi _{1}(s)\le M_{2}$,$s\in [-c\tau ,0]$,where $M_{1}>k_{1}$,$M_{2}>k_{2}$.

$$\begin{aligned} & f_{1}(\phi _{1},\psi _{1})-f_{1}(\phi _{2},\psi _{1})\\ & =[r_{1}-r_{1}a_{1}(\phi _{1}(0)+\phi _{2}(0))-r_{1}c_{1}\psi _{1}(0)] (\phi _{1}(0)-\phi _{2}(0))\\ & ~~~-d_{1}r_{1}\psi _{1}(-c\tau _{2})[\phi _{1}(0)-\phi _{2}(0)] -b_{1}r_{1}[\phi _{1}(0)\phi _{1}(-c\tau _{1})-\phi _{2}(0)\phi _{2} (-c\tau _{1})]\\ & \ge [r_{1}-2r_{1}a_{1}M_{1}-r_{1}c_{1}M_{2}-d_{1}r_{1}M_{2}] (\phi _{1}(0)-\phi _{2}(0))\\ & ~~~-b_{1}r_{1}\phi _{1}(-c\tau _{1})[\phi _{1}(0)-\phi _{2}(0)] -b_{1}r_{1}[\phi _{1}(-c\tau _{1})-\phi _{2}(-c\tau _{1})]\\ & \ge r_{1}[1-2a_{1}M_{1}-b_{1}M_{1}-c_{1}M_{2}-d_{1}M_{2}] (\phi _{1}(0)-\phi _{2}(0))\\ & ~~~-b_{1}r_{1}\phi _{2}(0)e^{\beta _{1}c\tau _{1}}e^{-\beta _{1}c\tau _{1}} [\phi _{1}(-c\tau _{1})-\phi _{2}(-c\tau _{1})]\\ & \ge r_{1}[1-2a_{1}M_{1}-b_{1}M_{1}-c_{1}M_{2}-d_{1}M_{2}-b_{1}M_{1} e^{\beta _{1}c\tau _{1}}] (\phi _{1}(0)-\phi _{2}(0)). \end{aligned}$$

If we choose $\beta _{1}>0$, such that

$$\beta _{1}-1>r_{1}[2a_{1}M_{1}+2b_{1}M_{1}+c_{1}M_{2}+d_{1}M_{2} -1],$$

then, for $\tau _{1}$ small enough, we obtain

$$\beta _{1}-1>r_{1}[2a_{1}M_{1}+b_{1}M_{1}+c_{1}M_{2}+d_{1}M_{2} b_{1}M_{1}e^{\beta _{1}c\tau _{1}}-1].$$

Thus

$$f_{1}(\phi _{1},\psi _{1})-f_{1}(\phi _{2},\psi _{1})+(\beta _{1}-1) (\phi _{1}(0)-\phi _{2}(0))\ge 0.$$

On the other hand,

$$f_{1}(\phi _{1},\psi _{1})-f_{1}(\phi _{1},\psi _{2}) \le 0.$$

In a similar way, there exists $\beta _{2}>0$ such that $f_{2}(\phi _{1},\psi _{1})-f_{2}(\phi _{1},\psi _{2})+(\beta _{2}-1) (\phi _{1}(0)-\psi _{2}(0))\ge 0$ and $f_{2}(\phi _{1},\psi _{1})-f_{2}(\phi _{2},\psi _{1})\le 0$. The proof is completed.  $\square$

we need to construct a weak upper and a weak lower solution of (13)satisfying the conditions in Theorem 2.1.

Define

$$\Delta _{1c}(\lambda )=D_{1}\lambda ^{2}+e^{-\lambda c}+r_{1}-1,$$

$$\Delta _{2c}(\lambda )=D_{2}\lambda ^{2}+e^{-\lambda c}+r_{2}-1,$$

then it easy obtain the following lemma

Lemma 7

Let $0<r_{1}<1$, $0<r_{2}<1$. Then there exists $c^{*}>0$ such that for $c>c^{*}$, $\Delta _{1c}(\lambda )$, $\Delta _{2c}(\lambda )$, respectively, has two positive real roots $\lambda _{1},\lambda _{2}$, $\lambda _{3},\lambda _{4}$ with $\lambda _{1}<\lambda _{2}$, $\lambda _{3}<\lambda _{4}$. Moreover,

$$\begin{aligned} \Delta _{1c}(\lambda )=\left\{ \begin{array}{lll}>0,~~~\text {for }\lambda<\lambda _{1};\\<0,~~~\text {for }\lambda \in (\lambda _{1},\lambda _{2});\\>0,~~~\text {for }\lambda>\lambda _{2}, \end{array} \right. ~\text {and}~~~ \Delta _{2c}(\lambda )=\left\{ \begin{array}{lll}>0,~~~\text {for }\lambda<\lambda _{3};\\ <0,~~~\text {for }\lambda \in (\lambda _{3},\lambda _{4});\\>0,~~~\text {for }\lambda >\lambda _{4}. \end{array} \right. \end{aligned}$$

The proof of the Lemma is easy and we omit it.

From now on, we assume that $c>c^{*}$ with $c^{*}$ given by lemma 7.

For fixed $\eta \in (1,\min \{2,\frac{\lambda _{2}}{\lambda _{1}}, \frac{\lambda _{4}}{\lambda _{3}},\frac{\lambda _{1}+\lambda _{3}}{\lambda _{1}} \frac{\lambda _{1}+\lambda _{3}}{\lambda _{3}}\})$, and a large constant $q>0$, we define two functions $g_{1}(t)=e^{\lambda _{1}t}-qe^{\eta \lambda _{1}t}$ and $g_{2}(t)=e^{\lambda _{3}t}-qe^{\eta \lambda _{3}t}$, then it easily learn that $g_{1}(t)$ and $g_{2}(t)$ have global maximum $m_{1}$, $m_{2}$, respectively, and there exist $t_{1}=\frac{1}{\lambda _{1}(\eta -1)}\ln \frac{1}{\eta q} <0$ and $t_{3}=\frac{1}{\lambda _{3}(\eta -1)}\ln \frac{1}{\eta q}<0$ with $e^{\lambda _{1}t_{1}}-qe^{\eta \lambda _{1}t_{1}} =m_{1}$, $e^{\lambda _{3}t_{3}}-qe^{\eta \lambda _{3}t_{3}} =m_{2}$. Therefore, for any given $\lambda >0$ there exist $\varepsilon _{2}>0$ and $\varepsilon _{4}>0$ such that $k_{1}-\varepsilon _{2}e^{-\lambda t_{1}}=m_{1}$, $k_{3}-\varepsilon _{4}e^{-\lambda t_{3}}=m_{2}$.

Since $a_{1}+b_{1}>c_{2}+d_{2}$, $a_{2}+b_{2}>c_{1}+d_{1}$, there exist $\varepsilon _{0}>0$,$\varepsilon _{1}>0$,$\varepsilon _{3}>0$ such that

$$\begin{aligned} & (a_{1}+b_{1})\varepsilon _{1}-(c_{1}+d_{1})\varepsilon _{4}>\varepsilon _{0}, (a_{1}+b_{1})\varepsilon _{2}-(c_{1}+d_{1})\varepsilon _{3}>\varepsilon _{0}, \end{aligned}$$

(15)

$$\begin{aligned} & (a_{2}+b_{2})\varepsilon _{3}-(c_{2}+d_{2})\varepsilon _{2}>\varepsilon _{0}, (a_{2}+b_{2})\varepsilon _{4}-(c_{2}+d_{2})\varepsilon _{1}>\varepsilon _{0}, \end{aligned}$$

(16)

Let $q>0$ large enough and $\lambda >0$ small enough be given, for the above constants and suitable constants $t_{2}$, $t_{4}$, define the following continuous functions

$$\begin{aligned} \overline{\phi }(t)=\left\{ \begin{array}{lll} e^{\lambda _{1}t},~~~~~~~~~~~~t\le t_{2},\\ k_{1}+\varepsilon _{1}e^{-\lambda t},~~~t\ge t_{2}, \end{array} \right. ~~~~~ \overline{\psi }(t)=\left\{ \begin{array}{lll} e^{\lambda _{3}t},~~~~~~~~~~~~t\le t_{4},\\ k_{2}+\varepsilon _{3}e^{-\lambda t},~~~t\ge t_{4}, \end{array} \right. \end{aligned}$$

$$\begin{aligned} \underline{\phi }(t)=\left\{ \begin{array}{lll} e^{\lambda _{1}t}-qe^{\eta \lambda _{1}t},~~~t\le t_{1},\\ k_{1}-\varepsilon _{2}e^{-\lambda t},~~~t\ge t_{1}, \end{array} \right. ~~~~~ \underline{\psi }(t)=\left\{ \begin{array}{lll} e^{\lambda _{3}t}-qe^{\eta \lambda _{3}t},~~~t\le t_{3},\\ k_{2}-\varepsilon _{4}e^{-\lambda t},~~~t\ge t_{3}. \end{array} \right. \end{aligned}$$

obviously, $M_{1}=\sup \limits _{t\in \mathbb {R}}\overline{\phi }(t) >k_{1}$, $M_{2}=\sup \limits _{t\in \mathbb {R}}\overline{\psi }(t) >k_{2}$, and $\overline{\phi }(t)$, $\underline{\phi }(t)$, $\overline{\psi }(t)$, $\underline{\psi }(t)$ satisfy (A1)-(A3), and

$$\min \{t_{2},t_{4}\}-c\max \{1,\tau _{1},\tau _{2},\tau _{3},\tau _{4}\}\ge \max \{t_{1},t_{3}\}.$$

In the following, we prove that $(\overline{\phi }(t),\overline{\psi }(t))$, $(\underline{\phi }(t),\underline{\psi }(t))$ are a pair of weak upper and weak lower solutions of (6), respectively.

Lemma 8

Let $0<r_{1}<1$, $0<r_{2}<1$ and suppose that (15), (16) are satisfied. Then $(\overline{\phi }(t),\overline{\psi }(t))$ is a weak upper solution of (13).

Proof

we first prove $\overline{\phi }(t)$ is a weak upper solution.

(i) For $t\le t_{2}$, $\overline{\phi }(t)=e^{\lambda _{1}t}$, $\overline{\phi }(t-c)=e^{\lambda _{1}(t-c)}$, $\overline{\phi }(t-c\tau _{1})\ge 0$,$\underline{\psi }(t)\ge 0$ $\underline{\psi }(t-c\tau _{2})\ge 0$, we have

$$\begin{aligned} \begin{array}{lll} & D_{1}\overline{\phi }^{''}(t)-\overline{\phi }(t)+ \overline{\phi }(t-c)\\ & \qquad + r_{1}\overline{\phi }(t) [1-a_{1}\overline{\phi }(t)-b_{1}\overline{\phi }(t-c\tau _{1}) -c_{1}\underline{\psi }(t)-d_{1}\underline{\psi }(t-c\tau _{2})]\\ & \quad \le D_{1}\overline{\phi }^{''}(t)-\overline{\phi }(t)+\overline{\phi } (t-c)+r_{1}\overline{\phi }(t)\\ & \quad =e^{\lambda _{1}t}(D_{1}\lambda _{1}^{2}+e^{-\lambda _{1}c}+r_{1}-1)=0. \end{array} \end{aligned}$$

(ii) For $t_{2}\le t\le t_{2}+c\tau _{1}$, $\overline{\phi }(t)=k_{1}+\varepsilon _{1}e^{-\lambda t}$, $\overline{\phi }(t-c\tau _{1})=e^{\lambda _{1}(t-c\tau _{1})}$, $\overline{\phi }(t-c)=e^{\lambda _{1}(t-c)}$, $\underline{\psi }(t)=k_{2}-\varepsilon _{4}e^{-\lambda t}$, $\underline{\psi }(t-c\tau _{2})=k_{2}-\varepsilon _{4}e^{-\lambda (t-c\tau _{2})}$, note that $k_{1}+\varepsilon _{1}e^{-\lambda t_{2}}=e^{\lambda _{1}t_{2}}$, we obtain

$$\begin{aligned} & D_{1}\overline{\phi }^{''}(t)-\overline{\phi }(t)+\overline{\phi }(t-c)\\ & \qquad + r_{1}\overline{\phi }(t) [1-a_{1}\overline{\phi }(t)-b_{1}\overline{\phi }(t-c\tau _{1}) -c_{1}\underline{\psi }(t)-d_{1}\underline{\psi }(t-c\tau _{2})]\\ & \quad =D_{1}\varepsilon _{1}\lambda ^{2}e^{-\lambda t}-(k_{1}+\varepsilon _{1}e^{-\lambda t})+e^{\lambda _{1}(t-c)}+r_{1}(k_{1}+\varepsilon _{1}e^{-\lambda t}) [1-a_{1}(k_{1}+\varepsilon _{1}e^{-\lambda t})\\ & \qquad -b_{1}e^{\lambda _{1}(t-c\tau _{1})}-c_{1}(k_{2}-\varepsilon _{4} e^{-\lambda t})-d_{1}(k_{2}-\varepsilon _{4}e^{-\lambda (t-c\tau _{2})})]\\ & \quad \le D_{1}\varepsilon _{1}\lambda ^{2}e^{-\lambda t}-(k_{1}+\varepsilon _{1}e^{-\lambda t})+e^{\lambda _{1}t_{2}} +r_{1}(k_{1}+\varepsilon _{1}e^{-\lambda t})[1-a_{1}(k_{1}+\varepsilon _{1}e^{-\lambda t})\\ & \qquad -b_{1}e^{\lambda _{1}(t_{2}-c\tau _{1})}-c_{1}(k_{2}-\varepsilon _{4} e^{-\lambda t})-d_{1}(k_{2}-\varepsilon _{4}e^{-\lambda (t-c\tau _{2})})]\\ & \quad = D_{1}\varepsilon _{1}\lambda ^{2}e^{-\lambda t}-\varepsilon _{1}e^{-\lambda t}+\varepsilon _{1}e^{-\lambda t_{2}}+r_{1}(k_{1}+\varepsilon _{1}e^{-\lambda t}) [b_{1}k_{1}-a_{1}\varepsilon _{1}e^{-\lambda t}\\ & \qquad -b_{1}e^{-\lambda _{1}c\tau _{1}}(k_{1}+\varepsilon _{1}e^{-\lambda t_{2}})+c_{1}\varepsilon _{4}e^{-\lambda t}+d_{1}\varepsilon _{4}e^{-\lambda (t-c\tau _{2})})]\\ & \quad :=I_{1}(\lambda ), \end{aligned}$$

since $\tau _{1}$ small enough, there exist $\varepsilon ^{*}$ $(0<\varepsilon ^{*}<\frac{\varepsilon _{0}}{b_{1}(k_{1}+\varepsilon _{1})})$, such that $1-\varepsilon ^{*}<e^{-\lambda _{1}c\tau _{1}}$, it follows that $(a_{1}+b_{1})\varepsilon _{1}-(c_{1}+d_{1})\varepsilon _{4}> \varepsilon _{0}$, we have

$$\begin{aligned} \begin{array}{lll} I_{1}(0)& \le r_{1}(k_{1}+\varepsilon _{1}) [b_{1}k_{1}-a_{1}\varepsilon _{1} -b_{1}(1-\varepsilon ^{*})(k_{1}+\varepsilon _{1})+c_{1} \varepsilon _{4}+d_{1}\varepsilon _{4}]\\ & =r_{1}(k_{1}+\varepsilon _{1}) [c_{1}\varepsilon _{4}+d_{1}\varepsilon _{4}-a_{1}\varepsilon _{1} -b_{1}\varepsilon _{1}+b_{1}\varepsilon ^{*}(k_{1}+\varepsilon _{1})]\\ & <0, \end{array} \end{aligned}$$

therefore, there exist a $\lambda _{1}^{*}>0$ such that $I_{1}(\lambda )<0$ for $\lambda \in (0,\lambda _{1}^{*})$.

(iii) For $t_{2}+c\tau _{1}\le t \le t_{2}+c$, $\overline{\phi }(t)=k_{1}+\varepsilon _{1}e^{-\lambda t}$, $\overline{\phi }(t-c\tau _{1})=k_{1}+\varepsilon _{1}e^{\lambda _{1} (t-c\tau _{1})}$, $\overline{\phi }(t-c)=e^{\lambda _{1}(t-c)}$, $\underline{\psi }(t)=k_{2}-\varepsilon _{4}e^{-\lambda t}$, $\underline{\psi }(t-c\tau _{2})=k_{2}-\varepsilon _{4}e^{-\lambda (t-c\tau _{2})}$, we have

$$\begin{aligned} & D_{1}\overline{\phi }^{''}(t)-\overline{\phi }(t)+\overline{\phi }(t-c)\\ & \qquad + r_{1}\overline{\phi }(t) [1-a_{1}\overline{\phi }(t)-b_{1}\overline{\phi }(t-c\tau _{1}) -c_{1}\underline{\psi }(t)-d_{1}\underline{\psi }(t-c\tau _{2})]\\ & \quad =D_{1}\varepsilon _{1}\lambda ^{2}e^{-\lambda t}-(k_{1}+\varepsilon _{1}e^{-\lambda t})+e^{\lambda _{1}(t-c)}+r_{1}(k_{1}+\varepsilon _{1}e^{-\lambda t}) [1-a_{1}(k_{1}+\varepsilon _{1}e^{-\lambda t})\\ & \qquad -b_{1}(k_{1}+\varepsilon _{1}e^{\lambda _{1}(t-c\tau _{1})})-c_{1} (k_{2}-\varepsilon _{4}e^{-\lambda t})-d_{1}(k_{2}-\varepsilon _{4}e^{-\lambda (t-c\tau _{2})})]\\ & \quad \le D_{1}\varepsilon _{1}\lambda ^{2}e^{-\lambda t}-(k_{1}+\varepsilon _{1}e^{-\lambda t})+e^{\lambda _{1}t_{2}} +r_{1}(k_{1}+\varepsilon _{1}e^{-\lambda t})[c_{1}\varepsilon _{4}e^{-\lambda t} +d_{1}\varepsilon _{4}e^{-\lambda (t-c\tau _{2})}\\ & \qquad -a_{1}\varepsilon _{1}e^{-\lambda t} -b_{1}\varepsilon _{1}e^{-\lambda (t-c\tau _{1})}]\\ & \quad = D_{1}\varepsilon _{1}\lambda ^{2}e^{-\lambda t}-\varepsilon _{1}e^{-\lambda t}+\varepsilon _{1}e^{-\lambda t_{2}}+r_{1}(k_{1}+\varepsilon _{1}e^{-\lambda t}) [c_{1}\varepsilon _{4}e^{-\lambda t} +d_{1}\varepsilon _{4}e^{-\lambda (t-c\tau _{2})}\\ & \qquad -a_{1}\varepsilon _{1}e^{-\lambda t} -b_{1}\varepsilon _{1}e^{-\lambda (t-c\tau _{1})}]\\ & \qquad :=I_{2}(\lambda ), \end{aligned}$$

since $(a_{1}+b_{1})\varepsilon _{1}-(c_{1}+d_{1})\varepsilon _{4}> \varepsilon _{0}$, we have $I_{2}(0)=r_{1}(k_{1}+\varepsilon _{1}) [(c_{1}+d_{1})\varepsilon _{4}-(a_{1}+b_{1})\varepsilon _{1}]<0$, therefore, there exist a $\lambda _{2}^{*}>0$ such that $I_{2}(\lambda )<0$ for $\lambda \in (0,\lambda _{2}^{*})$.

(iv)For $t \ge t_{2}+c$, $\overline{\phi }(t)=k_{1}+\varepsilon _{1}e^{-\lambda t}$, $\overline{\phi }(t-c)=k_{1}+\varepsilon _{1}e^{-\lambda (t-c)}$, $\overline{\phi }(t-c\tau _{1})=k_{1}+\varepsilon _{1}e^{-\lambda (t-c\tau _{1})}$, $\underline{\psi }(t)=k_{2}-\varepsilon _{4}e^{-\lambda t}$, $\underline{\psi }(t-c\tau _{2})=k_{2}-\varepsilon _{4}e^{-\lambda (t-c\tau _{2})}$, we obtain

$$\begin{aligned} & D_{1}\overline{\phi }^{''}(t)-\overline{\phi }(t)+\overline{\phi }(t-c)\\ & \qquad + r_{1}\overline{\phi }(t) [1-a_{1}\overline{\phi }(t)-b_{1}\overline{\phi }(t-c\tau _{1}) -c_{1}\underline{\psi }(t)-d_{1}\underline{\psi }(t-c\tau _{2})]\\ & \quad = D_{1}\varepsilon _{1}\lambda ^{2}e^{-\lambda t}-\varepsilon _{1}e^{-\lambda t}+\varepsilon _{1}e^{-\lambda (t-c)} +r_{1}(k_{1}+\varepsilon _{1}e^{-\lambda t})[c_{1}\varepsilon _{4}e^{-\lambda t} +d_{1}\varepsilon _{4}e^{-\lambda (t-c\tau _{2})}\\ & \qquad -a_{1}\varepsilon _{1}e^{-\lambda t} -b_{1}\varepsilon _{1}e^{-\lambda (t-c\tau _{1})}]\\ & \quad :=I_{3}(\lambda ), \end{aligned}$$

it follows that $(a_{1}+b_{1})\varepsilon _{1}-(c_{1}+d_{1})\varepsilon _{4}> \varepsilon _{0}$, we have $I_{3}(0)=r_{1}(k_{1}+\varepsilon _{1}) [(c_{1}+d_{1})\varepsilon _{4}-(a_{1}+b_{1})\varepsilon _{1}]<0$, which implies that there exist a $\lambda _{3}^{*}>0$ such that $I_{3}(\lambda )<0$ for $\lambda \in (0,\lambda _{3}^{*})$.

Taking $\lambda ^{*}=\min \{\lambda _{1}^{*},\lambda _{2}^{*},\lambda _{3}^{*}\}$, then $\lambda \in (0,\lambda ^{*})$, we have

$$\begin{aligned} & D_{1}\overline{\phi }^{''}(t)-\overline{\phi }(t)+\overline{\phi }(t-c)+ r_{1}\overline{\phi }(t) [1-a_{1}\overline{\phi }(t)\\ & \quad -b_{1}\overline{\phi }(t-c\tau _{1}) -c_{1}\underline{\psi }(t)-d_{1}\underline{\psi }(t-c\tau _{2})]\le 0. \end{aligned}$$

In a similar way, we can find a $\lambda ^{**}>0$,then $\lambda \in (0,\lambda ^{**})$, we have

$$\begin{aligned} & D_{2}\overline{\psi }^{''}(t)-\overline{\psi }(t)+\overline{\psi }(t-c)\\ & \quad + r_{2}\overline{\psi }(t) [1-a_{2}\overline{\psi }(t)-b_{2}\overline{\psi }(t-c\tau _{3})- c_{2}\underline{\phi }(t)-d_{2}\underline{\phi }(t-c\tau _{4})]\le 0. \end{aligned}$$

The proof is completed.  $\square$

Lemma 9

Let $0<r_{1}<1$, $0<r_{2}<1$. Suppose that (15),(16) are satisfied. Then $(\underline{\phi }(t),\underline{\psi }(t))$ is a weak lower solution of (13).

Proof

(i) For $t\le t_{1}$, $\underline{\phi }(t)=e^{\lambda _{1}t}-qe^{\eta \lambda _{1}t}$, $\underline{\phi }(t-c)=e^{\lambda _{1}(t-c)}-qe^{\eta \lambda _{1}(t-c)}$, $\underline{\phi }(t-c\tau _{1})=e^{\lambda _{1}(t-c\tau _{1})} -qe^{\eta \lambda _{1}(t-c\tau _{1})}$, $\overline{\psi }(t)=e^{\lambda _{3}t}$, $\overline{\psi }(t-c\tau _{2})=e^{\lambda _{3}(t-c\tau _{2})}$, we obtain

$$\begin{aligned} & D_{1}\underline{\phi }^{''}(t)-\underline{\phi }(t)+\underline{\phi } (t-c)+r_{1}\underline{\phi }(t) [1-a_{1}\underline{\phi }(t)-b_{1}\underline{\phi }(t-c\tau _{1})\\ & \qquad -c_{1}\overline{\psi }(t)-d_{1}\overline{\psi }(t-c\tau _{2})]\\ & \quad =e^{\lambda _{1}t}(D_{1}\lambda _{1}^{2}+e^{-\lambda _{1}c}+r_{1}-1) -qe^{\eta \lambda _{1}t} [D_{1}(\eta \lambda _{1})^{2} +e^{-\eta \lambda _{1}c}+r_{1}-1] -r_{1}(e^{\lambda _{1}t}\\ & \qquad -qe^{\eta \lambda _{1}t}) \{a_{1} (e^{\lambda _{1}t}-qe^{\eta \lambda _{1}t})+b_{1} [e^{\lambda _{1}(t-c\tau _{1})}-qe^{\eta \lambda _{1}(t-c\tau _{1})}] +c_{1}e^{\lambda _{3}t}+d_{1}e^{\lambda _{3}(t-c\tau _{2})}\}\\ & \qquad \ge -qe^{\eta \lambda _{1}t}\Delta _{1c}(\eta \lambda _{1})-r_{1}e^{\lambda _{1}t} [(a_{1}+b_{1})e^{\lambda _{1}t}+(c_{1}+d_{1})e^{\lambda _{3}t}]\\ & \qquad =-qe^{\eta \lambda _{1}t}[\Delta _{1c}(\eta \lambda _{1})+\frac{r_{1} (a_{1}+b_{1})}{q}e^{(2\lambda _{1}-\eta \lambda _{1})t} +\frac{r_{1}(c_{1}+d_{1})}{q}e^{(\lambda _{1}+\lambda _{3}-\eta \lambda _{1})t}]\\ & \qquad \ge -qe^{\eta \lambda _{1}t}[\Delta _{1c}(\eta \lambda _{1})+\frac{r_{1} (a_{1}+b_{1})}{q} +\frac{r_{1}(c_{1}+d_{1})}{q}], \end{aligned}$$

choose a sufficiently large number $q>0$ such that $q>-\frac{r_{1}}{\Delta _{1c}(\eta \lambda _{1})}(a_{1}+b_{1}+c_{1}+d_{1})$, then $-qe^{\eta \lambda _{1}t}[\Delta _{1c}(\eta \lambda _{1})+\frac{r_{1}(a_{1} +b_{1})}{q} +\frac{r_{1}(c_{1}+d_{1})}{q}]>0$.

(ii) For $t_{1}\le t\le t_{1}+c\tau _{1}$, $\underline{\phi }(t)=k_{1}-\varepsilon _{2}e^{-\lambda t}$, $\underline{\phi }(t-c\tau _{1})=e^{\lambda _{1}(t-c\tau _{1})}- qe^{\eta \lambda _{1}(t-c\tau _{1})}$, $\underline{\phi }(t-c)=e^{\lambda _{1}(t-c)}-qe^{\eta \lambda _{1}(t-c)}$, $\overline{\psi }(t)=e^{\lambda _{3}t}$, $\overline{\psi }(t-c\tau _{2 })=e^{\lambda _{3}(t-c\tau _{2})}$, note that $e^{\lambda _{1}t_{1}}-qe^{\eta \lambda _{1}t_{1}} =k_{1}-\varepsilon _{2}e^{-\lambda t_{1}}$, $e^{\lambda _{3}t_{4}}=k_{2}+\varepsilon _{3}e^{-\lambda t_{4}}$ and $t_{1}+c\le t_{4}$, we get

$$\begin{aligned} & D_{1}\underline{\phi }^{''}(t)-\underline{\phi }(t)+ \underline{\phi }(t-c)\\ & \qquad +r_{1}\underline{\phi }(t) [1-a_{1}\underline{\phi }(t)-b_{1}\underline{\phi }(t-c\tau _{1}) -c_{1}\overline{\psi }(t)-d_{1}\overline{\psi }(t-c\tau _{2})]\\ & \quad =-D_{1}\varepsilon _{2}\lambda ^{2}e^{-\lambda t}-(k_{1}-\varepsilon _{2}e^{-\lambda t}) +(e^{\lambda _{1}(t-c)}-qe^{\eta \lambda _{1}(t-c)}) +r_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t}) [1\\ & \qquad -a_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t})-b_{1}(e^{\lambda _{1}(t-c\tau _{1})}-qe^{\eta \lambda _{1}(t-c\tau _{1})}) -c_{1}e^{\lambda _{3}t}-d_{1}e^{\lambda _{3}(t-c\tau _{2})}]\\ & \quad \ge -D_{1}\varepsilon _{2}\lambda ^{2}e^{-\lambda t}-(k_{1}-\varepsilon _{2}e^{-\lambda (t_{1}+c)}) +(e^{\lambda _{1}(t_{1}-c)}-e^{\lambda _{1}t_{1}})+(e^{\lambda _{1}t_{1}} -qe^{\eta \lambda _{1}t_{1}})\\ & \qquad +r_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t}) [1-a_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t})-b_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t_{1}}) -c_{1}e^{\lambda _{3}t_{4}}-d_{1}e^{\lambda _{3}t_{4}}]\\ & \quad = -D_{1}\varepsilon _{2}\lambda ^{2}e^{-\lambda t}+\varepsilon _{2}e^{-\lambda (t_{1}+c)} +e^{\lambda _{1}t_{1}}(e^{-\lambda _{1}c}-1)-\varepsilon _{2}e^{-\lambda t_{1}}\\ & \qquad +r_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t})[a_{1}\varepsilon _{2}e^{-\lambda t}+b_{1}\varepsilon _{2} e^{-\lambda t_{1}}-(c_{1}+d_{1})\varepsilon _{3}e^{-\lambda t_{4}}]\\ & \quad :=I_{4}(\lambda ), \end{aligned}$$

we know that $-t_{1}$ is large enough if q is large enough by the definition of $t_{1}$, therefore, from $(a_{1}+b_{1})\varepsilon _{2}-(c_{1}+d_{1})\varepsilon _{3}>\varepsilon _{0}$, it is obvious that $I_{4}(0)=e^{\lambda _{1}t_{1}}(e^{-\lambda _{1}c}-1)+r_{1}(k_{1} -\varepsilon _{2}) [(a_{1}+b_{1})\varepsilon _{2}-(c_{1}+d_{1})\varepsilon _{3}]>0$ for large enough q, thus, there exists a $\lambda _{4}^{*}>0$ such that $I_{7}(\lambda )>0$ for $\lambda \in (0,\lambda _{4}^{*})$.

(iii) For $t_{1}+c\tau _{1}\le t\le t_{1}+c$, $\underline{\phi }(t)=k_{1}-\varepsilon _{2}e^{-\lambda t}$, $\underline{\phi }(t-c\tau _{1})=k_{1}-\varepsilon _{2} e^{-\lambda (t-c\tau _{1})}$, $\underline{\phi }(t-c)=e^{\lambda _{1}(t-c)}-qe^{\eta \lambda _{1}(t-c)}$, $\overline{\psi }(t)=e^{\lambda _{3}t}$, $\overline{\psi }(t-c\tau _{2 })=e^{\lambda _{3}(t-c\tau _{2})}$, note that $e^{\lambda _{1}t_{1}}-qe^{\eta \lambda _{1}t_{1}} =k_{1}-\varepsilon _{2}e^{-\lambda t_{1}}$, $e^{\lambda _{3}t_{4}}=k_{2}+\varepsilon _{3}e^{-\lambda t_{4}}$ and $t_{1}+c\le t_{4}$, we get

$$\begin{aligned} & D_{1}\underline{\phi }^{''}(t)-\underline{\phi }(t) +\underline{\phi }(t-c)\\ & \qquad +r_{1}\underline{\phi }(t) [1-a_{1}\underline{\phi }(t)-b_{1}\underline{\phi }(t-c\tau _{1}) -c_{1}\overline{\psi }(t)-d_{1}\overline{\psi }(t-c\tau _{2})]\\ & \quad =-D_{1}\varepsilon _{2}\lambda ^{2}e^{-\lambda t}-(k_{1}-\varepsilon _{2}e^{-\lambda t}) +(e^{\lambda _{1}(t-c)}-qe^{\eta \lambda _{1}(t-c)}) +r_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t}) [1\\ & \qquad -a_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t})-b_{1}(k_{1}-\varepsilon _{2}e^{-\lambda _{1}(t-c\tau _{1})}) -c_{1}e^{\lambda _{3}t}-d_{1}e^{\lambda _{3}(t-c\tau _{2})}]\\ & \quad \ge -D_{1}\varepsilon _{2}\lambda ^{2}e^{-\lambda t}-(k_{1}-\varepsilon _{2}e^{-\lambda (t_{1}+c)}) +(e^{\lambda _{1}(t_{1}+c\tau _{1}-c)}-e^{\lambda _{1}t_{1}}) +(e^{\lambda _{1}t_{1}}-qe^{\eta \lambda _{1}t_{1}})\\ & \qquad +r_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t}) [1-a_{1}(k_{1}-\varepsilon _{2}e^{-\lambda (t_{1}+c-c\tau _{1})})-b_{1}(k_{1}-\varepsilon _{2} e^{-\lambda (t_{1}+c-c\tau _{1})})\\ & \qquad -c_{1}e^{\lambda _{3}t_{4}}-d_{1}e^{\lambda _{3}t_{4}}]\\ & \quad = -D_{1}\varepsilon _{2}\lambda ^{2}e^{-\lambda t}+\varepsilon _{2}e^{-\lambda (t_{1}+c)} +e^{\lambda _{1}t_{1}}(e^{-\lambda _{1}(c-c\tau _{1})}-1) -\varepsilon _{2}e^{-\lambda t_{1}}\\ & \qquad +r_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t})[(a_{1}+b_{1})\varepsilon _{2}e^{-\lambda (t_{1}+c-c\tau _{1})} -(c_{1}+d_{1})\varepsilon _{3}e^{-\lambda t_{4}}]\\ & \quad :=I_{5}(\lambda ), \end{aligned}$$

we know that $-t_{1}$ is large enough if q is large enough by the definition of $t_{1}$, therefore, from $(a_{1}+b_{1})\varepsilon _{2}-(c_{1}+d_{1}) \varepsilon _{3}>\varepsilon _{0}$, it is obvious that $I_{5}(0)=e^{\lambda _{1}t_{1}}(e^{\lambda _{1}(c\tau _{1}-c)}-1) +r_{1}(k_{1}-\varepsilon _{2}) [(a_{1}+b_{1})\varepsilon _{2}-(c_{1}+d_{1})\varepsilon _{3}]>0$ for large enough q, thus, there exists a $\lambda _{5}^{*}>0$ such that $I_{5}(\lambda )>0$ for $\lambda \in (0,\lambda _{5}^{*})$.

(iv) For $t_{1}+c\le t\le t_{4}$, $\underline{\phi }(t)=k_{1}-\varepsilon _{2}e^{-\lambda t}$, $\underline{\phi }(t-c\tau _{1})=k_{1}-\varepsilon _{2} e^{-\lambda _{1}(t-c\tau _{1})}$, $\underline{\phi }(t-c)=k_{1}-\varepsilon _{2}e^{-\lambda (t-c)}$, $\overline{\psi }(t)=e^{\lambda _{3}t}$, $\overline{\psi }(t-c\tau _{2 })=e^{\lambda _{3}(t-c\tau _{2})}$, in view of $e^{\lambda _{3}t_{4}}=k_{2}+\varepsilon _{3}e^{-\lambda t_{4}}$, we get

$$\begin{aligned} & D_{1}\underline{\phi }^{''}(t)-\underline{\phi }(t) +\underline{\phi }(t-c)+r_{1}\underline{\phi }(t) [1-a_{1}\underline{\phi }(t)-b_{1}\underline{\phi }(t-c\tau _{1}\\ & \qquad -c_{1}\overline{\psi }(t)-d_{1}\overline{\psi }(t-c\tau _{2})]\\ & \quad =-D_{1}\varepsilon _{2}\lambda ^{2}e^{-\lambda t}-(k_{1}-\varepsilon _{2}e^{-\lambda t}) +(k_{1}-\varepsilon _{2}e^{-\lambda (t-c)}) +r_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t}) [1\\ & \qquad -a_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t})-b_{1}(k_{1}-\varepsilon _{2}e^{-\lambda _{1}(t-c\tau _{1})}) -c_{1}e^{\lambda _{3}t}-d_{1}e^{\lambda _{3}(t-c\tau _{2})}]\\ & \quad \ge -D_{1}\varepsilon _{2}\lambda ^{2}e^{-\lambda t}+\varepsilon _{2}e^{-\lambda t}-\varepsilon _{2}e^{-\lambda (t-c)} +r_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t})\\ & \qquad [1-a_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t})-b_{1}(k_{1}-\varepsilon _{2}e^{-\lambda (t-c\tau _{1})}) -c_{1}e^{\lambda _{3}t_{4}}-d_{1}e^{\lambda _{3}t_{4}}]\\ & \quad = -D_{1}\varepsilon _{2}\lambda ^{2}e^{-\lambda t}+\varepsilon _{2}e^{-\lambda t}-\varepsilon _{2}e^{-\lambda (t-c)} +r_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t})\\ & \qquad [a_{1}\varepsilon _{2}e^{-\lambda t}+b_{1}\varepsilon _{2}e^{-\lambda (t-c\tau _{1})}-(c_{1}+d_{1})\varepsilon _{3}e^{-\lambda t_{4}}]\\ & \quad :=I_{6}(\lambda ), \end{aligned}$$

therefore, from $(a_{1}+b_{1})\varepsilon _{2}-(c_{1}+d_{1})\varepsilon _{3}>\varepsilon _{0}$, it is obvious that $I_{6}(0)=r_{1}(k_{1}-\varepsilon _{2})[(a_{1}+b_{1}) \varepsilon _{2}-(c_{1}+d_{1})\varepsilon _{3}]>0$, thus, there exists a $\lambda _{6}^{*}>0$ such that $I_{6}(\lambda )$ for $\lambda \in (0,\lambda _{6}^{*})$.

(v) For $t_{4}\le t\le t_{4}+c\tau _{2}$, $\underline{\phi }(t)=k_{1}-\varepsilon _{2}e^{-\lambda t}$, $\underline{\phi }(t-c\tau _{1})=k_{1}-\varepsilon _{2} e^{-\lambda _{1}(t-c\tau _{1})}$, $\underline{\phi }(t-c)=k_{1}-\varepsilon _{2}e^{-\lambda (t-c)}$, $\overline{\psi }(t)=k_{2}+\varepsilon _{3}e^{-\lambda t}$, $\overline{\psi }(t-c\tau _{2 })=e^{\lambda _{3}(t-c\tau _{2})}$, in view of $e^{\lambda _{3}t_{4}}=k_{2}+\varepsilon _{3}e^{-\lambda t_{4}}$, we get

$$\begin{aligned} & D_{1}\underline{\phi }^{''}(t)-\underline{\phi }(t) +\underline{\phi }(t-c)+r_{1}\underline{\phi }(t) [1-a_{1}\underline{\phi }(t)-b_{1}\underline{\phi }(t-c\tau _{1})\\ & \qquad -c_{1}\overline{\psi }(t)-d_{1}\overline{\psi }(t-c\tau _{2})]\\ & \quad =-D_{1}\varepsilon _{2}\lambda ^{2}e^{-\lambda t}-(k_{1}-\varepsilon _{2}e^{-\lambda t}) +(k_{1}-\varepsilon _{2}e^{-\lambda (t-c)}) +r_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t}) [1\\ & \qquad -a_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t})-b_{1}(k_{1}-\varepsilon _{2}e^{-\lambda _{1}(t-c\tau _{1})}) -c_{1}(k_{2}+\varepsilon _{3}e^{-\lambda t}) -d_{1}e^{\lambda _{3}(t-c\tau _{2})}]\\ & \quad \ge -D_{1}\varepsilon _{2}\lambda ^{2}e^{-\lambda t}+\varepsilon _{2}e^{-\lambda t}-\varepsilon _{2}e^{-\lambda (t-c)} +r_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t})\\ & \qquad [1-a_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t})-b_{1}(k_{1}-\varepsilon _{2}e^{-\lambda (t-c\tau _{1})}) -c_{1}(k_{2}+\varepsilon _{3}e^{-\lambda t})-d_{1}e^{\lambda _{3}t_{4}}]\\ & \quad = -D_{1}\varepsilon _{2}\lambda ^{2}e^{-\lambda t}+\varepsilon _{2}e^{-\lambda t}-\varepsilon _{2}e^{-\lambda (t-c)} +r_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t})\\ & \qquad [a_{1}\varepsilon _{2}e^{-\lambda t}+b_{1}\varepsilon _{2}e^{-\lambda (t-c\tau _{1})}-c_{1}\varepsilon _{3}e^{-\lambda t}-d_{1}\varepsilon _{3}e^{-\lambda t_{4}}]\\ & \quad :=I_{7}(\lambda ), \end{aligned}$$

therefore, from $(a_{1}+b_{1})\varepsilon _{2}-(c_{1}+d_{1}) \varepsilon _{3}>\varepsilon _{0}$, it is obvious that $I_{7}(0)=r_{1}(k_{1}-\varepsilon _{2})[(a_{1}+b_{1}) \varepsilon _{2}-(c_{1}+d_{1})\varepsilon _{3}]>0$, thus, there exists a $\lambda _{7}^{*}>0$ such that $I_{7}(\lambda )$ for $\lambda \in (0,\lambda _{7}^{*})$.

(vi) For $t> t_{4}+c\tau _{2}$, $\underline{\phi }(t)=k_{1}-\varepsilon _{2}e^{-\lambda t}$, $\underline{\phi }(t-c\tau _{1})=k_{1}-\varepsilon _{2}e^{-\lambda (t-c\tau _{1})}$, $\underline{\phi }(t-c)=k_{1}-\varepsilon _{2}e^{-\lambda (t-c)}$, $\overline{\psi }(t)=k_{2}+\varepsilon _{3}e^{-\lambda t}$, $\overline{\psi }(t-c\tau _{2})=k_{2}+\varepsilon _{3}e^{-\lambda (t-c\tau _{2})}$, we get

$$\begin{aligned} & D_{1}\underline{\phi }^{''}(t)-\underline{\phi }(t) +\underline{\phi }(t-c)+r_{1}\underline{\phi }(t) [1-a_{1}\underline{\phi }(t)-b_{1}\underline{\phi }(t-c\tau _{1})\\ & \qquad -c_{1}\overline{\psi }(t)-d_{1}\overline{\psi }(t-c\tau _{2})]\\ & \quad = -D_{1}\varepsilon _{2}\lambda ^{2}e^{-\lambda t}+\varepsilon _{2}e^{-\lambda t}-\varepsilon _{2}e^{-\lambda (t-c)} +r_{1}(k_{1}-\varepsilon _{2}e^{-\lambda t})\\ & \qquad [a_{1}\varepsilon _{2}e^{-\lambda t}+b_{1}\varepsilon _{2}e^{-\lambda (t-c\tau _{1})}-c_{1}\varepsilon _{3}e^{-\lambda t}-d_{1}\varepsilon _{3}e^{-\lambda (t-c\tau _{2})})]\\ & \quad :=I_{8}(\lambda ), \end{aligned}$$

therefore, from $(a_{1}+b_{1})\varepsilon _{2}-(c_{1}+d_{1}) \varepsilon _{3}>\varepsilon _{0}$, it is obvious that $I_{8}(0)=r_{1}(k_{1}-\varepsilon _{2})[(a_{1}+b_{1}) \varepsilon _{2}-(c_{1}+d_{1})\varepsilon _{3}]>0$, thus, there exists a $\lambda _{8}^{*}>0$ such that $I_{8}(\lambda )$ for $\lambda \in (0,\lambda _{8}^{*})$.

Taking $\lambda ^{***}=\min \{\lambda _{4}^{*},\lambda _{5}^{*},\lambda _{6}^{*},\lambda _{7}^{*}, \lambda _{8}^{*}\}$, then $\lambda \in (0,\lambda ^{***})$, we have

$$D_{1}\underline{\phi }^{''}(t)-\underline{\phi }(t)+\underline{\phi } (t-c)+r_{1}\underline{\phi }(t) [1-a_{1}\underline{\phi }(t)-b_{1}\underline{\phi }(t-c\tau _{1}) -c_{1}\overline{\psi }(t)-d_{1}\overline{\psi }(t-c\tau _{2})]\ge 0.$$

Similarly, the rest inequalities are satisfied.The proof is completed.  $\square$

Therefore, by Theorem 2.1, we can get the following result.

Theorem 10

Let $0<r_{1}<1$, $0<r_{2}<1$, $\tau _{1},\tau _{3}$ are small enough and suppose that (15), (16) are satisfied. Then (13) has a traveling wave solution $(\phi (x+cn),\psi (x+cn))$ with wave speed c which connects (0, 0) and $(k_{1},k_{2})$.