1 Introduction

In population dynamics, Lotka-Volterra competitive, cooperative, and competitive-cooperative systems with diffusion have received great attention and have been studied extensively [1–7]. To illustrate and predict some ecological phenomena, various types of predator-prey model described by differential systems were proposed [8–10]. In studying the dynamics of predator-prey systems, one of the important topics is the existence of traveling wave solutions [11–19].

In this paper, we are concerned with the existence of traveling wave of the following competitive-cooperative system:

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u_{1}(x, t)}{\partial t}=d_{1}\frac{\partial^{2} u_{1}(x, t)}{\partial x^{2}}+r_{1}u_{1}(x, t)(1-a_{11}u_{1}(x, t-\tau_{11})-a_{12}u_{2}(x, t-\tau_{12}) \\ \hphantom{\frac{\partial u_{1}(x, t)}{\partial t}={}}{}-a_{13}u_{3}(x, t-\tau_{13})), \\ \frac{\partial u_{2}(x, t)}{\partial t}=d_{2}\frac{\partial^{2} u_{2}(x, t)}{\partial x^{2}}+r_{2}u_{2}(x, t)(1-a_{21}u_{1}(x, t-\tau_{21})-a_{22}u_{2}(x, t-\tau_{22}) \\ \hphantom{\frac{\partial u_{2}(x, t)}{\partial t}={}}{}+a_{23}u_{3}(x, t-\tau_{23})), \\ \frac{\partial u_{3}(x, t)}{\partial t}=d_{3}\frac{\partial^{2} u_{3}(x, t)}{\partial x^{2}}+r_{3}u_{3}(x, t)(1+a_{31}u_{1}(x, t-\tau_{31})+a_{32}u_{2}(x, t-\tau_{32}) \\ \hphantom{\frac{\partial u_{3}(x, t)}{\partial t}={}}{}-a_{33}u_{3}(x,t-\tau_{33})), \end{array}\displaystyle \right . $$
(1)

where all parameters \(d_{i}\), \(r_{i}\), \(a_{ij}\) are positive constants, \(\tau _{ij}\ge0\), \(i,j=1,2,3\), and the quantities \(u_{1}(x, t)\), \(u_{2}(x, t)\), \(u_{3}(x, t)\) can be interpreted as the population densities of the three species at space x and time t.

It is necessary to point out that, when any one of the quantities \(u_{1}(x, t)\), \(u_{2}(x, t)\), and \(u_{3}(x, t)\) are taken as zero, some cooperative system or competitive system can be derived from system (1), such as, when \(u_{1}=0\), system (1) becomes the two species cooperative system

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u_{2}}{\partial t}=d_{2}\frac{\partial^{2} u_{2}(x, t)}{\partial x^{2}}+r_{2}u_{2}(x, t)(1-a_{22}u_{2}(x, t-\tau_{22})+a_{23}u_{3}(x, t-\tau_{23})), \\ \frac{\partial u_{3}}{\partial t}=d_{3}\frac{\partial^{2} u_{3}(x, t)}{\partial x^{2}}+r_{3}u_{3}(x, t)(1+a_{32}u_{2}(x, t-\tau_{32})-a_{33}u_{3}(x, t-\tau_{33})), \end{array}\displaystyle \right . $$
(2)

considered by Huang and Zou [2]. When \(u_{2}=0\), system (1) is reduced to the two species predator-prey system

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u_{1}}{\partial t}=d_{1}\frac{\partial^{2} u_{1}(x, t)}{\partial x^{2}}+r_{1}u_{1}(x, t)(1-a_{11}u_{1}(x, t-\tau_{11})-a_{13}u_{3}(x, t-\tau_{13})), \\ \frac{\partial u_{3}}{\partial t}=d_{3}\frac{\partial^{2} u_{3}(x, t)}{\partial x^{2}}+r_{3}u_{3}(x, t)(1+a_{31}u_{1}(x, t-\tau_{31})-a_{33}u_{3}(x, t-\tau_{33})), \end{array}\displaystyle \right . $$
(3)

studied by Zhang and Li [17]. When \(u_{3}=0\) system (1) is reduced to the two species competing system

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u_{1}}{\partial t}=d_{1}\frac{\partial^{2} u_{1}(x, t)}{\partial x^{2}}+r_{1}u_{1}(x, t)(1-a_{11}u_{1}(x, t-\tau_{11})-a_{12}u_{2}(x, t-\tau_{12})), \\ \frac{\partial u_{2}}{\partial t}=d_{2}\frac{\partial^{2} u_{2}(x, t)}{\partial x^{2}}+r_{2}u_{2}(x, t)(1-a_{21}u_{1}(x, t-\tau_{21})-a_{22}u_{2}(x, t-\tau_{22})), \end{array}\displaystyle \right . $$
(4)

discussed by Lv and Wang [12].

This paper is organized as follows. In Section 2, we introduce some notations and lemmas which will be essential to our proofs. By applying the cross iteration method and Schauder’s fixed point theorem, we establish the existence result of traveling wave solutions for a general delayed reaction-diffusion system. In Section 3, by using the results given in Section 2 and constructing a pair of upper-lower solution, we obtain the existence of traveling wave solutions to the system (1).

2 Preliminaries

For convenience, we first give some notations and definitions of traveling wave solutions.

In this paper, we shall use the standard partial ordering in \(R^{3}\), namely, for \(u=(u_{1}, u_{2}, u_{3})^{T}\), \(v=(v_{1}, v_{2}, v_{3})^{T}\), we denote \(u\leq v\) if \(u_{i}\leq v_{i}\), \(i=1, 2, 3\); \(u< v\) if \(u\leq v\) but \(u\neq v\); and \(u\ll v\) if \(u\leq v\) but \(u_{i}\neq v_{i}\), \(i=1, 2, 3\). If \(u\neq v\), we denote \((u,v]=\{w\in R^{3}: u< w\leq v\}\), \([u,v)=\{w\in R^{3}: u\leq w< v\}\), and \([u, v]=\{w\in R^{3}: u\leq w\leq v\}\). We use \(|\cdot|\) to denote the Euclidean in \(R^{3}\) and \(\|\cdot\|\) to denote the supremum norm in \(C([-\tau,0], R^{3})\).

Definition 1

([15, 18])

A traveling wave solution of system (1) is a special solution of the form \(u(t,x)=\phi(x+ct)\), \(v(t, x)=\varphi(x+ct)\), \(w(t, x)=\psi (x+ct)\), where \(\phi,\varphi,\psi \in C^{2}(R,R)\) are the wave profiles that propagate through the one-dimensional spatial domain at a constant velocity \(c>0\).

To show the existence of a traveling wave solution to system (1), we first discuss the following general reaction-diffusion system:

$$ \left \{ \textstyle\begin{array}{l} \frac{\partial u(x, t)}{\partial t}=d_{1}\frac{\partial^{2} u(x, t)}{\partial x^{2}}+f_{1}(u(x, t), v(x, t), w(x, t)), \\ \frac{\partial v(x, t)}{\partial t}=d_{2}\frac{\partial^{2} v(x, t)}{\partial x^{2}}+f_{2}(u(x, t), v(x, t), w(x, t)), \\ \frac{\partial w(x, t)}{\partial t}=d_{3}\frac{\partial^{2} w(x, t)}{\partial x^{2}}+f_{3}(u(x, t), v(x, t), w(x, t)). \end{array}\displaystyle \right . $$
(5)

Substituting \(u(x, t)=\phi(x+ct)\), \(v(x,t)=\varphi(x+ct)\), \(w(x, t)=\psi(x+ct)\) into (5) and denote the traveling wave coordinate \(x+ct\) still by t, then (5) has a traveling wave solution if and only if the following system:

$$ \left \{ \textstyle\begin{array}{l} d_{1}\phi''(t)-c\phi'(t)+f_{c1}(\phi_{t}, \varphi_{t},\psi _{t})=0, \\ d_{2}\varphi''(t)-c\varphi'(t)+f_{c2}(\phi_{t}, \varphi_{t},\psi_{t})=0, \\ d_{3}\psi''(t)-c\psi'(t)+f_{c3}(\phi_{t}, \varphi_{t},\psi_{t})=0, \end{array}\displaystyle \right . $$
(6)

with asymptotic boundary conditions

$$ \begin{aligned} &\lim_{t\rightarrow-\infty}\phi(t)= \phi_{-},\qquad \lim_{t\rightarrow-\infty}\varphi(t)= \varphi_{-},\qquad \lim_{t\rightarrow-\infty}\psi(t)= \psi_{-}, \\ &\lim_{t\rightarrow+\infty}\phi(t)=\phi_{+},\qquad \lim _{t\rightarrow+\infty}\varphi(t)=\varphi_{+},\qquad \lim _{t\rightarrow+\infty}\psi(t)=\psi_{+}, \end{aligned} $$
(7)

has a solution \((\phi(t),\varphi(t),\psi(t)) \) on R, where \((\phi _{-}, \varphi_{-}, \psi_{-})\) and \((\phi_{+}, \varphi_{+}, \psi_{+})\) are steady states of (1) and the functions \(f_{ci}: X_{c}=C([-c\tau, 0], R^{3}) \rightarrow R^{3}\), \(i=1, 2, 3\), are defined by

$$\begin{aligned}& f_{ci}(\phi, \varphi, \psi)=f_{i}\bigl(\phi^{c}, \varphi^{c},\psi ^{c}\bigr),\qquad \phi^{c}(s)= \phi(cs), \\ & \varphi^{c}(s)=\varphi(cs),\qquad \psi^{c}(s)= \psi(cs), \quad s\in[-\tau, 0]. \end{aligned}$$

Without loss of generality, we can assume

$$(\phi_{-}, \varphi_{-}, \psi_{-})=(0, 0, 0), \qquad (\phi_{+}, \varphi_{+}, \psi_{+})=(k_{1}, k_{2}, k_{3}), $$

and we seek for traveling wave solution connecting these two steady states. In order to address traveling waves of (6) and (7), we make the following assumptions:

  1. (A1)

    \(f_{i}(0, 0, 0)=f_{i}(k_{1}, k_{2}, k_{3})=0\) for \(i=1, 2, 3\);

  2. (A2)

    there exist three positive constants \(L_{i}>0 \) (\(i=1, 2, 3\)), such that

    $$\begin{aligned}& \bigl\vert f_{1}(\phi_{1},\varphi_{1}, \psi_{1})-f_{1}(\phi_{2},\varphi _{2}, \psi_{2})\bigr\vert \leq L_{1}\|\Phi-\Psi\|, \\& \bigl\vert f_{2}(\phi_{1},\varphi_{1}, \psi_{1})-f_{2}(\phi_{2},\varphi _{2}, \psi_{2})\bigr\vert \leq L_{2}\|\Phi-\Psi\|, \\& \bigl\vert f_{3}(\phi_{1},\varphi_{1}, \psi_{1})-f_{3}(\phi_{2},\varphi _{2}, \psi_{2})\bigr\vert \leq L_{3}\|\Phi-\Psi\|, \end{aligned}$$

for \(\Phi=(\phi_{1}, \varphi_{1}, \psi_{1})\), \(\Psi=(\phi_{2}, \varphi_{2}, \psi_{2}) \in C([-\tau, 0], R^{3})\) with \(0\leq\phi_{i}(s)\leq M_{1}\), \(0\leq\varphi_{i}(s)\leq M_{2}\), \(0\leq\psi_{i}(s)\leq M_{3}\), \(i=1, 2\), where \(M_{j}\ge k_{j}\) (\(j=1, 2, 3\)) are positive constants.

The reaction terms satisfy the following partial quasi-monotonicity conditions (PQM), different from [15, 18, 19].

(PQM):

There exist three positive constants \(\beta_{1}, \beta_{2}, \beta_{3} > 0\) such that

$$ \begin{aligned} &f_{c1}(\phi_{1}, \varphi_{1},\psi_{1})-f_{c1}(\phi_{2}, \varphi _{1},\psi_{1})+\beta_{1}\bigl[ \phi_{1}(0)-\phi_{2}(0)\bigr]\geq0, \\ &f_{c1}(\phi_{1},\varphi_{1}, \psi_{1})-f_{c1}(\phi_{1},\varphi _{2}, \psi_{2})\leq0, \\ &f_{c2}(\phi_{1},\varphi_{1}, \psi_{1})-f_{c2}(\phi_{1},\varphi _{2}, \psi_{2})+\beta_{2}\bigl[\varphi_{1}(0)- \varphi_{2}(0)\bigr]\geq0, \\ &f_{c2}(\phi_{1},\varphi_{1}, \psi_{1})-f_{c2}(\phi_{2},\varphi _{1}, \psi_{1})\leq0, \\ &f_{c3}(\phi_{1},\varphi_{1}, \psi_{1})-f_{c3}(\phi_{1},\varphi _{2}, \psi_{2})+\beta_{3}\bigl[\psi_{1}(0)- \psi_{2}(0)\bigr]\geq0, \\ &f_{c3}(\phi_{1},\varphi_{1}, \psi_{1})-f_{c3}(\phi_{2},\varphi _{1}, \psi_{1})\leq0, \end{aligned} $$
(8)

where \(\phi_{i}, \varphi_{i}, \psi_{i} \in C([-\tau,0],R)\), \(i=1, 2\), \(0\leq\phi_{2}(s)\leq\phi_{1}(s)\leq M_{1}\), \(0\leq\varphi_{2}(s)\leq\varphi_{1}(s)\leq M_{2}\), \(0\leq\psi_{2}(s)\leq\psi_{1}(s)\leq M_{3}\), \(s \in[-\tau,0]\).

We need the following definition of upper and lower solutions.

Definition 2

([15, 18])

A pair of continuous functions \(\overline{\rho}=(\overline{\phi}, \overline{\varphi}, \overline{\psi})\) and \(\underline{\rho}=(\underline{\phi}, \underline{\varphi}, \underline{\psi})\) are called a pair of upper and lower solutions of the system (1) if ρ̅ and \(\underline{\rho}\) are twice differentiable almost everywhere in R and they are essentially bounded on R, and we have

$$ \left \{ \textstyle\begin{array}{l} d_{1}\overline{\phi}''-c\overline{\phi}'+f_{c1}(\overline{\phi}_{t}, \underline{\varphi}_{t}, \underline{\psi}_{t})\leq0,\quad \mbox{a.e. in } R, \\ d_{2}\overline{\varphi}''-c\overline{\varphi}'+f_{c2}(\underline {\phi}_{t}, \overline{\varphi}_{t}, \overline{\psi}_{t})\leq0,\quad \mbox{a.e. in } R, \\ d_{3}\overline{\psi}''-c\overline{\psi}'+f_{c3}(\underline{\phi}_{t}, \overline{\varphi}_{t}, \overline{\psi}_{t})\leq0,\quad \mbox{a.e. in } R \end{array}\displaystyle \right . $$
(9)

and

$$ \left \{ \textstyle\begin{array}{l} d_{1}\underline{\phi}''-c\underline{\phi}'+f_{c1}(\underline{\phi}_{t}, \overline{\varphi}_{t}, \overline{\psi}_{t})\geq0,\quad \mbox{a.e. in } R, \\ d_{2}\underline{\varphi}''-c\underline{\varphi}'+f_{c2}(\overline {\phi}_{t}, \underline{\varphi}_{t}, \underline{\psi}_{t})\geq0,\quad \mbox{a.e. in } R, \\ d_{3}\underline{\psi}''-c\underline{\psi}'+f_{c3}(\overline{\phi}_{t}, \underline{\varphi}_{t}, \underline{\psi}_{t})\geq0,\quad \mbox{a.e. in } R. \end{array}\displaystyle \right . $$
(10)

Let

$$C_{k}:=\bigl\{ (\phi, \varphi, \psi)|(0, 0, 0)\leq(\phi, \varphi, \psi )\leq(M_{1}, M_{2}, M_{3}), \mbox{for } t \in R \bigr\} . $$

We shall combine Schauder’s fixed point theorem with the method of upper and lower solutions to establish the existence of solutions. For this purpose, we need to introduce a topology in \(C(R,R^{3})\).

Let \(\mu > 0\) and let \(C(R,R^{3})\) be equipped with the exponential decay norm defined by

$$|\Phi|_{\mu}=\sup_{t \in R}e^{-\mu|t|}\bigl\vert \Phi(t)\bigr\vert _{R^{3}}. $$

Define

$$B_{\mu}\bigl(R, R^{3}\bigr)=\bigl\{ \Phi\in C\bigl(R, R^{3}\bigr): |\Phi|_{\mu}< \infty\bigr\} . $$

Then it is easy to check that \((B_{\mu}(R, R^{3}),|\cdot|_{\mu})\) is a Banach space. We shall look for the traveling wave solution of system (6) in the following profile set:

$$\begin{aligned}& \Gamma\bigl((\underline{\phi},\underline{\varphi},\underline{\psi }),(\overline{ \phi},\overline{\varphi},\overline{\psi})\bigr) \\& \quad =\left \{ \textstyle\begin{array}{rl} (\mathrm{i}) &\, \underline{\phi}(t)\leq\phi(t)\leq\overline{\phi}(t), \underline{\varphi}(t)\leq\varphi(t)\leq\overline{\varphi}(t), \underline{\psi}(t)\leq\psi(t)\leq\overline{\psi}(t); \\ (\mathrm{ii})&\, e^{\beta_{1}s}[\phi(t)-\underline{\phi}(t)], e^{\beta_{1}s}[\phi(t)-\overline{\phi}(t)],e^{\beta_{2}s}[\varphi (t)-\underline{\varphi}(t)], e^{\beta_{2}s}[\varphi(t)-\overline{\varphi}(t)], \\ &\, e^{\beta_{3}s}[\psi(t)-\underline{\psi}(t)], e^{\beta_{3}s}[\psi(t)-\overline{\psi}(t)] \mbox{ are nondecreasing for } t \in R \end{array}\displaystyle \right \}. \end{aligned}$$

It is easy to see that \(\Gamma((\underline{\phi },\underline{\varphi},\underline{\psi}),( \overline{\phi },\overline{\varphi},\overline{\psi}))\) is nonempty, convex, closed, and bounded.

In the following, we assume that there exist a pair of upper and lower solutions \((\overline{\phi}(t), \overline{\varphi}(t), \overline {\psi}(t))\), \((\underline{\phi}(t),\underline{\varphi}(t),\underline{\psi }(t))\) of (6) satisfying the conditions (P1) and (P2):

  1. (P1)

    \((0,0,0)\leq(\underline{\phi}(t),\underline{\varphi }(t),\underline{\psi}(t))\leq(\overline{\phi }(t), \overline{\varphi}(t), \overline{\psi}(t))\leq(M_{1}, M_{2},M_{3})\), \(t \in R\).

  2. (P2)

    \(\lim_{t\rightarrow -\infty}(\underline{\phi}(t),\underline{\varphi}(t),\underline {\psi}(t))=(0,0,0)\), \(\lim_{t\rightarrow +\infty}(\overline{\phi}(t),\overline{\varphi}(t),\overline{\psi }(t))=(k_{1},k_{2},k_{3})\).

Define the operators \(H_{i}:C(R,R^{3})\rightarrow C(R,R^{3})\) by

$$ H_{i}(\phi,\varphi,\psi) (t)=f_{ci}( \phi_{t},\varphi_{t},\psi _{t})+ \beta_{i}\theta_{i}(t),\quad \phi,\varphi,\psi\in C(R,R), i=1,2,3, $$
(11)

where

$$\theta_{i}(t)= \left \{ \textstyle\begin{array}{l@{\quad}l} \phi(t), & \mbox{if } i=1, \\ \varphi(t), & \mbox{if } i=2, \\ \psi(t), & \mbox{if } i=3, \end{array}\displaystyle \right . $$

and the constants \(\beta_{i}>0\) are as in inequalities (8). The operators \(H_{i}\), \(i=1,2,3\) satisfy the following properties.

Lemma 1

Assume that (A1) and (8) hold, for \(t\in R\) with \(0\leq\phi _{2}(t)\leq\phi_{1}(t)\leq M _{1}\), \(0\leq\varphi_{2}(t)\leq\varphi_{1}(t)\leq M_{2}\), \(0\leq\psi_{2}(t)\leq\psi_{1}(t)\leq M_{3}\), then

$$\begin{aligned}& H_{1}(\phi_{2},\varphi_{1},\psi_{1}) \leq H_{1}(\phi_{1},\varphi _{1}, \psi_{1}) ,\qquad H_{1}(\phi_{1}, \varphi_{1},\psi_{1})\leq H_{1}( \phi_{1},\varphi_{2},\psi_{2}), \\& H_{2}(\phi_{1},\varphi_{2},\psi_{2}) \leq H_{2}(\phi_{1},\varphi _{1}, \psi_{1}) ,\qquad H_{2}(\phi_{1}, \varphi_{1},\psi_{1})\leq H_{2}( \phi_{2},\varphi_{1},\psi_{1}), \\& H_{3}(\phi_{1},\varphi_{2},\psi_{2}) \leq H_{3}(\phi_{1},\varphi _{1}, \psi_{1}) ,\qquad H_{3}(\phi_{1}, \varphi_{1},\psi_{1})\leq H_{3}( \phi_{2},\varphi_{1},\psi_{1}). \end{aligned}$$

Proof

From (8), a direct calculation shows that

$$\begin{aligned}& H_{1}(\phi_{1},\varphi_{1}, \psi_{1})-H_{1}(\phi_{2},\varphi _{1}, \psi_{1})=f_{c1}(\phi _{1},\varphi_{1}, \psi_{1})-f_{c1}(\phi_{2},\varphi_{1}, \psi _{1})+\beta_{1}\bigl[\phi_{1}(0)- \phi_{2}(0)\bigr]\geq0, \\& H_{1}(\phi_{1},\varphi_{1}, \psi_{1})-H_{1}(\phi_{1},\varphi _{2}, \psi_{2})=f_{c1}(\phi _{1},\varphi_{1}, \psi_{1})-f_{c1}(\phi_{1},\varphi_{2}, \psi _{2})\leq0, \\& H_{2}(\phi_{1},\varphi_{1},\psi_{1})-H_{2}( \phi_{1},\varphi _{2},\psi_{2})=f_{c2}( \phi _{1},\varphi_{1},\psi_{1})-f_{c1}( \phi_{1},\varphi_{2},\psi _{2})+\beta_{2} \bigl[\varphi_{1}(0)-\varphi_{2}(0)\bigr]\geq0, \\& H_{2}(\phi_{1},\varphi_{1},\psi_{1})-H_{2}( \phi_{2},\varphi _{1},\psi_{1})=f_{c2}( \phi _{1},\varphi_{1},\psi_{1})-f_{c2}( \phi_{2},\varphi_{1},\psi _{1})\leq0, \\& H_{3}(\phi_{1},\varphi_{1},\psi_{1})-H_{3}( \phi_{1},\varphi _{2},\psi_{2})=f_{c3}( \phi _{1},\varphi_{1},\psi_{1})-f_{c3}( \phi_{1},\varphi_{2},\psi _{2})+\beta_{3} \bigl[\psi_{1}(0)-\psi_{2}(0)\bigr]\geq0, \\& H_{3}(\phi_{1},\varphi_{1},\psi_{1})-H_{3}( \phi_{2},\varphi _{1},\psi_{1})=f_{c3}( \phi _{1},\varphi_{1},\psi_{1})-f_{c3}( \phi_{2},\varphi_{1},\psi _{1})\leq0. \end{aligned}$$

 □

From the definitions of \(H_{1}\), \(H_{2}\), and \(H_{3}\) in (11), system (6) can be rewritten as

$$ d_{i}\theta_{i}''(t)-c \theta_{i}'(t)-\beta_{1}\theta_{i}(t)+H_{i}( \phi ,\varphi,\psi) (t)=0, \quad i=1,2,3. $$
(12)

We define

$$\begin{aligned}& \lambda_{1}=\frac{c-\sqrt{c^{2}+4\beta_{1}d_{1}}}{2d_{1}},\qquad \lambda _{2}=\frac{c+\sqrt{c^{2}+4\beta_{1}d_{1}}}{2d_{1}}, \\& \lambda_{3}=\frac{c-\sqrt{c^{2}+4\beta_{2}d_{2}}}{2d_{2}},\qquad \lambda _{4}=\frac{c+\sqrt{c^{2}+4\beta_{2}d_{2}}}{2d_{2}}, \\& \lambda_{5}=\frac{c-\sqrt{c^{2}+4\beta_{3}d_{3}}}{2d_{3}},\qquad \lambda _{6}=\frac{c+\sqrt{c^{2}+4\beta_{3}d_{3}}}{2d_{3}}. \end{aligned}$$

For \((\phi,\varphi,\psi)\in C_{k}(R,R^{3})\), we define \(F=(F_{1},F_{2},F_{3}):C_{k}(R,R^{3})\rightarrow C(R,R^{3})\) by

$$\begin{aligned}& F_{1}(\phi,\varphi,\psi) (t) \\& \quad =\frac{1}{d_{1}(\lambda_{2}-\lambda _{1})}\biggl[ \int_{-\infty}^{t}e^{\lambda_{1}(t-s)}H_{1}(\phi, \varphi,\psi ) (s)\,ds+ \int_{t}^{+\infty}e^{\lambda_{2}(t-s)}H_{1}(\phi, \varphi ,\psi) (s)\,ds\biggr], \\& F_{2}(\phi,\varphi,\psi) (t) \\& \quad =\frac{1}{d_{2}(\lambda_{4}-\lambda _{3})}\biggl[ \int_{-\infty}^{t}e^{\lambda_{3}(t-s)}H_{2}(\phi, \varphi,\psi ) (s)\,ds+ \int_{t}^{+\infty}e^{\lambda_{4}(t-s)}H_{2}(\phi, \varphi ,\psi) (s)\,ds\biggr], \\& F_{3}(\phi,\varphi,\psi) (t) \\& \quad =\frac{1}{d_{3}(\lambda_{6}-\lambda _{5})}\biggl[ \int_{-\infty}^{t}e^{\lambda_{5}(t-s)}H_{3}(\phi, \varphi,\psi ) (s)\,ds+ \int_{t}^{+\infty}e^{\lambda_{6}(t-s)}H_{3}(\phi, \varphi ,\psi) (s)\,ds\biggr]. \end{aligned}$$

It is easy to see that \(F_{i}(\phi,\varphi,\psi)\) (\(i=1,2,3\)) satisfy

$$ d_{i}F_{i}''(\phi, \varphi,\psi)-cF_{i}'(\phi,\varphi,\psi)-\beta _{i}F_{i}(\phi,\varphi,\psi)+H_{i}(\phi,\varphi, \psi)=0. $$
(13)

Corresponding to Lemma 1, we have the same results of F.

Lemma 2

Assume that (A2) holds, then \(F=(F_{1},F_{2},F_{3})\) is continuous with respective to the norm \(|\cdot|\) in \(B_{\mu}(R,R^{3})\).

Lemma 3

Assume that (A2) and (8) hold, then

$$F\bigl(\Gamma\bigl((\underline{\phi},\underline{\varphi},\underline{\psi }),( \overline{\phi},\overline{\varphi},\overline{\psi})\bigr)\bigr)\subset \Gamma \bigl((\underline{\phi},\underline{\varphi},\underline{\psi }),(\overline{\phi}, \overline{\varphi},\overline{\psi})\bigr). $$

Lemma 4

Assume that (8) holds, then

$$F:\Gamma\bigl((\underline{\phi},\underline{\varphi},\underline{\psi }),( \overline{\phi},\overline{\varphi},\overline{\psi })\bigr)\rightarrow \Gamma \bigl((\underline{\phi},\underline{\varphi},\underline{\psi }),(\overline{\phi}, \overline{\varphi},\overline{\psi})\bigr) $$

is compact.

Remark 1

The proofs of Lemmas 2-4 are similar to those of Lemmas 3.4-3.6 in [19], and we omit them here.

Theorem 1

Assume that (A1), (A2), and (8) hold. Suppose there is a pair of upper and lower solutions \({\Phi}=(\overline{\phi},\overline{\varphi},\overline{\psi})\), and \({\Psi}=(\underline{\phi},\underline{\varphi},\underline{\psi})\) for (6) satisfying (P1) and (P2), then system (1) has a traveling wave solution.

Proof

Combining Lemmas 1-4 with Schauder’s fixed point theorem, we know that there exists a fixed point \((\phi^{\ast}(t), \varphi^{\ast}(t), \psi^{\ast}(t))\) of F in \(\Gamma((\underline{\phi},\underline{\varphi},\underline{\psi }),(\overline{\phi},\overline{\varphi},\overline{\psi}))\), which gives a solution of (6).

From (P2) and the fact that

$$(0,0,0)\leq (\underline{\phi},\underline{\varphi},\underline{\psi})\leq \bigl( \phi^{\ast}(t), \varphi^{\ast}(t), \psi^{\ast}(t)\bigr) \leq (\overline{\phi},\overline{\varphi},\overline{\psi}) \leq (M_{1},M_{2},M_{3}), $$

we know that

$$\lim_{t\rightarrow-\infty}\bigl(\phi^{\ast}(t), \varphi^{\ast }(t), \psi^{\ast}(t)\bigr)=(0,0,0);\qquad \lim_{t\rightarrow+\infty}\bigl( \phi^{\ast}(t), \varphi^{\ast}(t), \psi^{\ast}(t) \bigr)=(k_{1},k_{2},k_{3}). $$

Therefore, the fixed point \((\phi^{\ast}(t), \varphi^{\ast}(t), \psi^{\ast}(t))\) satisfies the asymptotic boundary conditions (7). □

3 Existence of traveling waves

In this section, we will apply Theorem 1 to establish the existence of traveling wave solutions for system (1). Assuming that

$$\begin{aligned}& D=\left \vert \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\displaystyle \right \vert >0, \qquad D_{1}=\left \vert \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1 & a_{12} & a_{13} \\ 1 & a_{22} & a_{23} \\ 1 & a_{32} & a_{33} \end{array}\displaystyle \right \vert >0, \\& D_{2}=\left \vert \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} a_{11} & 1 & a_{13} \\ a_{21} & 1 & a_{23} \\ a_{31} & 1 & a_{33} \end{array}\displaystyle \right \vert >0, \qquad D_{3}=\left \vert \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} a_{11} & a_{12} & 1 \\ a_{21} & a_{22} & 1 \\ a_{31} & a_{32} & 1 \end{array}\displaystyle \right \vert >0. \end{aligned}$$

We are interested in looking for a traveling wave solution of (1) connecting \((0, 0, 0)\) and a positive equilibrium \((k_{1}, k_{2}, k_{3})\). Here \(k_{i}=\frac{D_{i}}{D}\) (\(i=1,2,3\)) are the roots of the following equations:

$$ \left \{ \textstyle\begin{array}{l} a_{11}k_{1}+a_{12}k_{2}+a_{13}k_{3}=1, \\ a_{21}k_{1}+a_{22}k_{2}+a_{23}k_{3}=1, \\ a_{31}k_{1}+a_{32}k_{2}+a_{33}k_{3}=1. \end{array}\displaystyle \right . $$
(14)

Substituting \(s=x+ct\) into (1) and denoting the variable s still by t, then the corresponding wave profile equations are

$$ \left \{ \textstyle\begin{array}{l} d_{1}\phi''(t)-c\phi'(t)+r_{1}\phi(t)(1-a_{11}\phi(t-\tau _{11})-a_{12}\varphi(t-\tau_{12})-a_{13}\psi(t-\tau_{13}))=0, \\ d_{2}\varphi''(t)-c\varphi'(t)+r_{2}\varphi(t)(1-a_{21}\phi(t-\tau _{21})-a_{22}\varphi(t-\tau_{22})+a_{23}\psi(t-\tau_{23}))=0, \\ d_{3}\psi''(t)-c\psi'(t)+r_{3}\psi(t)(1+a_{31}\phi(t-\tau _{31})+a_{32}\varphi(t-\tau_{32})-a_{33}\psi(t-\tau_{33}))=0. \end{array}\displaystyle \right . $$
(15)

Lemma 5

Assume that \(\tau_{ii}\) (\(i=1,2,3\)) are small enough, then the functions \((f_{1},f_{2},f_{3})\) satisfy (PQM).

Proof

For any \(\phi_{1}(s), \phi_{2}(s), \varphi_{1}(s), \varphi_{2}(s), \psi_{1}(s), \psi_{2}(s) \in C([-\tau, 0], R)\),

  1. (i)

    \(0\leq\phi_{2}(s)\leq\phi_{1}(s)\leq M_{1}\), \(0\leq\varphi_{2}(s)\leq\varphi_{1}(s)\leq M_{2}\), \(0\leq\psi_{2}(s)\leq\psi_{1}(s)\leq M_{3}\), \(s\in[-\tau, 0]\);

  2. (ii)

    \(e^{\beta_{1}s}(\phi_{1}(s)-\phi_{2}(s))\), \(e^{\beta_{2}s}(\varphi_{1}(s)-\varphi_{2}(s))\), and \(e^{\beta_{3}s}(\psi_{1}(s)-\psi_{2}(s))\) are nondecreasing in \(s\in[-\tau, 0]\).

If \(\tau_{11}\) is small enough, we can choose \(\beta_{1}>0\) satisfying

$$\begin{aligned}& f_{c1}(\phi_{1}, \varphi_{1}, \psi_{1})-f_{c1}(\phi_{2}, \varphi_{1}, \psi_{1}) \\& \quad = r_{1}\phi_{1}(0) \bigl(1-a_{11} \phi_{1}(-\tau_{11})-a_{12}\varphi _{1}(-\tau_{12})-a_{13}\psi_{1}(- \tau_{13})\bigr) \\& \qquad {} -r_{1}\phi_{2}(0) \bigl(1-a_{11} \phi_{2}(-\tau_{11})-a_{12}\varphi _{1}(- \tau_{12})-a_{13}\psi_{1}(-\tau_{13}) \bigr) \\& \quad = r_{1}\bigl(\phi_{1}(0)-\phi_{2}(0) \bigr)-a_{11}r_{1}\bigl(\phi_{1}(0)\phi _{1}(-\tau_{11})-\phi_{2}(0)\phi_{2}(- \tau_{11})\bigr) \\& \qquad {} -a_{12}r_{1}\varphi_{1}(- \tau_{12}) \bigl(\phi_{1}(0)-\phi _{2}(0) \bigr)-a_{13}r_{1}\psi_{1}(-\tau_{13}) \bigl(\phi_{1}(0)-\phi_{2}(0)\bigr) \\& \quad \geq r_{1}(1-a_{12}M_{2}-a_{13}M_{3}-a_{11}M_{1}) \bigl(\phi_{1}(0)-\phi _{2}(0)\bigr) \\& \qquad {}-a_{11}r_{1} \phi_{2}(0) \bigl(\phi_{1}(-\tau_{11})- \phi_{2}(-\tau _{11})\bigr) \\& \quad \geq r_{1}\bigl(1-a_{12}M_{2}-a_{13}M_{3}-a_{11}M_{1}-a_{11}M_{1}e^{\beta _{1}\tau_{11}} \bigr) \bigl(\phi_{1}(0)-\phi_{2}(0)\bigr). \end{aligned}$$

Let

$$\beta_{1}\geq -r_{1}\bigl(1-a_{12}M_{2}-a_{13}M_{3}-a_{11}M_{1}-a_{11}M_{1}e^{\beta _{1}\tau_{11}} \bigr), $$

then it is easy to show that \(f_{c1}(\phi_{1},\varphi_{1},\psi_{1})-f_{c1}(\phi_{2},\varphi _{1},\psi_{1})+\beta_{1}(\phi_{1}(0)-\phi_{2}(0))\geq0\), and

$$\begin{aligned}& f_{c1}(\phi_{1}, \varphi_{1}, \psi_{1})-f_{c1}(\phi_{1}, \varphi_{2}, \psi_{2}) \\& \quad = r_{1}\phi_{1}(0) \bigl(1-a_{11} \phi_{1}(-\tau_{11})-a_{12}\varphi _{1}(-\tau_{12})-a_{13}\psi_{1}(- \tau_{13})\bigr) \\& \qquad {} -r_{1}\phi_{1}(0) \bigl(1-a_{11} \phi_{1}(-\tau_{11})-a_{12}\varphi _{2}(- \tau_{12})-a_{13}\psi_{2}(-\tau_{13}) \bigr) \\& \quad = r_{1}\phi_{1}(0) \bigl(a_{12}\bigl( \varphi_{2}(-\tau_{12})-\varphi _{1}(- \tau_{12})\bigr)+a_{13}\bigl(\psi_{2}(- \tau_{13})-\psi_{1}(-\tau _{13})\bigr)\bigr) \\& \quad \leq 0. \end{aligned}$$

For \(f_{c2}\), we have

$$\begin{aligned}& f_{c2}(\phi_{1}, \varphi_{1}, \psi_{1})-f_{c2}(\phi_{1},\varphi _{2}, \psi_{2}) \\& \quad = r_{2}\varphi_{1}(0) \bigl(1-a_{21} \phi_{1}(-\tau_{21})-a_{22}\varphi _{1}(-\tau_{22})-a_{23}\psi_{1}(- \tau_{23})\bigr) \\& \qquad {} -r_{2}\varphi_{2}(0) \bigl(1-a_{21} \phi_{1}(-\tau_{21})-a_{22}\varphi _{2}(- \tau_{22})-a_{23}\psi_{2}(-\tau_{23}) \bigr) \\& \quad = r_{2}\bigl(\varphi_{1}(0)-\varphi_{2}(0) \bigr)-a_{21}r_{2}\bigl(\varphi _{1}(0) \phi_{1}(-\tau_{21})-\varphi_{2}(0) \phi_{1}(-\tau_{21})\bigr) \\& \qquad {} -a_{22}r_{2}\bigl(\varphi_{1}(0) \varphi_{1}(-\tau_{22})-\varphi _{2}(0) \varphi_{2}(-\tau_{22})\bigr) \\& \qquad {}-a_{23}r_{2} \bigl(\varphi_{1}(0)\psi _{1}(-\tau_{23})- \varphi_{2}(0)\psi_{2}(-\tau_{23})\bigr) \\& \quad \geq r_{2}(1-a_{21}M_{1}) \bigl( \varphi_{1}(0)-\varphi _{2}(0)\bigr)-a_{22}r_{2} \bigl(\varphi_{2}(0)\varphi_{1}(-\tau_{22})-\varphi _{2}(0)\varphi_{2}(-\tau_{22}) \\& \qquad {}+\bigl( \varphi_{1}(0) -\varphi_{2}(0)\bigr)\varphi_{1}(- \tau_{22})\bigr)+a_{23}r_{2}\bigl(\varphi _{2}(0)\psi_{1}(-\tau_{23})-\varphi_{2}(0) \psi_{2}(-\tau _{23}) \\& \qquad {}+\bigl(\varphi_{1}(0)- \varphi_{2}(0)\bigr)\psi_{1}(-\tau_{23})\bigr) \\& \quad \geq r_{1}\bigl(1-a_{21}M_{1}-a_{23}M_{3}-a_{21}M_{1}-a_{22}M_{2}e^{\beta _{2}\tau_{22}}-a_{23}M_{3}e^{\beta_{3}\tau_{33}} \bigr) \bigl(\varphi _{1}(0)-\varphi_{2}(0)\bigr). \end{aligned}$$

Let \(\beta_{2}\geq r_{2}(1-a_{21}M_{1}-a_{23}M_{3}-a_{21}M_{1}-a_{22}M_{2}e^{\beta _{2}\tau_{22}}-a_{23}M_{3}e^{\beta_{3}\tau_{33}})\), then

$$f_{c2}(\phi_{1},\varphi_{1}, \psi_{1})-f_{c2}(\phi_{1},\varphi _{2}, \psi_{2})+\beta_{2}\bigl[\varphi_{1}(0)- \varphi_{2}(0)\bigr]\geq0 $$

and

$$\begin{aligned}& f_{c2}(\phi_{1}, \varphi_{1}, \psi_{1})-f_{c2}(\phi_{2}, \varphi_{1}, \psi_{1}) \\& \quad = r_{2}\varphi_{1}(0) \bigl(1-a_{21} \phi_{1}(-\tau_{21})-a_{22}\varphi _{1}(-\tau_{22})-a_{23}\psi_{1}(- \tau_{23})\bigr) \\& \qquad {} -r_{2}\varphi_{1}(0) \bigl(1-a_{21} \phi_{2}(-\tau_{21})-a_{22}\varphi _{1}(- \tau_{22})-a_{23}\psi_{1}(-\tau_{23}) \bigr) \\& \quad = r_{2}\varphi_{1}(0)a_{21}\bigl( \phi_{2}(-\tau_{21})-\phi(-\tau _{21})\bigr) \\& \quad \leq 0. \end{aligned}$$

In a similar way for \(f_{c3}\), we let \(\beta_{3}>r_{3}(1-a_{33}M_{3}-a_{33}M_{3}e^{\beta_{3}\tau_{33}})\), then \(f_{c3}(\phi_{1},\varphi_{1},\psi_{1})-f_{c3}(\phi_{1},\varphi _{2},\psi_{2})+\beta_{3}[\psi_{1}(0)-\psi_{2}(0)]\geq0\), and \(f_{c3}(\phi_{1},\varphi_{1},\psi_{1})-f_{c3}(\phi_{2},\varphi _{1},\psi_{1})\leq0\). This completes the proof. □

Let

$$c>c^{*}=\max\bigl(2\sqrt{d_{1}r_{1}},2\sqrt {d_{2}r_{2}(1+a_{23}M_{3})},2 \sqrt{d_{3}r_{3}(1+a_{31}M_{1}+a_{32}M_{2})} \bigr). $$

There exist \(\lambda_{i}> 0\) (\(i=1,3,5\)) so that

$$\begin{aligned}& d_{1}\lambda_{1}^{2}-c\lambda_{1}+r_{1}=0, \\& d_{2}\lambda _{3}^{2}-c\lambda_{3}+r_{2}(1+a_{23}M_{3})=0, \\& d_{2}\lambda_{5}^{2}-c\lambda_{5}+r_{3}(1+a_{31}M_{1}+a_{32}M_{2})=0. \end{aligned}$$

We find that there exist \(\varepsilon_{i}> 0\) (\(i=0,1,2,3,4,5,6\)) satisfying

$$ \left \{ \textstyle\begin{array}{l} a_{11}\varepsilon_{1}-a_{12}\varepsilon_{4}-a_{13}\varepsilon _{6}>\varepsilon_{0}, \\ -a_{21}\varepsilon_{2}+a_{22}\varepsilon_{3}-a_{23}\varepsilon _{5}>\varepsilon_{0}, \\ a_{31}\varepsilon_{2}-a_{32}\varepsilon_{3}+\varepsilon _{5}>\varepsilon_{0}, \\ \varepsilon_{2}-a_{21}\varepsilon_{3}-a_{13}\varepsilon _{5}>\varepsilon_{0}, \\ -a_{11}\varepsilon_{1}+\varepsilon_{4}+a_{13}\varepsilon _{5}>\varepsilon_{0}, \\ 1-k_{3}+\varepsilon_{6}>\varepsilon_{0}. \end{array}\displaystyle \right . $$
(16)

For the above constants and suitable constants \(t_{i} >0\) (\(i=1,2,3,4,5,6\)), we define the continuous functions \(\overline{\Phi}=(\overline{\phi}(t),\overline{\varphi }(t),\overline{\psi}(t))\) and \(\underline{\Psi}=(\underline{\phi}(t),\underline{\varphi }(t),\underline{\psi}(t))\) as follows:

$$\begin{aligned}& \overline{\phi}(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} e^{\lambda_{1}t}, & t\leq t_{1}, \\ k_{1}+\varepsilon_{1}e^{-\lambda t}, & t>t_{1}, \end{array}\displaystyle \right .\qquad \underline{\phi}(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} 0, & t\leq t_{2}, \\ k_{1}-\varepsilon_{2}e^{-\lambda t}, & t>t_{2},\end{array}\displaystyle \right . \\& \overline{\varphi}(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} e^{\lambda_{3}t}, & t\leq t_{3}, \\ k_{2}+\varepsilon_{3}e^{-\lambda t}, & t>t_{3},\end{array}\displaystyle \right .\qquad \underline{\varphi}(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} 0, & t\leq t_{4}, \\ k_{2}-\varepsilon_{4}e^{-\lambda t}, & t>t_{4},\end{array}\displaystyle \right . \\& \overline{\psi}(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} e^{\lambda_{5}t}, & t\leq t_{5}, \\ k_{3}+\varepsilon_{5}e^{-\lambda t}, & t>t_{5},\end{array}\displaystyle \right .\qquad \underline{\psi}(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} 0, & t\leq t_{6}, \\ k_{3}-\varepsilon_{6}e^{-\lambda t}, & t>t_{6},\end{array}\displaystyle \right . \end{aligned}$$

where \(\lambda>0\) is a constant to be chosen later and

$$\min\{t_{1}, t_{3}, t_{5}\}-c\max\{ \tau_{ij}, i,j=1,2,3\}\geq \max\{ t_{2}, t_{4}, t_{6}\},\quad t_{4}>t_{6}. $$

Lemma 6

Assume that \(D>0\), \(D_{i}>0\) (\(i=1,2,3\)) and (16) hold, then \(\overline{\Phi}=(\overline{\phi}(t),\overline{\varphi }(t),\overline{\psi}(t))\) is an upper solution of system (15).

Proof

When \(t> t_{1}+c\tau_{11}\), \(\overline{\phi}(t)=k_{1}+\varepsilon_{1}e^{-\lambda t}\), we have

$$\begin{aligned}& d_{1}\overline{\phi}''(t)-c\overline{ \phi}'(t)+r_{1}\overline {\phi}(t)\bigl[1-a_{11} \overline{\phi}(t-c\tau_{11})-a_{12}\underline {\varphi}(t-c \tau_{12})-a_{13}\underline{\psi}(t-c\tau_{13}) \bigr] \\& \quad = d_{1}\varepsilon_{1}\lambda^{2}e^{-\lambda t}+c \varepsilon_{1}\lambda e^{-\lambda t}+r_{1} \bigl(k_{1}+\varepsilon_{1}e^{-\lambda t}\bigr) \bigl(1-a_{11}\bigl(k_{1}+\varepsilon_{1}e^{-\lambda(t-c\tau_{11})} \bigr) \\& \qquad {} -a_{12}\bigl(k_{2}-\varepsilon_{4}e^{-\lambda (t-c\tau_{12})} \bigr)-a_{13}\bigl(k_{3}-\varepsilon_{6}e^{-\lambda (t-c\tau_{13})} \bigr)\bigr) \\& \quad =: I_{1}(\lambda). \end{aligned}$$

Obviously,

$$\begin{aligned} I_{1}(0) =& r_{1}(k_{1}+\varepsilon_{1}) \bigl(1-a_{11}(k_{1}+\varepsilon _{1})-a_{12}(k_{2}- \varepsilon_{4})-a_{13}(k_{3}-\varepsilon_{6}) \bigr) \\ =&r_{1}(k_{1}+\varepsilon_{1}) (-a_{11}\varepsilon _{1}+a_{12} \varepsilon_{4}+a_{13}\varepsilon_{6}). \end{aligned}$$

It is easy to see that \(I_{1}(0)<0\) and there exists \(\lambda^{\ast}_{1}>0\), such that

$$d_{1}\overline{\phi}''(t)-c\overline{ \phi}'(t)+r_{1}\overline{\phi }(t)\bigl[1-a_{11} \overline{\phi}(t-c\tau_{11})-a_{12}\underline{\varphi }(t-c \tau_{12})-a_{13}\underline{\psi}(t-c\tau_{13}) \bigr]\leq0, $$

for all \(\lambda \in (0,\lambda^{\ast}_{1})\).

If \(t\leq t_{1}\), \(\overline{\phi}(t)=e^{\lambda_{1} t}\), we have

$$\begin{aligned}& d_{1}\overline{\phi}''(t)-c\overline{ \phi}'(t)+r_{1}\overline {\phi}(t)\bigl[1-a_{11} \overline{\phi}(t-c\tau_{11})-a_{12}\underline {\varphi}(t-c \tau_{12})-a_{13}\underline{\psi}(t-c\tau_{13}) \bigr] \\& \quad \leq d_{1}\lambda^{2}_{1}e^{\lambda_{1}t}-c \lambda_{1}e^{\lambda _{1}t}+r_{1}e^{\lambda_{1}t}=0. \end{aligned}$$

If \(t_{1}< t\leq t_{1}+c\tau_{11}\), then we have

$$\begin{aligned}& d_{1}\overline{\phi}''(t)-c\overline{ \phi}'(t)+r_{1}\overline {\phi}(t)\bigl[1-a_{11} \overline{\phi}(t-c\tau_{11})-a_{12}\underline {\varphi}(t-c \tau_{12})-a_{13}\underline{\psi}(t-c\tau_{13}) \bigr] \\& \quad = d_{1}\varepsilon_{1}\lambda^{2}e^{-\lambda t}+c \varepsilon_{1}\lambda e^{-\lambda t}+r_{1} \bigl(k_{1}+\varepsilon_{1}e^{-\lambda t}\bigr) \bigl(1-a_{11}e^{\lambda_{1}(t_{1}-c\tau _{11})} \\& \qquad {} -a_{12}\bigl(k_{2}- \varepsilon_{4}e^{-\lambda t}\bigr)-a_{13}\bigl(k_{3}-\varepsilon_{6}e^{-\lambda t} \bigr)\bigr) \\& \quad =: I_{2}(\lambda). \end{aligned}$$

For small enough \(\tau_{11}\), there exists \(\varepsilon^{\ast}_{1}\) (\(0<\varepsilon^{\ast}_{1}<\frac{\varepsilon _{0}}{a_{11}(k_{1}+\varepsilon_{1})}\)) such that \(e^{-\lambda_{1}c\tau_{11}}>1-\varepsilon^{\ast}_{1}\). Thus we have

$$\begin{aligned} I_{2}(0) =& r_{1}(k_{1}+\varepsilon_{1}) \bigl(1-a_{11}e^{\lambda _{1}(t_{1}-c\tau_{11})}-a_{12}(k_{2}- \varepsilon _{4})-a_{13}(k_{3}- \varepsilon_{6})\bigr) \\ =&r_{1}(k_{1}+\varepsilon_{1}) \bigl(a_{11}k_{1}+a_{12}\varepsilon _{4}+a_{13}\varepsilon_{6}-a_{11}e^{-\lambda_{1}c\tau _{11}}(k_{1}+ \varepsilon_{1})\bigr) \\ \leq&r_{1}(k_{1}+\varepsilon_{1}) \bigl(a_{11}k_{1}+a_{12}\varepsilon _{4}+a_{13}\varepsilon_{6}-a_{11} \bigl(1-\varepsilon^{\ast }_{1}\bigr) (k_{1}+ \varepsilon_{1})\bigr) \\ < &r_{1}(k_{1}+\varepsilon_{1}) \bigl(- \varepsilon_{0}+a_{11}\varepsilon ^{\ast}_{1}(k_{1}+ \varepsilon_{1})\bigr) \\ < &0. \end{aligned}$$

Therefore, there exists a \(\lambda^{\ast}_{2}\), such that for all \(\lambda \in (0,\lambda^{\ast}_{2})\), we have

$$d_{1}\overline{\phi}''(t)-c\overline{ \phi}'(t)+r_{1}\overline{\phi }(t)\bigl[1-a_{11} \overline{\phi}(t-c\tau_{11})-a_{12}\underline{\varphi }(t-c \tau_{12})-a_{13}\underline{\psi}(t-c\tau_{13}) \bigr]\leq0. $$

From the above argument, we see that

$$d_{1}\overline{\phi}''(t)-c\overline{ \phi}'(t)+r_{1}\overline{\phi }(t)\bigl[1-a_{11} \overline{\phi}(t-c\tau_{11})-a_{12}\underline{\varphi }(t-c \tau_{12})-a_{13}\underline{\psi}(t-c\tau_{13}) \bigr]\leq0, $$

for small enough \(\lambda \in (0,\overline{\lambda}^{\ast}_{1})\), where \(\overline{\lambda}^{\ast}_{1}=\min\{\lambda^{\ast}_{1},\lambda ^{\ast}_{2}\}\).

When \(t> t_{3}+c\tau_{22}\), \(\overline{\varphi}(t)=k_{2}+\varepsilon_{3}e^{-\lambda t}\), we have

$$\begin{aligned}& d_{2}\overline{\varphi}''(t)-c\overline{ \varphi }'(t)+r_{2}\overline{\varphi}(t) \bigl[1-a_{21}\underline{\phi}(t-c\tau _{21})-a_{22} \overline{\varphi}(t-c\tau_{22})-a_{23}\overline{\psi }(t-c \tau_{23})\bigr] \\& \quad \leq d_{2}\varepsilon_{3}\lambda^{2}e^{-\lambda t}+c \varepsilon_{3}\lambda e^{-\lambda t}+r_{3} \bigl(k_{2}+\varepsilon_{3}e^{-\lambda t}\bigr) \bigl(1-a_{21}\bigl(k_{1}-\varepsilon_{2}e^{-\lambda t} \bigr) \\& \qquad {} -a_{22}\bigl(k_{2}+\varepsilon_{3}e^{-\lambda t} \bigr)+a_{23}(k_{3}+\varepsilon_{5})\bigr) \\& \quad =: I_{3}(\lambda). \end{aligned}$$

Obviously,

$$\begin{aligned} I_{3}(0) =& r_{3}(k_{2}+\varepsilon_{3}) \bigl(1-a_{21}(k_{1}-\varepsilon _{2})-a_{22}(k_{2}+ \varepsilon_{3})-a_{23}(k_{3}+\varepsilon_{5}) \bigr) \\ =&r_{3}(k_{2}+\varepsilon_{3}) (a_{21} \varepsilon _{2}-a_{22}\varepsilon_{3}+a_{23} \varepsilon_{5}). \end{aligned}$$

It is easy to see that \(I_{3}(0)<0\) and there exists \(\lambda^{\ast}_{3}>0\), such that

$$d_{2}\overline{\varphi}''(t)-c\overline{ \varphi}'(t)+r_{2}\overline {\varphi}(t) \bigl[1-a_{21}\underline{\phi}(t-c\tau_{21})-a_{22} \overline {\varphi}(t-c\tau_{22})-a_{23}\overline{\psi}(t-c \tau_{23})\bigr]\leq0, $$

for all \(\lambda \in (0,\lambda^{\ast}_{3})\).

If \(t\leq t_{3}\), \(\overline{\varphi}(t)=e^{\lambda_{3} t}\), we have

$$\begin{aligned}& d_{2}\overline{\varphi}''(t)-c\overline{ \varphi }'(t)+r_{2}\overline{\varphi}(t) \bigl[1-a_{21}\underline{\phi}(t-c\tau _{21})-a_{22} \overline{\varphi}(t-c\tau_{22})-a_{23}\overline{\psi }(t-c \tau_{23})\bigr] \\& \quad \leq d_{2}\lambda^{2}_{3}e^{\lambda_{3}t}-c \lambda_{3}e^{\lambda _{3}t}+r_{2}e^{\lambda_{3}t}(1+a_{23}M_{3})=0. \end{aligned}$$

If \(t_{3}< t\leq t_{3}+c\tau_{22}\), then we have

$$\begin{aligned}& d_{2}\overline{\varphi}''(t)-c\overline{ \varphi }'(t)+r_{2}\overline{\varphi}(t) \bigl[1-a_{21}\underline{\phi}(t-c\tau _{21})-a_{22} \overline{\varphi}(t-c\tau_{22})+a_{23}\overline{\psi }(t-c \tau_{23})\bigr] \\& \quad < d_{2}\varepsilon_{3}\lambda^{2}e^{-\lambda t}+c \varepsilon_{3}\lambda e^{-\lambda t}+r_{2} \bigl(k_{2}+\varepsilon_{3}e^{-\lambda t}\bigr) \bigl(1-a_{21}\bigl(k_{1}-\varepsilon_{2}e^{-\lambda t} \bigr) \\& \qquad {}-a_{22}e^{\lambda _{3}(t-c\tau_{22})} +a_{23}(k_{3}+\varepsilon_{5})\bigr) \\& \quad =: I_{4}(\lambda). \end{aligned}$$

For small enough \(\tau_{22}\), there exists \(\varepsilon^{\ast}_{2}\) (\(0<\varepsilon^{\ast}_{2}<\frac{\varepsilon _{0}}{a_{22}(k_{2}+\varepsilon_{3})}\)) such that \(e^{-\lambda_{3}c\tau_{22}}>1-\varepsilon^{\ast}_{2}\). Thus we have

$$\begin{aligned} I_{4}(0) \leq& r_{2}(k_{2}+\varepsilon _{3}) \bigl(1-a_{21}(k_{1}-\varepsilon_{2})-a_{22}e^{-\lambda_{3}c\tau _{22}}(k_{2}+ \varepsilon_{3})+a_{23}(k_{3}+\varepsilon_{5}) \bigr) \\ \leq&r_{2}(k_{2}+\varepsilon_{3}) \bigl(1-a_{21}(k_{1}-\varepsilon _{2})-a_{22} \bigl(1-\varepsilon^{\ast}_{2}\bigr) (k_{2}+ \varepsilon _{3})+a_{23}(k_{3}+\varepsilon_{5}) \bigr) \\ < &r_{2}(k_{2}+\varepsilon_{3}) \bigl(- \varepsilon_{0}+(k_{2}+\varepsilon _{3}) \varepsilon^{\ast}_{2}\bigr) \\ < &0. \end{aligned}$$

Therefore, there exists a \(\lambda^{\ast}_{4}\), such that for all \(\lambda \in (0,\lambda^{\ast}_{4})\)

$$d_{2}\overline{\varphi}''(t)-c\overline{ \varphi}'(t)+r_{2}\overline {\varphi}(t) \bigl[1-a_{21}\underline{\phi}(t-c\tau_{21}) -a_{22} \overline{\varphi}(t-c\tau_{22})+a_{23}\overline{\psi }(t-c \tau_{23})\bigr]\leq0. $$

From the above argument, we see that

$$d_{2}\overline{\varphi}''(t)-c\overline{ \varphi}'(t)+r_{2}\overline {\varphi}(t) \bigl[1-a_{21}\underline{\phi}(t-c\tau_{21}) -a_{22} \overline{\varphi}(t-c\tau_{22})+a_{23}\overline{\psi }(t-c \tau_{23})\bigr]\leq0, $$

for small enough \(\lambda \in(0,\overline{\lambda}^{\ast}_{2})\), where \(\overline{\lambda}^{\ast}_{2}=\min\{\lambda^{\ast}_{3},\lambda ^{\ast}_{4}\}\).

Similarly, for all \(t \in R\), there exists a \(\overline{\lambda}^{\ast}_{3}>0\), such that, for \(\lambda \in (0,\overline{\lambda}^{\ast}_{2})\), we have

$$d_{3}\overline{\psi}''(t)-c\overline{ \psi}'(t)+r_{3}\overline{\psi }(t)\bigl[1+a_{31} \underline{\phi}(t-c\tau_{31})+a_{32}\overline{\varphi }(t-c \tau_{32})-a_{33}\overline{\psi}(t-c\tau_{33})\bigr] \leq0. $$

From all of the above argument, we see that \(\overline{\Phi}=(\overline{\phi}(t),\overline{\varphi }(t),\overline{\psi}(t))\) is an upper solution of (15) for small enough \(\lambda \in (0,\hat{\lambda}_{1})\), where \(\hat{\lambda}_{1}=\min\{\lambda^{\ast}_{1},\lambda^{\ast }_{2},\lambda^{\ast}_{3}\}\). □

Lemma 7

Assume that \(D>0\), \(D_{i}>0\) (\(i=1,2,3\)), and (16) hold, then \(\underline{\Psi}(\underline{\phi},\underline{\varphi},\underline {\psi})\) is a lower solution of system (15).

Proof

If \(t\leq t_{2}\),

$$d_{1}\underline{\phi}''(t)-c\underline{ \phi}'(t)+r_{1}\underline {\phi}(t) \bigl[1-a_{11} \underline{\phi}(t-c\tau_{11})-a_{12}\overline{\varphi }(t-c \tau_{12})-a_{13}\overline{\psi}(t-c\tau_{13}) \bigr]=0. $$

If \(t> t_{2}+c\tau_{11}\),

$$\begin{aligned}& d_{1}\underline{\phi}''(t)-c\underline{ \phi}'(t)+r_{1}\underline {\phi}(t)\bigl[1-a_{11} \underline{\phi}(t-c\tau_{11})-a_{12}\overline {\varphi}(t-c \tau_{12})-a_{13}\overline{\psi}(t-c\tau_{13})\bigr] \\& \quad \geq -d_{1}\varepsilon_{2}\lambda^{2}e^{-\lambda t}-c \varepsilon_{2}\lambda e^{-\lambda t}+r_{1} \bigl(k_{1}-\varepsilon_{2}e^{-\lambda t}\bigr) \bigl(1-a_{11}\bigl(k_{1}-\varepsilon_{2}e^{-\lambda t} \bigr) \\& \qquad {}-a_{12}(k_{2}+\varepsilon_{3})-a_{13}(k_{3}+ \varepsilon_{5})\bigr) \\& \quad =: I_{5}(\lambda). \end{aligned}$$

Obviously,

$$\begin{aligned} I_{5}(0) =&r_{1}(k_{1}-\varepsilon_{2}) \bigl(1-a_{11}(k_{1}-\varepsilon _{2})-a_{12}(k_{2}+ \varepsilon_{3})-a_{13}(k_{3}+\varepsilon_{5}) \bigr) \\ =&r_{1}(k_{1}-\varepsilon_{2}) (a_{11}\varepsilon _{2}-a_{12}\varepsilon_{3}-a_{13} \varepsilon_{5}). \end{aligned}$$

\(a_{11}\varepsilon_{2}-a_{12}\varepsilon_{3}-a_{13}\varepsilon _{5}>\varepsilon_{0}\) implies that \(I_{5}(0)>0\) and there exists \(\lambda^{\ast}_{4}>0\) such that

$$d_{1}\underline{\phi}''(t)-c\underline{ \phi}'(t)+r_{1}\underline {\phi}(t)\bigl[1-a_{11} \underline{\phi}(t-c\tau_{11})-a_{12}\overline {\varphi}(t-c \tau_{12})-a_{13}\overline{\psi}(t-c\tau_{13})\bigr] \geq0, $$

for all \(\lambda \in (0,\lambda^{\ast}_{5})\).

If \(t_{2}< t\leq t_{2}+c\tau_{11}\),

$$\begin{aligned} \begin{aligned} &d_{1}\underline{\phi}''(t)-c\underline{ \phi}'(t)+r_{1}\underline {\phi}(t)\bigl[1-a_{11} \underline{\phi}(t-c\tau_{11})-a_{12}\overline {\varphi}(t-c \tau_{12})-a_{13}\overline{\psi}(t-c\tau_{13})\bigr] \\ &\quad \geq -d_{1}\varepsilon_{2}\lambda^{2}e^{-\lambda t}-c \varepsilon_{2}\lambda e^{-\lambda t}+r_{1} \bigl(k_{1}-\varepsilon_{2}e^{-\lambda t}\bigr) \bigl(1-a_{12}(k_{2}+\varepsilon_{3})-a_{13}(k_{3}+ \varepsilon_{5})\bigr) \\ &\quad =: I_{6}(\lambda). \end{aligned} \end{aligned}$$

It is easy to see that \(I_{6}>I_{5}>0\) and

$$d_{1}\underline{\phi}''(t)-c\underline{ \phi}'(t)+r_{1}\underline {\phi}(t) \bigl[1-a_{11} \underline{\phi}(t-c\tau_{11})-a_{12}\overline{\varphi }(t-c \tau_{12})-a_{13}\overline{\psi}(t-c\tau_{13})\bigr] \ge0. $$

Similarly, for all \(t \in R\), there exists a \(\overline{\lambda}^{\ast}_{5}>0\), such that for \(\lambda \in (0,\overline{\lambda}^{\ast}_{5})\), we have

$$d_{2}\underline{\varphi}''(t)-c\underline{ \varphi }'(t)+r_{2}\underline{\varphi}(t) \bigl[1-a_{21}\overline{\phi}(t-c\tau_{21})-a_{22} \underline{\varphi }(t-c\tau_{22})+a_{23}\underline{\psi}(t-c \tau_{23})\bigr]\ge0. $$

For all \(t \in R\), there exists a \(\overline{\lambda}^{\ast}_{6}>0\), such that for \(\lambda \in (0,\overline{\lambda}^{\ast}_{6})\), we have

$$d_{3}\underline{\psi}''(t)-c\underline{ \psi}'(t)+r_{3}\underline {\psi}(t) \bigl[1+a_{31} \overline{\phi}(t-c\tau_{31})+a_{32}\underline{\varphi }(t-c \tau_{32})-a_{33}\underline{\psi}(t-c\tau_{33}) \bigr]\ge0. $$

From all of the above arguments, we see that \(\underline{\Psi}(\underline{\phi},\underline{\varphi},\underline {\psi})\) is a lower solution of (15) for small enough \(\lambda \in (0,\hat{\lambda}_{2})\), where \(\hat{\lambda}_{2}=\min\{\lambda^{\ast}_{4},\lambda^{\ast }_{5},\lambda^{\ast}_{6}\}\). □

Theorem 2

If \(D>0\), \(D_{i}>0\) (\(i=1,2,3\)), and (16) holds for every \(c>c^{*}=\max\{2\sqrt{d_{1}r_{1}}, 2\sqrt {d_{2}r_{2}(1+a_{23}M_{3})},2\sqrt {d_{3}r_{3}(1+a_{31}M_{1}+a_{32}M_{2})}\}\), system (1) has a traveling wave solution with speed c connecting the trivial steady-state solution \((0,0,0)\) and the position steady state \((k_{1},k_{2},k_{3})\).