In this section, we assume that \(\Omega\subset R^{1}\). Denote \(f_{1}(u_{1},u_{2},u_{3})=u_{1}(r_{1}-a_{1}u_{2}-b_{1}u_{1})\), \(f_{2}(u_{1},u_{2},u_{3})=u_{2}(r_{2}-b_{2}u_{2}-a_{2}u_{3} +a_{3}\int_{\Omega}\int_{-\infty }^{t}K_{1}(x,y,t-s)u_{1}(s,y)\,ds\,dy)\), \(f_{3}(u_{1},u_{2},u_{3})=u_{3}(-\alpha-b_{3}u_{3} +a_{4}\int_{\Omega}\int_{-\infty}^{t}K_{2}(x,y,t-s)u_{2}(s,y)\,ds\,dy)\). Let \((u_{1}(t, x), u_{2}(t, x), u_{3}(t, x)) = (\phi(x + c t), \varphi(x +c t), \psi(x + c t))\) be a traveling wave solution of (1.1), where \(\phi,\varphi,\psi\in C^{2}(R,R^{2})\) and \(c > 0\) is a constant accounting for the wave speed, and denote the traveling wave coordinate \(x+ct \) still by t. Then the system (1.1) can be rewritten in the form
$$\begin{aligned}& d_{1}\phi''(t)-c \phi'(t)+f_{c1}(\phi_{t}, \varphi_{t}, \psi_{t})=0, \\& d_{2}\varphi''(t)-c\varphi'(t)+f_{c2}( \phi_{t}, \varphi_{t}, \psi_{t})=0, \\& d_{3}\psi''(t)-c\psi'(t)+f_{c3}( \phi_{t}, \varphi_{t}, \psi_{t})=0, \end{aligned}$$
(3.1)
where \(f_{ci}\) (\(i=1, 2, 3\)) are defined by
$$\begin{aligned}& f_{ci}(\phi,\varphi,\psi)=f_{i}\bigl(\phi^{c}, \varphi^{c},\psi^{c}\bigr),\qquad \phi^{c}(s)=\phi(cs),\qquad \varphi^{c}(s)=\varphi(cs), \\& \psi^{c}(s)=\psi(cs),\quad s\in(-\infty, 0], i=1,2,3. \end{aligned}$$
If (3.1) has a solution satisfying the following asymptotic boundary conditions:
$$\begin{aligned}& \lim_{t\rightarrow-\infty}\phi(t)=\phi_{-},\qquad \lim _{t\rightarrow-\infty}\varphi(t)=\varphi_{-},\qquad \lim _{t\rightarrow-\infty}\psi(t)=\psi_{-}, \\& \lim_{t\rightarrow+\infty}\phi(t)=\phi_{+},\qquad \lim _{t\rightarrow+\infty}\varphi(t)=\varphi_{+},\qquad \lim _{t\rightarrow+\infty}\psi(t)=\psi_{+}, \end{aligned}$$
then system (1.1) has a traveling wave solution (see [15, 16]). Without loss of generality, we assume that \((\phi_{-}, \varphi_{-},\psi_{-} ) = (0, 0, 0)\) and \((\phi_{+}, \varphi_{+},\psi_{+} )= (k_{1}^{\ast}, k_{2}^{\ast}, k_{3}^{\ast})\).
According to basic theory of the existence of traveling wave solutions (see [15, 16]), we mainly need to check that the system (1.1) satisfies partial quasi-monotonicity conditions, that is, there exist three positive constants \(\rho_{1}, \rho_{2}, \rho_{3}> 0\) such that
$$\begin{aligned}& f_{1}(\phi_{1},\varphi_{1}, \psi_{1})-f_{1}(\phi_{2},\varphi _{1}, \psi_{2}) +\rho_{1}\bigl[\phi_{1}(0)- \phi_{2}(0)\bigr]\geq0, \\& f_{1}(\phi_{1},\varphi_{1},\psi_{1})-f_{1}( \phi_{1},\varphi _{2},\psi_{1}) \leq0, \\& f_{2}(\phi_{1},\varphi_{1},\psi_{1})-f_{2}( \phi_{2},\varphi _{2},\psi_{1}) +\rho_{2} \bigl[\varphi_{1}(0)-\varphi_{2}(0)\bigr]\geq0, \\& f_{2}(\phi_{1},\varphi_{1},\psi_{1})-f_{2}( \phi_{1},\varphi _{1},\psi_{2}) \leq0, \\& f_{3}(\phi_{1},\varphi_{1},\psi_{1})-f_{3}( \phi_{2},\varphi _{2},\psi_{2}) +\rho_{3} \bigl[\psi_{1}(0)-\psi_{2}(0)\bigr]\geq0, \end{aligned}$$
(3.2)
with \(0\leq\phi_{2}(s)\leq\phi_{1}(s)\leq M_{1}\), \(0\leq \varphi_{2}(s)\leq\varphi_{1}(s)\leq M_{2}\), \(0\leq\psi_{2}(s)\leq \psi_{1}(s)\leq M_{3}\), and we also need to check that a pair of continuous functions \((\overline{\phi}, \overline{\varphi}, \overline{\psi})\) and \((\underline{\phi},\underline{\varphi},\underline{\psi})\) is a pair of upper-lower solution of system (3.1), that is,
$$\begin{aligned}& d_{1}\overline{\phi}''(t)-c \overline{\phi}'(t)+f_{c1}(\overline {\phi}_{t}, \underline{\varphi}_{t},\overline{\psi}_{t})\leq0, \\& d_{2}\overline{\varphi}''(t)-c\overline{ \varphi }'(t)+f_{c2}(\overline{\varphi}_{t}, \overline{\varphi}_{t},\underline{\psi}_{t})\leq0, \\& d_{3}\overline{\psi}''(t)-c\overline{ \psi}'(t)+f_{c3}(\overline {\psi}_{t}, \overline{ \varphi}_{t},\overline{\psi}_{t})\leq0 \end{aligned}$$
(3.3)
and
$$\begin{aligned}& d_{1}\underline{\phi}''(t)-c \underline{\phi}'(t)+f_{c1}(\underline { \phi}_{t}, \overline{\varphi}_{t},\underline{ \psi}_{t})\geq0, \\& d_{2}\underline{\varphi}''(t)-c\underline{ \varphi }'(t)+f_{c2}(\underline{\varphi}_{t}, \underline{\varphi}_{t},\overline{\psi}_{t})\geq0, \\& d_{3}\underline{\psi}''(t)-c\underline{ \psi}'(t)+f_{c3}(\underline {\psi}_{t}, \underline{\varphi}_{t},\underline{\psi}_{t})\geq0, \end{aligned}$$
(3.4)
where \((0, 0, 0)\leq (\underline{\phi},\underline{\varphi},\underline{\psi})\leq (\overline{\phi},\overline{\varphi},\overline{\psi})\leq (M_{1},M_{2},M_{3})\), \(t\in R\).
Lemma 3.1
\(f_{c1}(\phi_{t},\varphi_{t},\psi_{t})\), \(f_{c2}(\phi_{t},\varphi_{t},\psi_{t})\), and
\(f_{c3}(\phi_{t},\varphi_{t},\psi_{t})\)
of system (1.1) satisfy (3.2).
Proof
Let \(\phi_{1}(s)\), \(\phi_{2}(s)\), \(\varphi_{1}(s)\), \(\varphi_{2}(s)\), \(\psi_{1}(s)\), \(\psi_{2}(s)\) satisfy \(0\leq\phi_{2}(s)\leq\phi_{1}(s)\leq M_{1}\), \(0\leq \varphi_{2}(s)\leq\varphi_{1}(s)\leq M_{2}\), \(0\leq \psi_{2}(s)\leq\psi_{1}(s)\leq M_{3}\), \(s\in(-\infty, 0]\).
For any \(\phi_{i},\varphi_{i},\psi_{i}\in((-\infty, 0], R)\), \(i = 1, 2\), we have
$$\begin{aligned}& f_{c1}(\phi_{1t},\varphi_{1t}, \psi_{1t})-f_{c1}(\phi_{2t},\varphi _{1t}, \psi_{2t}) \\& \quad =\phi_{1}(0) \bigl(r_{1}-a_{1} \varphi_{1}(0)-b_{1}\phi_{1}(0)\bigr)- \phi_{2}(0) \bigl(r_{1} -a_{1}\varphi_{1}(0)-b_{1} \phi_{2}(0)\bigr) \\& \quad \geq r_{1}\bigl(\phi_{1}(0)-\phi_{2}(0) \bigr)-a_{1}\varphi_{1}(0) \bigl(\phi_{1}(0)- \phi_{2}(0)\bigr) -2b_{1}M_{1}\bigl( \phi_{1}(0)-\phi_{2}(0)\bigr) \\& \quad \geq(-a_{1}M_{2}-2b_{1}M_{1}) \bigl(\phi_{1}(0)-\phi_{2}(0)\bigr). \end{aligned}$$
(3.5)
Let \(\rho_{1}=a_{1}M_{2}+2b_{1}M_{1}>0\), then it is easy to see that
$$ \begin{aligned} &f_{c1}(\phi_{1t}, \varphi_{1t},\psi_{1t})-f_{c1}(\phi_{2t}, \varphi _{1t},\psi_{2t}) +\rho_{1}\bigl( \phi_{1}(0)-\phi_{2}(0)\bigr)\geq0, \\ &f_{c1}(\phi_{1t},\varphi_{1t}, \psi_{1t})-f_{c1}(\phi_{1t},\varphi _{2t}, \psi_{1t}) \\ &\quad =\phi_{1}(0) \bigl(r_{1}-a_{1} \varphi_{1}(0)-b_{1}\phi_{1}(0)\bigr)- \phi_{1}(0) \bigl(r_{1} -a_{1}\varphi_{2}(0)-b_{1} \phi_{1}(0)\bigr) \\ &\quad =-a_{1}\phi_{1}(0) \bigl(\varphi_{1}(0)- \varphi_{2}(0)\bigr)\leq0. \end{aligned} $$
(3.6)
For \(f_{c2}(\phi_{t},\varphi_{t},\psi_{t})\), we have
$$\begin{aligned}& f_{c2}(\phi_{1t},\varphi_{1t}, \psi_{1t})-f_{c2}(\phi_{2t},\varphi _{2t}, \psi_{1t}) \\& \quad =\varphi_{1}(0) \bigl(r_{2}-b_{2} \varphi_{1}(0)-a_{2}\psi _{1}(0)+a_{3} \phi_{1}(0)\bigr)-\varphi_{2}(0) \bigl(r_{2} -b_{2}\varphi_{2}(0) \\& \qquad {}-a_{2}\psi_{1}(0)+a_{3} \phi_{2}(0)\bigr) \\& \quad \geq r_{2}\bigl(\varphi_{1}(0)-\varphi_{2}(0) \bigr)-2b_{2}M_{2}\bigl(\varphi _{1}(0)- \varphi_{2}(0)\bigr) -a_{2}M_{3}\bigl( \varphi_{1}(0)-\varphi_{2}(0)\bigr) \\& \qquad {}+a_{3}\phi_{1}(0) \bigl(\varphi_{1}(0)- \varphi_{2}(0)\bigr) \\& \quad \geq(-a_{2}M_{3}-2b_{2}M_{2}) \bigl(\varphi_{1}(0)-\varphi_{2}(0)\bigr). \end{aligned}$$
(3.7)
Let \(\rho_{2}=a_{2}M_{3}+2b_{2}M_{2}>0\), then it is easy to see that
$$\begin{aligned}& f_{c2}(\phi_{1t},\varphi_{1t}, \psi_{1t})-f_{c2}(\phi_{2t},\varphi _{2t}, \psi_{1t}) +\rho_{2}\bigl(\varphi_{1}(0)- \varphi_{2}(0)\bigr)\geq0, \\& f_{c2}(\phi_{1t},\varphi_{1t}, \psi_{1t})-f_{c2}(\phi_{1t},\varphi _{1t}, \psi_{2t}) \\& \quad =\varphi_{1}(0) \bigl(r_{2}-b_{1} \varphi_{1}(0)-a_{2}\psi _{1}(0)+a_{3} \phi_{1}(0)\bigr)-\varphi_{1}(0) \bigl(r_{2} -b_{1}\varphi_{1}(0) \\& \qquad {}-a_{2}\psi_{2}(0)+a_{3} \phi_{1}(0)\bigr) \\& \quad =-a_{2}\varphi_{1}(0) \bigl(\psi_{1}(0)- \psi_{2}(0)\bigr). \end{aligned}$$
(3.8)
For \(f_{c3}(\phi_{t},\varphi_{t},\psi_{t})\), we have
$$\begin{aligned}& f_{c3}(\phi_{1t},\varphi_{1t}, \psi_{1t})-f_{c3}(\phi_{2t},\varphi _{2t}, \psi_{2t}) \\& \quad =\psi_{1}(0) \bigl(-\alpha-b_{3}\psi_{1}(0)+a_{4} \varphi_{1}(0)\bigr) \\& \qquad {}-\psi_{2}(0) \bigl(-\alpha-b_{3} \psi_{2}(0)+a_{4}\varphi_{2}(0)\bigr) \\& \quad \geq-\alpha \bigl(\psi_{1}(0)-\psi_{2}(0) \bigr)-2b_{3}M_{3}\bigl(\psi_{1}(0)- \psi_{2}(0)\bigr) \\& \quad =(-\alpha-2b_{3}M_{3}) \bigl(\psi_{1}(0)- \psi_{2}(0)\bigr). \end{aligned}$$
(3.9)
Let \(\rho_{3}=2b_{3}M_{3}+\alpha>0 \), then it is easy to see that
$$f_{c3}(\phi_{1t},\varphi_{1t}, \psi_{1t})-f_{c3}(\phi_{2t},\varphi _{2t}, \psi_{2t}) +\rho_{3}\bigl(\psi_{1}(0)- \psi_{2}(0)\bigr)\geq0. $$
This completes the proof of Lemma 3.1. □
We assume that \(c^{2} > c^{2}_{\ast}\triangleq\max\{ 4d_{1}r_{1}, 4d_{2}(r_{2}+a_{3}M_{1}), 4d_{3}(\alpha+a_{4}M_{2})\} \), \(c_{\ast}> 0\). Using this assumption, one can see that there exist \(\eta_{i}>0\) (\(i = 1, 2, 3\)) such that
$$\begin{aligned}& d_{1}\eta_{1}^{2}-c \eta_{1}+r_{1}=0, \\& d_{2}\eta_{2}^{2}-c \eta_{2}+r_{2}+a_{3}M_{1}=0, \\& d_{3}\eta_{3}^{2}-c\eta_{3}+ \alpha+a_{4}M_{2}=0. \end{aligned}$$
(3.10)
Assume that \(r_{1}>\max\{a_{1}k_{2}^{\ast}, a_{1}k_{1}^{\ast}\}\) and \(\alpha>\frac{b_{2}b_{3}k_{2}^{\ast}}{a_{2}}\), then one can choose positive constants \(\varepsilon_{5}\) and \(\varepsilon_{6}\) such that
$$ \begin{aligned} & a_{1}\bigl(\varepsilon_{5}-k_{2}^{\ast} \bigr)>b_{1}k_{1}^{\ast}, \qquad a_{2}\bigl( \varepsilon_{6}-k_{3}^{\ast}\bigr)>b_{2}k_{2}^{\ast}, \qquad \alpha >b_{3}\bigl(\varepsilon_{6}-k_{3}^{\ast} \bigr), \\ & r_{1}b_{2}>a_{1}a_{2}\bigl( \varepsilon_{6}-k_{3}^{\ast}\bigr), \qquad r_{2}+b_{2}\bigl(\varepsilon_{5}-k_{2}^{\ast} \bigr)>a_{2}k_{3}^{\ast}, \end{aligned} $$
(3.11)
and
$$ \begin{aligned} &\bigl(k_{1}^{\ast}+ \varepsilon_{1}\bigr) \bigl(r_{1}-a_{1} \bigl(k_{2}^{\ast}-\varepsilon _{5} \bigr)-b_{1}\bigl(k_{1}^{\ast}+\varepsilon_{1} \bigr)\bigr)< 0, \\ &\bigl(k_{2}^{\ast}+\varepsilon_{2}\bigr) \bigl(r_{2}-b_{2}\bigl(k_{2}^{\ast}+ \varepsilon _{2}\bigr)-a_{2}\bigl(k_{3}^{\ast}- \varepsilon_{6}\bigr)+a_{3}M_{1}\bigr)< 0, \\ &\bigl(k_{3}^{\ast}+\varepsilon_{3}\bigr) \bigl(- \alpha-b_{3}\bigl(k_{3}^{\ast }+\varepsilon_{3} \bigr)+a_{4}M_{2}\bigr)< 0, \\ &\bigl(k_{1}^{\ast}-\varepsilon_{4}\bigr) \bigl[r_{1}-a_{1}\bigl(k_{2}^{\ast}+ \varepsilon _{2}\bigr)-b_{1}\bigl(k_{1}^{\ast}- \varepsilon_{4}\bigr)\bigr]> 0, \\ &\bigl(k_{2}^{\ast}-\varepsilon_{5}\bigr) \bigl[r_{2}-b_{2}\bigl(k_{2}^{\ast}- \varepsilon _{5}\bigr)-a_{2}\bigl(k_{3}^{\ast}+ \varepsilon_{3}\bigr)\bigr]> 0, \\ &\bigl(k_{3}^{\ast}-\varepsilon_{6}\bigr) \bigl(- \alpha-b_{3}\bigl(k_{3}^{\ast }-\varepsilon_{6} \bigr)\bigr)> 0, \end{aligned} $$
(3.12)
for \(\varepsilon_{i}\) (\(i=1,2,3\)) being relatively big.
For the above constants and suitable constants \(\tilde{t}_{i}> 0\) (\(i = 1, 2, 3, 4, 5, 6\)) satisfying \(\tilde{t}_{5}<\min\{\tilde{t}_{1},\tilde{t}_{3}\}\), \(\tilde{t}_{2}>\max\{\tilde{t}_{4}, \tilde{t}_{6}\}\), we define the continuous functions \(\overline{\Phi}(t) = (\overline{\phi}(t), \overline{\varphi}(t), \overline{\psi}(t))\) and \(\underline{\Phi}(t) =(\underline{\phi}(t), \underline{\varphi}(t), \underline{\psi}(t))\) as follows:
$$\begin{aligned}& \overline{\phi}(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} k_{1}^{\ast}e^{\eta_{1}t}, & t\leq\tilde{t}_{1}, \\ k_{1}^{\ast}+\varepsilon_{1}e^{-\eta t},& t>\tilde{t}_{1}, \end{array}\displaystyle \right . \qquad \overline{\varphi}(t)= \left \{ \textstyle\begin{array}{l@{\quad}l} k_{2}^{\ast}e^{\eta_{2}t}, & t\leq\tilde{t}_{2}, \\ k_{2}^{\ast}+\varepsilon_{2}e^{-\eta t},& t>\tilde{t}_{2}, \end{array}\displaystyle \right . \\& \overline{\psi}(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} k_{3}^{\ast}e^{\eta_{3}t}, & t\leq\tilde{t}_{3}, \\ k_{3}^{\ast}+\varepsilon_{3}e^{-\eta t},& t>\tilde{t}_{3}, \end{array}\displaystyle \right .\qquad \underline{\phi}(t)= \left \{ \textstyle\begin{array}{l@{\quad}l} 0, & t\leq\tilde{t}_{4}, \\ k_{1}^{\ast}-\varepsilon_{4}e^{-\eta t},& t>\tilde{t}_{4}, \end{array}\displaystyle \right . \\& \underline{\varphi}(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} 0, & t\leq\tilde{t}_{5}, \\ k_{2}^{\ast}-\varepsilon_{5}e^{-\eta t},& t>\tilde{t}_{5}, \end{array}\displaystyle \right . \qquad \underline{\psi}(t)= \left \{ \textstyle\begin{array}{l@{\quad}l} 0, & t\leq\tilde{t}_{6}, \\ k_{3}^{\ast}-\varepsilon_{6}e^{-\eta t},& t>\tilde{t}_{6}, \end{array}\displaystyle \right . \end{aligned}$$
where
$$ \begin{aligned} &\tilde{t}_{2}=\frac{1}{\eta_{2}}\ln \frac {r_{1}}{a_{1}k_{2}^{\ast}},\qquad \tilde{t}_{6}=\frac{1}{\eta_{2}}\ln \frac{a_{2}(\varepsilon _{6}-k_{3}^{\ast})}{b_{2}k_{2}^{\ast}}, \\ &\tilde{t}_{5}=\max \biggl\{ \frac{1}{\eta_{1}}\ln\frac {a_{1}(\varepsilon_{5}-k_{2}^{\ast})}{b_{1}k_{1}^{\ast}}, \frac{1}{\eta_{3}}\ln\frac{r_{2}+b_{2}(\varepsilon_{5}-k_{2}^{\ast })}{a_{2}k_{3}^{\ast}} \biggr\} . \end{aligned} $$
(3.13)
Lemma 3.2
Assume that
\(r_{1}>\max\{a_{1}k_{2}^{\ast}, a_{1}k_{1}^{\ast}\}\)
and
\(\alpha>\frac{b_{2}b_{3}k_{2}^{\ast}}{a_{2}}\), then
\((\overline{\phi},\overline{\varphi},\overline{\psi})\)
is an upper solution of system (3.1).
Proof
If \(t\leq\tilde{t}_{5}<\tilde{t}_{1} \), then \(\overline{\phi}(t)=k_{1}^{\ast}e^{\eta_{1}t} \), and \(\underline{\varphi}(t)=0 \). Therefore, we have
$$\begin{aligned}& d_{1}\overline{\phi}''(t)-c \overline{\phi}'(t)+\overline{\phi}\bigl(r_{1}-a_{1} \underline{\varphi }(t)-b_{1}\overline{\phi}(t)\bigr) \\& \quad \leq \bigl(d_{1}\eta_{1}^{2}-c \eta_{1}+r_{1}\bigr)k_{1}^{\ast}e^{\eta_{1}t}=0. \end{aligned}$$
(3.14)
If \(t>\tilde{t}_{5}\) and \(t\leq\tilde{t}_{1}\), then \(\overline{\phi}(t)=k_{1}^{\ast}e^{\eta_{1} t}\) and \(\underline{\varphi}(t)=k_{2}^{\ast}-\varepsilon_{5}e^{-\eta t}\). Therefore, we obtain
$$ d_{1}\overline{\phi}''(t)-c \overline{\phi}'(t)+\overline{\phi}(t) \bigl(r_{1} -a_{1}\underline{\varphi}(t)-b_{1}\overline{\phi}(t) \bigr)=I_{1}(\eta), $$
(3.15)
where \(I_{1}(\eta)=k_{1}^{\ast}e^{\eta_{1} t}[a_{1}(\varepsilon_{5}e^{-\eta t}-k_{2}^{\ast})-b_{1}k_{1}^{\ast}e^{\eta_{1} t}]\), \(I_{1}(0)=k_{1}^{\ast}e^{\eta_{1} t}[a_{1}(\varepsilon_{5}-k_{2}^{\ast})-b_{1}k_{1}^{\ast}e^{\eta_{1} t}]< 0\) when \(t>\tilde{t}_{5}\). Therefore, there exists \(\eta_{1}^{\ast}>0\) such that \(d_{1}\overline{\phi}''(t)-c \overline{\phi}'(t)+\overline{\phi}(t)(r_{1} -a_{1}\underline{\varphi}(t)-b_{1}\overline{\phi}(t))\leq0\) for all \(\eta\in(0,\eta_{1}^{\ast})\).
If \(t > \tilde{t}_{1}> \tilde{t}_{5}\), then \(\overline{\phi}(t)=k_{1}^{\ast}+\varepsilon_{1}e^{-\eta t}\) and \(\underline{\varphi}(t)=k_{2}^{\ast}-\varepsilon_{5}e^{-\eta t}\). Therefore, we have
$$ d_{1}\overline{\phi}''(t)-c \overline{\phi}'(t)+\overline{\phi}(t) \bigl(r_{1} -a_{1}\underline{\varphi}(t)-b_{1}\overline{\phi}(t) \bigr)=I_{2}(\eta), $$
(3.16)
where \(I_{2}(\eta)=(d_{1}\varepsilon_{1}\eta^{2}+c\varepsilon_{1}\eta )e^{-\eta t}+(k_{1}^{\ast}+\varepsilon_{1}e^{-\eta t})(r_{1}-a_{1}(k_{2}^{\ast}-\varepsilon_{5}e^{-\eta t})-b_{1}(k_{1}^{\ast}+\varepsilon_{1}e^{-\eta t}))\). It follows from (3.12) that \(I_{2}(0)=(k_{1}^{\ast}+\varepsilon_{1})(r_{1}-a_{1}(k_{2}^{\ast }-\varepsilon_{5})-b_{1}(k_{1}^{\ast}+\varepsilon_{1}))< 0\). Therefore, there exists \(\eta_{2}^{\ast}>0\) such that \(d_{1}\overline{\phi}''(t)-c \overline{\phi}'(t)+\overline{\phi}(t)(r_{1} -a_{1}\underline{\varphi}(t)-b_{1}\overline{\phi}(t))\leq0\) for all \(\eta\in(0,\eta_{2}^{\ast})\).
If \(t\leq\tilde{t}_{6}< \tilde{t}_{2} \), then \(\overline{\varphi}(t)=k_{2}^{\ast}e^{\eta_{2}t} \), and \(\underline{\psi}(t)=0 \). It follows that
$$\begin{aligned}& d_{2}\overline{\varphi}''(t)-c \overline{\varphi}'(t)+f_{c2}\bigl(\overline{\phi}(t), \overline {\varphi}(t),\underline{\psi}(t)\bigr) \\& \quad \leq\bigl(d_{2}\eta_{2}^{2}-c \eta_{2}+r_{2}\bigr)k_{2}^{\ast}e^{\eta_{2}t} +k_{2}^{\ast}e^{\eta_{2}t}\bigl(-b_{2}k_{2}^{\ast}e^{\eta _{2}t}+a_{3}M_{1} \bigr) \\& \quad \leq \bigl(d_{2}\eta_{2}^{2}-c \eta_{2}+r_{2}+a_{3}M_{1} \bigr)k_{2}^{\ast}e^{\eta_{2}t}=0. \end{aligned}$$
(3.17)
If \(t>\tilde{t}_{6}\) and \(t\leq\tilde{t}_{2}\), then \(\overline{\varphi}(t)=k_{2}^{\ast}e^{\eta_{2} t}\) and \(\underline{\psi}=k_{3}^{\ast}-\varepsilon_{6}e^{-\eta t}\). By calculating, we have
$$ d_{2}\overline{\varphi}''(t)-c \overline{\varphi}'(t)+f_{c2}\bigl(\overline{\phi}(t), \overline{\varphi}(t),\underline{\psi}(t)\bigr)\leq I_{3}(\eta), $$
(3.18)
where \(I_{3}(\eta)=k_{2}^{\ast}e^{\eta_{2} t}(a_{2}(\varepsilon_{6}e^{-\eta t}-k_{3}^{\ast})-b_{2}k_{2}^{\ast}e^{\eta_{2} t})\). Thus, \(I_{3}(0)=k_{2}^{\ast}e^{\eta_{2} t}(a_{2}(\varepsilon_{6} -k_{3}^{\ast})-b_{2}k_{2}^{\ast}e^{\eta_{2} t})< 0\) when \(t>\tilde{t}_{6}\). Hence, there exists \(\eta_{3}^{\ast}>0\) such that \(d_{2}\overline{\varphi}''(t)-c \overline{\varphi}'(t)+f_{c2}(\overline{\phi}(t), \overline{\varphi}(t),\underline{\psi}(t))\leq0\) for all \(\eta\in(0,\eta_{3}^{\ast})\).
If \(t>\tilde{t}_{2} \), then \(\overline{\varphi}(t)=k_{2}^{\ast}+\varepsilon_{2}e^{-\eta t} \) and \(\underline{\psi}(t)=k_{3}^{\ast}-\varepsilon_{6}e^{-\eta t} \). By calculating, we have
$$ d_{2} \overline{\varphi}''(t)-c \overline{\varphi}'(t)+f_{c2}\bigl(\overline{\phi}(t), \overline{\varphi}(t),\underline{\psi}(t)\bigr)\leq I_{4}(\eta), $$
(3.19)
where \(I_{4}(\eta)=(d_{2}\varepsilon_{2}\eta^{2}+c\varepsilon_{2}\eta )e^{-\eta t}+(k_{2}^{\ast}+\varepsilon_{2}e^{-\eta t})(r_{2}+a_{3}M_{1}-b_{2}(k_{2}^{\ast}+\varepsilon_{2}e^{-\eta t})-a_{2}(k_{3}^{\ast}-\varepsilon_{6}e^{-\eta t}))\). It follows from (3.12) that \(I_{4}(0)=(k_{2}^{\ast}+\varepsilon _{2})(r_{2}+a_{3}M_{1}-b_{2}(k_{2}^{\ast}+\varepsilon_{2}) -a_{2}(k_{3}^{\ast}-\varepsilon_{6}))< 0\). Hence, there exists \(\eta_{4}^{\ast}>0\) such that \(d_{2}\overline{\varphi}''(t)-c \overline{\varphi}'(t)+f_{c2}(\overline{\phi}(t), \overline{\varphi}(t),\underline{\psi}(t))\leq0\) for all \(\eta\in(0,\eta_{4}^{\ast})\).
If \(t\leq\tilde{t}_{3}\), then \(\overline{\psi}(t)=k_{3}^{\ast}e^{\eta_{3}t}\). By calculating, we have
$$\begin{aligned}& d_{3}\overline{\psi}''(t)-c \overline{\psi}'(t)+f_{c3}\bigl(\overline{\phi}(t), \overline{\varphi }(t),\overline{\psi}(t)\bigr) \\& \quad \leq\bigl(d_{3}\eta_{3}^{2}-c \eta_{3}\bigr)k_{3}^{\ast}e^{\eta_{3}t} +k_{3}^{\ast}e^{\eta_{3}t}\bigl(-\alpha-b_{3}k_{3}^{\ast}e^{\eta _{3}t}+a_{4}M_{2} \bigr) \\& \quad \leq \bigl(d_{3}\eta_{3}^{2}-c \eta_{3}-\alpha+a_{4}M_{2}\bigr)k_{3}^{\ast}e^{\eta_{3}t}=0. \end{aligned}$$
(3.20)
If \(t>\tilde{t}_{3}\), then \(\overline{\psi}(t)=k_{3}^{\ast}+\varepsilon_{3}e^{-\eta_{3}t}\). By calculating, we get
$$ d_{3}\overline{\psi}''(t)-c \overline{\psi}'(t)+f_{c3}\bigl(\overline{\phi}(t), \overline{\varphi}(t),\overline{\psi}(t)\bigr)\leq I_{5}(\eta), $$
(3.21)
where \(I_{5} (\eta)=(d_{3}\varepsilon_{3}\eta^{2}+c\varepsilon_{3}\eta)e^{-\eta t}+(k_{3}^{\ast}+\varepsilon_{3}e^{-\eta t})(-\alpha-b_{3}(k_{3}^{\ast}+\varepsilon_{2}e^{-\eta t})+a_{4}M_{2})\). It follows from (3.12) that \(I_{5}(0)=(k_{3}^{\ast}+\varepsilon_{3})(-\alpha-b_{3}(k_{3}^{\ast }+\varepsilon_{3}) +a_{4}M_{2})< 0\). Hence, there exists \(\eta_{5}^{\ast}>0\) such that \(d_{3}\overline{\psi}''(t)-c \overline{\psi}'(t)+f_{c3}(\overline{\phi}(t), \overline{\varphi}(t),\underline{\psi}(t))\leq0\) for all \(\eta\in(0,\eta_{5}^{\ast})\).
Finally, for any \(\eta\in(0,\min\{\eta_{1}^{\ast}, \eta_{2}^{\ast}, \eta_{3}^{\ast}, \eta_{4}^{\ast}, \eta_{5}^{\ast} \})\), we see that (3.3) holds. This completes the proof. □
Lemma 3.3
Assume that
\(r_{1}>\max\{a_{1}k_{2}^{\ast}, a_{1}k_{1}^{\ast}\}\)
and
\(\alpha>\frac{b_{2}b_{3}k_{2}^{\ast}}{a_{2}}\), then
\((\underline{\phi}(t), \underline{\varphi}(t), \underline{\psi}(t))\)
is a pair of lower solution of system (3.1).
Proof
If \(t\leq\tilde{t}_{4}\), then \(\underline{\phi}(t)=0\). We have \(d_{1}\underline{\phi}''(t)-c\underline{\phi}'(t)+\underline{\phi}(t) (r_{1}-a_{1}\overline{\varphi}(t)-b_{1}\underline{\phi}(t))=0\).
If \(\tilde{t}_{4}< t\leq\tilde{t}_{2}\), then \(\underline{\phi}(t)=k_{1}^{\ast}-\varepsilon_{4}e^{-\eta t}\), \(\overline{\varphi}=k_{2}^{\ast}e^{\eta_{2} t}\). We have
$$\begin{aligned}& d_{1}\underline{\phi}''(t)-c \underline{\phi}'(t)+\underline{\phi}(t) \bigl(r_{1}-a_{1} \overline{\varphi}(t)-b_{1}\underline{\phi}(t)\bigr) \\& \quad = (-d_{1}\eta-c)\varepsilon_{4}\eta e^{-\eta t}+ \bigl(k_{1}^{\ast}-\varepsilon_{4}e^{-\eta t} \bigr) \bigl(r_{1}-a_{1}k_{2}^{\ast}e^{\eta_{2} t}-b_{1} \bigl(k_{1}^{\ast}-\varepsilon_{4}e^{-\eta t} \bigr)\bigr) \\& \quad \triangleq I_{6}(\eta). \end{aligned}$$
(3.22)
Using (3.13), we see that \(I_{6}(0)=(k_{1}^{\ast}-\varepsilon_{4})(r_{1}-a_{1}k_{2}^{\ast }e^{\eta_{2}t} - b_{1}(k_{1}^{\ast}-\varepsilon_{4}))> 0\) when \(\tilde{t}_{4}< t\leq\tilde{t}_{2}\). Therefore, there exists \(\eta_{6}^{\ast}\) such that \(d_{1}\underline{\phi}''(t)-c\underline{\phi}'(t)+\underline{\phi}(t) (r_{1}-a_{1}\underline{\varphi}(t)-b_{1}\underline{\phi}(t))\geq0\) for \(\eta\in(0,\eta_{6}^{\ast})\).
If \(t>\tilde{t}_{2}\), then \(\underline{\phi}(t)=k_{1}^{\ast}-\varepsilon_{4}e^{-\eta t}\), \(\overline{\varphi}=k_{2}^{\ast}+\varepsilon_{2} e^{-\eta t}\). We have
$$\begin{aligned}& d_{1}\underline{\phi}''(t)-c \underline{\phi}'(t)+\underline{\phi}(t) \bigl(r_{1}-a_{1} \overline{\varphi}(t)-b_{1}\underline{\phi}(t)\bigr) \\& \quad \geq(-d_{1}\eta-c)\varepsilon_{4}\eta e^{-\eta t}+\bigl(k_{1}^{\ast}-\varepsilon_{4}e^{-\eta t} \bigr) \bigl(r_{1}-a_{1}\bigl(k_{2}^{\ast}+ \varepsilon_{2}e^{-\eta t}\bigr)-b_{1} \bigl(k_{1}^{\ast}-\varepsilon_{4}e^{-\eta t} \bigr)\bigr) \\& \quad \triangleq I_{7}(\eta). \end{aligned}$$
(3.23)
Using (3.12), we see that \(I_{7}(0)=(k_{1}^{\ast}-\varepsilon_{4})(r_{1}-a_{1}(k_{2}^{\ast} + \varepsilon_{2})- b_{1}(k_{1}^{\ast}-\varepsilon_{4}))> 0\). Therefore, there exists \(\eta_{7}^{\ast}\) such that \(d_{1}\underline{\phi}''(t)-c\underline{\phi}'(t)+\underline{\phi}(t) (r_{1}-a_{1}\underline{\varphi}(t)-b_{1}\underline{\phi}(t))\geq0\) for \(\eta\in(0,\eta_{7}^{\ast})\).
If \(t\leq\tilde{t}_{5}\), then \(\underline{\varphi}=0\). Therefore we have \(d_{2}\underline{\varphi}''(t)-c\underline{\varphi }'+f_{c2}(\underline{\phi}(t), \underline{\varphi}(t),\overline{\psi}(t))=0\).
If \(\tilde{t}_{5}< t\leq\tilde{t}_{3}\), then \(\underline{\varphi}(t) =k_{2}^{\ast}-\varepsilon_{5}e^{-\eta t}\), \(\overline{\psi}=k_{3}^{\ast}e^{\eta_{3}t}\). We get
$$\begin{aligned}& d_{2}\underline{\varphi}''(t)-c \underline{\varphi }'+f_{c2}\bigl(\underline{\phi}(t), \underline{\varphi}(t),\overline{\psi}(t)\bigr) \\& \quad = (-d_{2}\eta-c)\varepsilon_{5}\eta e^{-\eta t}+ \bigl(k_{2}^{\ast}-\varepsilon_{5}e^{-\eta t} \bigr) \bigl(r_{2}-b_{2}\bigl(k_{2}^{\ast}- \varepsilon_{5}e^{-\eta t}\bigr)-a_{2}k_{3}^{\ast}e^{\eta_{3} t} \bigr) \\& \quad \triangleq I_{8}(\eta). \end{aligned}$$
(3.24)
Using (3.13), we see that \(I_{8}(0)=(k_{2}^{\ast}-\varepsilon_{5})(r_{2}-b_{2}(k_{2}^{\ast} - \varepsilon_{5})- a_{2}k_{3}^{\ast}e^{\eta_{3}t})> 0\) when \(t>\tilde{t}_{5}\). Hence, there exists \(\eta_{8}^{\ast}\) such that \(d_{2}\underline{\varphi}''(t)-c\underline{\varphi }'(t)+f_{c2}(\underline{\phi}(t), \underline{\varphi}(t),\overline{\psi}(t))\geq0\) for all \(\eta\in (0,\eta_{8}^{\ast})\).
If \(t>\tilde{t}_{3}\), then \(\underline{\varphi}(t) =k_{2}^{\ast}-\varepsilon_{5}e^{-\eta t}\). We get
$$\begin{aligned} \begin{aligned}[b] &d_{2}\underline{\varphi}''(t)-c \underline{\varphi }'+f_{c2}\bigl(\underline{\phi}(t), \underline{\varphi}(t),\overline{\psi}(t)\bigr) \\ &\quad \geq(-d_{2}\eta-c)\varepsilon_{5}\eta e^{-\eta t}+\bigl(k_{2}^{\ast}-\varepsilon_{5}e^{-\eta t} \bigr) \bigl(r_{2}-b_{2}\bigl(k_{2}^{\ast}- \varepsilon_{5}e^{-\eta t}\bigr)-a_{2} \bigl(k_{3}^{\ast}+\varepsilon_{3}e^{-\eta t} \bigr)\bigr) \\ &\quad \triangleq I_{9}(\eta). \end{aligned} \end{aligned}$$
(3.25)
Using (3.12), we see that \(I_{9}(0)=(k_{2}^{\ast}-\varepsilon_{5})(r_{2}-b_{2}(k_{2}^{\ast} - \varepsilon_{5})- a_{2}(k_{3}^{\ast}+\varepsilon_{3}))> 0\). Hence, there exists \(\eta_{9}^{\ast}\) such that \(d_{2}\underline{\varphi}''(t)-c\underline{\varphi }'(t)+f_{c2}(\underline{\phi}(t), \underline{\varphi}(t),\overline{\psi}(t))\geq0\) for all \(\eta\in (0,\eta_{9}^{\ast})\).
If \(t\leq\tilde{t}_{6}\), then \(\underline{\psi}=0\). Therefore, we have \(d_{3}\underline{\psi}''(t)-c\underline{\psi}'+f_{c3}(\underline {\phi}(t), \underline{\varphi}(t),\underline{\psi}(t))=0\).
If \(t>\tilde{t}_{6}\), then \(\underline{\psi}(t) =k_{3}^{\ast}-\varepsilon_{6}e^{-\eta t}\). Hence, we get
$$\begin{aligned}& d_{3}\underline{\psi}''(t)-c \underline{\psi}'+f_{c3}\bigl(\underline {\phi}(t), \underline{\varphi}(t),\overline{\psi}(t)\bigr) \\& \quad \geq(-d_{3}\eta-c)\varepsilon_{6}\eta e^{-\eta t}+\bigl(k_{3}^{\ast}-\varepsilon_{6}e^{-\eta t} \bigr) \bigl(-\alpha-b_{3}\bigl(k_{3}^{\ast}- \varepsilon_{6}e^{-\eta t}\bigr)\bigr) \\& \quad \triangleq I_{10}(\eta). \end{aligned}$$
(3.26)
Using (3.12), we see that \(I_{10}(0)=(k_{3}^{\ast}-\varepsilon_{6})(-\alpha-b_{3}(k_{3}^{\ast }-\varepsilon_{6}))> 0\). Hence, there exists \(\eta_{10}^{\ast}\) such that \(d_{3}\underline{\psi}''(t)-c\underline{\psi}'(t)+f_{c3}(\underline {\phi}(t), \underline{\varphi}(t),\overline{\psi}(t))\geq0 \) for all \(\eta \in(0,\eta_{10}^{\ast})\).
Finally, for any \(\eta\in(0, \min\{\eta_{6}^{\ast}, \eta_{7}^{\ast}, \eta_{8}^{\ast}, \eta_{9}^{\ast}, \eta_{10}^{\ast}\}) \), we see that (3.4) holds. This completes the proof. □
By using Lemmas 3.1-3.3, we have the following conclusion.
Theorem 3.1
Assume that
\(r_{1}>\max\{a_{1}k_{2}^{\ast}, a_{1}k_{1}^{\ast}\}\)
and
\(\alpha>\frac{b_{2}b_{3}k_{2}^{\ast}}{a_{2}}\), then, for any
\(c > c^{\ast} > 0\), system (1.1) always has a traveling wave solution with speed c connecting the trivial steady state
\((0, 0)\)
and the positive steady state
\((k_{1}^{\ast}, k_{2}^{\ast}, k_{3}^{\ast})\).