We now investigate the existence of traveling wave solutions and first give the following definition.
Definition 4.1
A traveling wave solution of (2.1) is a special solution taking the form \(u(x,t)=\varPhi(x+ct)\in C^{2}(\mathbb{R},\mathbb{R}^{3})\) with
$$u(x,t)=\bigl(u_{1}(x,t),u_{2}(x,t),u_{3}(x,t)\bigr), \qquad \varPhi(\xi)=\bigl(\phi_{1}(\xi),\phi _{2}(\xi),\phi_{3}(\xi)\bigr),\quad \xi=x+ct, $$
in which Φ is the wave profile that propagates through the one-dimension spatial domain \(\mathbb{R}\) at the constant wave speed \(c > 0\).
By definition, \(\varPhi(\xi)=(\phi_{1}(\xi),\phi_{2}(\xi),\phi_{3}(\xi))\) must satisfy
$$ \textstyle\begin{cases} d_{1}\phi''_{1}(\xi)-c\phi'_{1}(\xi)+r_{1}\phi_{1}(\xi)[1-\phi_{1}(\xi)-a_{12}\phi _{2}(\xi)-a_{13}\phi_{3}(\xi)]=0,\\ d_{2}\phi''_{2}(\xi)-c\phi'_{2}(\xi)+r_{2}\phi_{2}(\xi)[1-a_{21}\phi_{1}(\xi)-\phi _{2}(\xi)-a_{23}\phi_{3}(\xi)]=0,\\ d_{3}\phi''_{3}(\xi)-c\phi'_{3}(\xi)+r_{3}\phi_{3}(\xi)[-1+a_{31}\phi_{1}(\xi )+a_{32}\phi_{2}(\xi)-\phi_{3}(\xi)]=0. \end{cases} $$
(4.1)
Since we shall discuss the dynamical behavior that \(u_{2}\) and \(u_{3}\) invade the habitat of \(u_{1}\), \(\varPhi(\xi)\) will satisfy the following asymptotic boundary conditions:
$$ \begin{aligned} &\lim_{\xi\to-\infty}\bigl(\phi_{1}(\xi), \phi_{2}(\xi),\phi_{3}(\xi)\bigr) =(1,0,0), \\ &\lim_{\xi\to+\infty}\bigl(\phi_{1}(\xi),\phi_{2}( \xi),\phi_{3}(\xi)\bigr) =(k_{1},k_{2},k_{3}). \end{aligned} $$
(4.2)
In population dynamics, positive solutions of (4.1)–(4.2) describe the following biological process: at any fixed location \(x\in \mathbb{R}\), there was only one prey a long time ago (\(t \to-\infty\) such that \(x+ct\to-\infty\)), and the predator and two preys will coexist after a long-term species interaction (\(t\to+\infty\) such that \(x+ct\to+\infty\)).
Letting \(\varPsi(\xi)=(\varphi_{1},\varphi_{2},\varphi_{3})(\xi)=(1-\phi_{1},\phi _{2},\phi_{3})(\xi)\), we obtain
$$ \textstyle\begin{cases} d_{1}\varphi''_{1}(\xi)-c\varphi'_{1}(\xi) +r_{1}[1-\varphi_{1}(\xi)][a_{12}\varphi_{2}(\xi)+a_{13}\varphi_{3}(\xi) -\varphi_{1}(\xi)]=0,\\\quad \xi\in\mathbb{R},\\ d_{2}\varphi''_{2}(\xi)-c\varphi'_{2}(\xi) +r_{2}\varphi_{2}(\xi)[1-a_{21}+a_{21}\varphi_{1}(\xi)-\varphi_{2}(\xi) -a_{23}\varphi_{3}(\xi)]=0,\\\quad \xi\in\mathbb{R},\\ d_{3}\varphi''_{3}(\xi)-c\varphi'_{3}(\xi) +r_{3}\varphi_{3}(\xi)[a_{31}-1+a_{32}\varphi_{2}(\xi)- a_{31}\varphi_{1}(\xi)-\varphi_{3}(\xi)]=0,\\\quad \xi\in\mathbb{R}. \end{cases} $$
(4.3)
Due to (4.2), we have
$$ \begin{aligned} &\lim_{\xi\to-\infty}\bigl(\varphi_{1}(\xi), \varphi_{2}(\xi),\varphi_{3}(\xi )\bigr)=(0,0,0), \\ &\lim_{\xi\to+\infty}\bigl(\varphi_{1}(\xi), \varphi_{2}(\xi),\varphi_{3}(\xi )\bigr)=(1-k_{1},k_{2},k_{3}). \end{aligned} $$
(4.4)
Define a constant
$$ \beta\geq r_{2}\bigl[a_{21}+a_{23}(a_{31}+a_{32}-1) \bigr]+r_{3}(a_{31}+a_{32}) $$
(4.5)
such that
$$\beta\varphi_{1}+r_{1}[1-\varphi_{1}] [a_{12}\varphi_{2} +a_{13}\varphi_{3}- \varphi_{1}] $$
is nondecreasing with respect to \(0\leq\varphi_{1}\leq1\), \(0\leq\varphi _{2}\leq1\), \(0\leq\varphi_{3}\leq a_{31}+a_{32}-1\),
$$\beta\varphi_{2}+r_{2}\varphi_{2}[1-a_{21}+a_{21} \varphi_{1} -\varphi_{2}-a_{23} \varphi_{3}] $$
is nondecreasing with respect to \(0\leq\varphi_{1}\leq1\), \(0\leq\varphi _{2}\leq1\), while it is nonincreasing with respect to \(0\leq\varphi _{3}\leq a_{31}+a_{32}-1\), and
$$\beta\varphi_{3}+r_{3}\varphi_{3}[a_{31}-1+a_{32} \varphi_{2} -a_{31}\varphi_{1}- \varphi_{3}] $$
is nonincreasing with respect to \(0\leq\varphi_{1}\leq1\), while it is nondecreasing with respect to \(0\leq\varphi_{2}\leq1\), \(0\leq\varphi _{3}\leq a_{31}+a_{32}-1\).
For \(\varPsi(\xi)=(\varphi_{1},\varphi_{2},\varphi_{3})(\xi)\in X_{[0,M]}\) with \(M=(1,1,a_{31}+a_{32}-1)\), denote
$$ \textstyle\begin{cases} F_{1}(\varphi_{1},\varphi_{2},\varphi_{3})(\xi)=\beta\varphi_{1}(\xi) +r_{1}[1-\varphi_{1}(\xi)][a_{12}\varphi_{2}(\xi)+a_{13}\varphi_{3}(\xi) -\varphi_{1}(\xi)],\\\quad \xi\in\mathbb{R},\\ F_{2}(\varphi_{1},\varphi_{2},\varphi_{3})(\xi)=\beta\varphi_{2}(\xi) +r_{2}\varphi_{2}(\xi)[1-a_{21}+a_{21}\varphi_{1}(\xi)-\varphi_{2}(\xi) -a_{23}\varphi_{3}(\xi)],\\\quad \xi\in\mathbb{R},\\ F_{3}(\varphi_{1},\varphi_{2},\varphi_{3})(\xi)=\beta\varphi_{3}(\xi) +r_{3}\varphi_{3}(\xi)[a_{31}-1+a_{32}\varphi_{2}(\xi)- a_{31}\varphi_{1}(\xi)-\varphi_{3}(\xi)],\\\quad \xi\in\mathbb{R}. \end{cases} $$
Then (4.3) can be rewritten as
$$ d_{i}\varphi''_{i}( \xi)-c\varphi'_{i}(\xi)-\beta\varphi_{i}(\xi) +F_{i}(\varPsi) (\xi)=0,\quad i=1,2,3. $$
(4.6)
Define constants
$$\lambda_{i1}(c)=\frac{c-\sqrt{c^{2}+4\beta d_{i}}}{2d_{i}},\qquad \lambda _{i2}(c)= \frac{c+\sqrt{c^{2}+4\beta d_{i}}}{2d_{i}},\quad i=1,2,3. $$
Then \(\beta>0\) implies \(\lambda_{i1}<0<\lambda_{i2}\) and
$$d_{i}\lambda^{2}_{i1}-c\lambda_{i1}- \beta=0,\qquad d_{i}\lambda^{2}_{i2}-c\lambda _{i2}-\beta=0,\quad i=1,2,3. $$
For \(\varPsi(\xi)=(\varphi_{1},\varphi_{2},\varphi_{3})(\xi)\in X_{[0,M]}\), define an operator \(P=(P_{1},P_{2},P_{3}):X_{[\mathbf{0},\mathbf{M}]} \to X\) (see Wu and Zou [49]) by
$$ P_{i}(\varPsi) (\xi)=\frac{1}{d_{i}(\lambda_{i2} -\lambda_{i1})} \biggl[ \int_{-\infty} ^{\xi} e^{\lambda_{i1}(\xi-s)}+ \int_{\xi}^{+\infty}e^{\lambda_{i2}(\xi-s)} \biggr] F_{i}(\varPsi) (s)\,ds, $$
(4.7)
where \(i=1,2,3\), \(\xi\in\mathbb{R}\). Then a fixed point of operator P is a solution of (4.3) or (4.6). On the other hand, a solution of (4.3) or (4.6) is a fixed point of operator P (see Huang [22]).
In the following, we will establish the existence of a nontrivial positive solution of (4.3) by combining Schauder’s fixed point theorem with the method of upper and lower solutions (for quasimonotone systems, we refer to [20, 34, 46, 49]). We now introduce the definition of upper and lower solutions of (4.3).
Definition 4.2
\(\overline{\varPsi}(\xi)=(\overline{\varphi}_{1},\overline{\varphi}_{2}, \overline{\varphi}_{3})(\xi)\), \(\underline{\varPsi}(\xi)=(\underline{\varphi}_{1},\underline{\varphi}_{2}, \underline{\varphi}_{3})(\xi)\in X_{[0,M]}\) are a pair of upper and lower solutions of (4.3), if \(\overline{\varPsi}'',\overline{\varPsi}', \underline{\varPsi}'',\underline{\varPsi}'\) are bounded and continuous for each \(\xi\in\mathbb{R}\setminus\mathbb{T}\) with \(\mathbb{T}=\{ T_{1},T_{2},\dots,T_{m}\}\) and they satisfy
$$ \textstyle\begin{cases} d_{1}\overline{\varphi}''_{1}(\xi) -c\overline{\varphi}'_{1}(\xi) +r_{1}[1-\overline{\varphi}_{1}(\xi)][a_{12}\overline{\varphi}_{2}(\xi) +a_{13}\overline{\varphi}_{3}(\xi)-\overline{\varphi}_{1}(\xi)]\leq0,\\ d_{2}\overline{\varphi}''_{2}(\xi) -c\overline{\varphi}'_{2}(\xi) +r_{2}\overline{\varphi}_{2}(\xi)[1-a_{21}+ a_{21}\overline{\varphi}_{1}(\xi)-\overline{\varphi}_{2}(\xi) -a_{23}\underline{\varphi}_{3}(\xi)]\leq0,\\ d_{3}\overline{\varphi}''_{3}(\xi)-c\overline{\varphi}'_{3}(\xi) +r_{3}\overline{\varphi}_{3}(\xi)[a_{31}-1 +a_{32}\overline{\varphi}_{2}(\xi)- a_{31}\underline{\varphi}_{1}(\xi) -\overline{\varphi}_{3}(\xi)]\leq0,\\ d_{1}\underline{\varphi}''_{1}(\xi) -c\underline{\varphi}'_{1}(\xi) +r_{1}[1-\underline{\varphi}_{1}(\xi)] [a_{12}\underline{\varphi}_{2}(\xi) +a_{13}\underline{\varphi}_{3}(\xi) -\underline{\varphi}_{1}(\xi)]\geq0,\\ d_{2}\underline{\varphi}''_{2}(\xi)-c\underline{\varphi}'_{2}(\xi) +r_{2}\underline{\varphi}_{2}(\xi)[1-a_{21} +a_{21}\underline{\varphi}_{1}(\xi)-\underline{\varphi}_{2}(\xi) -a_{23}\overline{\varphi}_{3}(\xi)]\geq0,\\ d_{3}\underline{\varphi}''_{3}(\xi)-c\underline{\varphi}'_{3}(\xi) +r_{3}\underline{\varphi}_{3}(\xi)[a_{31}-1 +a_{32}\underline{\varphi}_{2}(\xi) -a_{31}\overline{\varphi}_{1}(\xi) -\underline{\varphi}_{3}(\xi)]\geq0. \end{cases} $$
(4.8)
Lemma 4.3
Assume that (4.3) has a pair of upper and lower solutions satisfying
-
(1)
\(\underline{\varPsi}(\xi)\leq\overline{\varPsi}(\xi)\), \(\xi\in \mathbb{R}\);
-
(2)
\(\underline{\varPsi}'(\xi_{-})\leq\overline{\varPsi}'(\xi_{+})\), \(\overline{\varPsi}'(\xi_{+})\leq\overline{\varPsi}'(\xi_{-})\), \(\xi\in\mathbb {T}\), herein
$$\underline{\varPsi}'(\xi_{\pm})=\lim _{t \to\xi_{\pm}}\underline{ \varPsi }'(t),\qquad \overline{\varPsi}'(\xi_{\pm})= \lim _{t \to\xi_{\pm}}\overline {\varPsi}'(t). $$
Then (4.3) has a positive solution
\(\varPsi(\xi)\)
such that
\(\underline{\varPsi}(\xi)\leq\varPsi(\xi)\leq\overline{\varPsi}(\xi)\).
Proof
We prove this lemma by Schauder’s fixed point theorem. Since a similar result has been proved in several earlier papers mentioned above [31], we only give the scheme.
Define
$$\begin{aligned} &B_{\mu}\bigl(\mathbb{R},\mathbb{R}^{3}\bigr)=\Bigl\{ u\in X: \sup _{\xi\in\mathbb {R}}\bigl\{ \bigl\Vert \mathbf{u}(\xi) \bigr\Vert e^{-\mu \vert \xi \vert }\bigr\} < \infty\Bigr\} , \\ &\bigl\vert \mathbf{u}(\xi) \bigr\vert _{\mu}=\sup _{\xi\in\mathbb{R}} \bigl\{ \bigl\Vert \mathbf {u}(\xi) \bigr\Vert e^{-\mu \vert \xi \vert }\bigr\} , \end{aligned}$$
where
$$ \mu\in\bigl(0,\min\{-\lambda_{11},-\lambda_{21},- \lambda_{31}\}\bigr), $$
then \((B_{\mu}(\mathbb{R},\mathbb{R}^{3}),|\cdot|_{\mu})\) is a Banach space. Let
$$ \varLambda=\bigl\{ \varPsi(\xi)\in X_{[\mathbf{0},\mathbf{M}]}: \underline{\varPsi}(\xi)\leq\varPsi(\xi) \leq\overline{\varPsi}(\xi)\bigr\} . $$
Obviously, Λ is nonempty and convex. It is also closed and bounded with respect to the decay norm \(|\cdot|_{\mu}\).
We now verify that \(P:\varLambda\to\varLambda\). For \(\varPsi(\xi)=(\varphi_{1},\varphi_{2},\varphi_{3})(\xi)\in\varLambda\) and each fixed \(\xi\in\mathbb{R}\), the definition of operator P and the choice of β imply that it suffices to prove that
$$ \textstyle\begin{cases} \underline{\varphi}_{1}(\xi)\leq P_{1}(\underline{\varphi}_{1},\underline {\varphi}_{2}, \underline{\varphi}_{3})(\xi)\leq P_{1}(\overline{\varphi}_{1},\overline {\varphi}_{2}, \overline{\varphi}_{3})(\xi) \leq\overline{\varphi}_{1}(\xi),\\ \underline{\varphi}_{2}(\xi)\leq P_{2}(\underline{\varphi}_{1},\underline {\varphi}_{2}, \overline{\varphi}_{3})(\xi)\leq P_{2}(\overline{\varphi}_{1},\overline{\varphi}_{2}, \underline{\varphi}_{3})(\xi) \leq\overline{\varphi}_{2}(\xi),\\ \underline{\varphi}_{3}(\xi)\leq P_{3}(\overline{\varphi}_{1},\underline {\varphi}_{2}, \underline{\varphi}_{3})(\xi)\leq P_{3}(\underline{\varphi}_{1},\overline{\varphi}_{2}, \overline{\varphi}_{3})(\xi) \leq\overline{\varphi}_{3}(\xi). \end{cases} $$
(4.9)
Without loss of generality, we assume that \(T_{1}< T_{2}<\cdots<T_{m}\) and denote \(T_{0}=-\infty\), \(T_{m+1}=+\infty\). If \(\xi\in\mathbb {R}\backslash\mathbb{T}\), namely, \(\xi\in(T_{k},T_{k+1})\) with some \(k\in \{0,1,\dots,m\}\), then
$$\begin{aligned} & P_{1}(\underline{\varphi}_{1},\underline{ \varphi}_{2},\underline{\varphi }_{3}) (\xi) \\ &\quad = \frac{1}{d_{1}(\lambda_{12}-\lambda_{11})} \biggl[ \int^{\xi}_{-\infty }e^{\lambda_{11}(\xi-s)} + \int^{+\infty}_{\xi}e^{\lambda_{12}(\xi-s)} \biggr] F_{1}(\underline{\varphi}_{1},\underline{ \varphi}_{2},\underline{\varphi }_{3}) (s)\,ds \\ &\quad = \frac{1}{d_{1}(\lambda_{12}-\lambda_{11})} \biggl[ \int^{\xi}_{-\infty }e^{\lambda_{11}(\xi-s)} + \int^{+\infty}_{\xi}e^{\lambda_{12}(\xi-s)} \biggr] \bigl[\beta \underline{\varphi}_{1}(s)+c\underline{\varphi }'_{1}(s)-d_{1} \underline{\varphi}''_{1}(s) \bigr]\,ds \\ &\quad \geq \underline{\varphi}_{1}(\xi)+\frac{1}{\lambda_{12}-\lambda_{11}} \Biggl[\sum _{j=1}^{k}e^{\lambda_{11}(\xi-T_{j})} \bigl(\underline{ \varphi}'_{1}(T_{j+})-\underline{\varphi }'_{1}(T_{j-}) \bigr) \\ &\qquad {}+ \sum _{j=k+1}^{m}e^{\lambda_{12}(\xi-T_{j})} \bigl( \underline {\varphi}'_{1}(T_{j+}) -\underline{ \varphi}'_{1}(T_{j-}) \bigr) \Biggr] \\ &\quad \geq \underline{\varphi}_{1}(\xi). \end{aligned}$$
Because ξ was arbitrary and due to the continuity, we have \(P_{1}(\underline{\varphi}_{1},\underline{\varphi}_{2},\underline{\varphi }_{3})(\xi)\geq\underline{\varphi}_{1}(\xi)\) in \(\mathbb{R}\). In a similar way, we can verity the remainder of (4.9).
Note that the compactness in [24] is independent of the monotonicity, then \(P:\varLambda\to\varLambda\) is compact in the sense of the decay norm \(|\cdot|_{\mu}\) by a discussion similar to that in [24].
By Schauder’s fixed point theorem, there exists \(\varPsi(\xi)=(\varphi _{1},\varphi_{2},\varphi_{3})(\xi)\in\varLambda\) which is a positive solution of (4.3) satisfying \(\underline{\varPsi}(\xi)\leq\varPsi(\xi)\leq \overline{\varPsi}(\xi)\). The proof is complete. □
Next, we construct the upper and lower solutions of (4.3), and we assume that
For any fixed
$$c>\max\bigl\{ 2\sqrt{d_{2}r_{2} (1-a_{21}) }, 2 \sqrt{d_{3}r_{3} (a_{31}-1)}\bigr\} :=c^{*}, $$
we define positive constants \(\gamma_{21}< \gamma_{22},\gamma_{31}<\gamma_{32}\) such that
$$\begin{aligned} &d_{2}\gamma^{2}_{21}-c\gamma_{21}+r_{2}(1-a_{21})=d_{2} \gamma^{2}_{22}-c\gamma _{22}+r_{2}(1-a_{21})=0, \\ &d_{3}\gamma^{2}_{31}-c\gamma_{31}+r_{3}(a_{31}-1)=d_{3} \gamma^{2}_{32}-c\gamma _{32}+r_{3}(a_{31}-1)=0. \end{aligned}$$
Further choose \(\epsilon>0\) such that
$$\gamma_{21}+\epsilon< \min\{2\gamma_{21}, \gamma_{22},\gamma_{21}+\gamma _{31}\}, $$
and
$$\gamma_{31}+\epsilon< \min\{2\gamma_{31}, \gamma_{32}, \gamma_{21}+\gamma _{31}\}. $$
Let
$$\gamma_{11}=\min \{\gamma_{21},\gamma_{31} \}. $$
We now assume that
$$ d_{1}\gamma_{11}^{2} -c \gamma_{11} < 0. $$
(4.11)
Define \(\varGamma=(\gamma_{21}, \gamma_{22}) \cap(\gamma_{31},\gamma_{32})\), then Γ is nonempty if c is large enough or other parameters satisfy suitable conditions. In particular, when Γ is nonempty, we further assume that there exists \(\gamma\in\varGamma\) such that
$$ d_{1}\gamma^{2} -c\gamma< 0,\quad \gamma< \gamma_{21}+\gamma_{31}. $$
(4.12)
Remark 4.4
Assume that all the parameters in (2.1) are fixed. Then there exists \(c'\ge c^{*}\) such that (4.11)–(4.12) hold.
For any given \(c>c^{*}\), we now fix these constants and define continuous functions as follows:
$$\begin{aligned} &\overline{\varphi}_{1}(\xi)=\min\bigl\{ 1,e^{\gamma_{11}\xi}+pe^{\gamma\xi} \bigr\} ,\qquad \underline{\varphi}_{1}(\xi)=0, \\ &\overline{\varphi}_{2}(\xi)=\min\bigl\{ 1,e^{\gamma_{21}\xi}+pe^{\gamma\xi} \bigr\} ,\qquad \underline{\varphi}_{2}(\xi)=\max\bigl\{ 0,e^{\gamma_{21}\xi}-qe^{(\gamma _{21}+\epsilon)\xi} \bigr\} , \\ &\overline{\varphi}_{3}(\xi)=\min\bigl\{ a_{31}+a_{32}-1,e^{\gamma_{31}\xi }+pa_{32}e^{\gamma\xi} \bigr\} ,\qquad \underline{\varphi}_{3}(\xi)=\max\bigl\{ 0,e^{\gamma_{31}\xi}-qe^{(\gamma _{31}+\epsilon)\xi} \bigr\} , \end{aligned}$$
in which \(p>1\), \(q>1\) will be clarified in the following lemma.
Lemma 4.5
Assume that
\(c>c^{*}\). Further suppose that
Γ
is nonempty such that (4.11)–(4.12) hold. Then there exist
\(p, q\)
such that
\(\overline{\varPsi}(\xi)=(\overline{\varphi}_{1}(\xi ),\overline{\varphi}_{2}(\xi),\overline{\varphi}_{3}(\xi))\)
and
\(\underline {\varPsi}(\xi)=(\underline{\varphi}_{1}(\xi), \underline{\varphi}_{2}(\xi ),\underline{\varphi}_{3}(\xi))\)
are a pair of upper and lower solutions of (4.3).
Proof
It suffices to verify (4.8) one by one.
-
(1)
(i) If \(\overline{\varphi}_{1}(\xi)=1< e^{\gamma_{11}\xi }+pe^{\gamma\xi}\), then
$$d_{1}\overline{\varphi}''_{1}( \xi)-c\overline{\varphi}'_{1}(\xi) +r_{1} \bigl[1-\overline{\varphi}_{1}(\xi)\bigr] \bigl[a_{12} \overline{\varphi}_{2}(\xi)+ a_{13}\overline{ \varphi}_{3}(\xi)-\overline{\varphi}_{1}(\xi)\bigr]=0. $$
(ii) If \(\overline{\varphi}_{1}(\xi)=e^{\gamma_{11}\xi}+pe^{\gamma\xi }<1\), then \(\xi<0\) and
$$\overline{\varphi}_{2}(\xi)\leq e^{\gamma_{21}\xi}+pe^{\gamma\xi },\qquad \overline{\varphi}_{3}(\xi)\leq e^{\gamma_{31}\xi}+pa_{32}e^{\gamma _{31}\xi} $$
so that
$$\begin{aligned} & a_{12}\overline{\varphi}_{2}(\xi) +a_{13} \overline{\varphi}_{3}(\xi)-\overline{\varphi}_{1}(\xi) \\ &\quad \leq a_{12}\bigl(e^{\gamma_{21}\xi}+pe^{\gamma\xi}\bigr) +a_{13}\bigl(e^{\gamma_{31}\xi}+pa_{32}e^{\gamma\xi}\bigr)- \bigl(e^{\gamma_{11}\xi }+pe^{\gamma\xi}\bigr) \\ &\quad \leq (a_{12}+a_{13}-1)e^{\gamma_{11}\xi}+(a_{12}+a_{13}a_{32}-1)pe^{\gamma\xi } \\ &\quad < 0 \end{aligned}$$
because \(\gamma_{11}= \min\{\gamma_{21},\gamma_{31}\}\), \(a_{12}+a_{13}a_{32}<1\) and \(a_{12}+a_{13}<1\). Therefore, we have
$$\begin{aligned} & d_{1}\overline{\varphi}''_{1}( \xi)-c\overline{\varphi}'_{1}(\xi) +r_{1} \bigl[1-\overline{\varphi}_{1}(\xi)\bigr] \bigl[a_{12} \overline{\varphi}_{2}(\xi) +a_{13}\overline{ \varphi}_{3}(\xi)-\overline{\varphi}_{1}(\xi)\bigr] \\ &\quad \leq d_{1}\overline{\varphi}''_{1}( \xi)-c\overline{\varphi}'_{1}(\xi) \\ &\quad =\bigl(d_{1}\gamma^{2}_{11}-c \gamma_{11}\bigr)e^{\gamma_{11}\xi}+\bigl(d_{1}\gamma ^{2}-c\gamma\bigr)pe^{\gamma\xi} \\ &\quad < 0. \end{aligned}$$
-
(2)
(i) If \(\overline{\varphi}_{2}(\xi)=1< e^{\gamma_{21}\xi }+pe^{\gamma\xi}\), then \(\overline{\varphi}_{1}(\xi)\leq1\) so that
$$\begin{aligned} & d_{2}\overline{\varphi}''_{2}( \xi)-c\overline{\varphi}'_{2}(\xi) +r_{2} \overline{\varphi}_{2}(\xi)\bigl[1-a_{21}+a_{21} \overline{\varphi}_{1}(\xi )-\overline{\varphi}_{2}(\xi) -a_{23}\underline{\varphi}_{3}(\xi)\bigr] \\ & \quad \leq d_{2}\overline{\varphi}''_{2}( \xi)-c\overline{\varphi}'_{2}(\xi) +r_{2} \overline{\varphi}_{2}(\xi)\bigl[1-a_{21}+a_{21} \overline{\varphi}_{1}(\xi )-\overline{\varphi}_{2}(\xi) \bigr] \\ &\quad \leq r_{2}(1-a_{21}+a_{21}-1) \\ &\quad = 0. \end{aligned}$$
(ii) If \(\overline{\varphi}_{2}(\xi)=e^{\gamma_{21}\xi}+pe^{\gamma\xi}<1\), then
$$\begin{aligned} & d_{2}\overline{\varphi}''_{2}( \xi)-c\overline{\varphi}'_{2}(\xi) +r_{2} \overline{\varphi}_{2}(\xi)\bigl[1-a_{21}+a_{21} \overline{\varphi}_{1}(\xi)- \overline{\varphi}_{2}( \xi)-a_{23}\underline{\varphi}_{3}(\xi)\bigr] \\ &\quad \leq d_{2}\overline{\varphi}''_{2}( \xi)-c\overline{\varphi}'_{2}(\xi) +r_{2} \overline{\varphi}_{2}(\xi)\bigl[1-a_{21}+a_{21} \overline{\varphi}_{1}(\xi )-\overline{\varphi}_{2}(\xi) \bigr] \\ &\quad \leq d_{2}\bigl(\gamma^{2}_{21}e^{\gamma_{21}\xi}+p \gamma^{2}e^{\gamma\xi}\bigr) -c\bigl(\gamma_{21}e^{\gamma_{21}\xi}+ p\gamma e^{\gamma\xi}\bigr) \\ &\qquad {}+r_{2}\bigl(e^{\gamma_{21}\xi} +pe^{\gamma\xi}\bigr) \bigl[1-a_{21}+a_{21}\bigl(e^{\gamma_{11}\xi}+p e^{\gamma\xi } \bigr)-\bigl(e^{\gamma_{21}\xi}+p e^{\gamma\xi}\bigr)\bigr]. \end{aligned}$$
If \(\gamma_{11}=\gamma_{21}\le\gamma_{31}\), then
$$a_{21}\bigl(e^{\gamma_{11}\xi}+p e^{\gamma\xi}\bigr)\le \bigl(e^{\gamma_{21}\xi}+p e^{\gamma\xi}\bigr) $$
so that
$$\begin{aligned} & d_{2}\overline{\varphi}''_{2}( \xi)-c\overline{\varphi}'_{2}(\xi) +r_{2} \overline{\varphi}_{2}(\xi)\bigl[1-a_{21}+a_{21} \overline{\varphi}_{1}(\xi)- \overline{\varphi}_{2}( \xi)-a_{23}\underline{\varphi}_{3}(\xi)\bigr] \\ &\quad \leq d_{2}\bigl(\gamma^{2}_{21}e^{\gamma_{21}\xi}+p \gamma^{2}e^{\gamma\xi}\bigr) -c\bigl(\gamma_{21}e^{\gamma_{21}\xi}+ p\gamma e^{\gamma\xi}\bigr)+r_{2}\bigl(e^{\gamma _{21}\xi} +pe^{\gamma\xi}\bigr)[1-a_{21}] \\ &\quad \leq 0 \end{aligned}$$
by the definitions of \(\gamma_{21}\) and γ. Otherwise, \(\gamma_{11}=\gamma_{31}<\gamma_{21}\) so that
$$\gamma< \gamma_{11}+\gamma_{21} $$
and
$$\begin{aligned} & d_{2}\overline{\varphi}''_{2}( \xi)-c\overline{\varphi}'_{2}(\xi) +r_{2} \overline{\varphi}_{2}(\xi)\bigl[1-a_{21}+a_{21} \overline{\varphi}_{1}(\xi)- \overline{\varphi}_{2}( \xi)-a_{23}\underline{\varphi}_{3}(\xi)\bigr] \\ &\quad \leq d_{2}\bigl(\gamma^{2}_{21}e^{\gamma_{21}\xi}+p \gamma^{2}e^{\gamma\xi}\bigr) -c\bigl(\gamma_{21}e^{\gamma_{21}\xi}+ p\gamma e^{\gamma\xi}\bigr)\\ &\qquad {}+r_{2}\bigl(e^{\gamma _{21}\xi} +pe^{\gamma\xi}\bigr)\bigl[1-a_{21}+a_{21}e^{\gamma_{11}\xi} \bigr] \\ &\quad \leq \bigl[d_{2}\gamma^{2}-c\gamma+r_{2}(1-a_{21}) \bigr]pe^{\gamma\xi}+r_{2}a_{21}e^{(\gamma _{11}+\gamma_{21})\xi}+pr_{2}a_{21}e^{(\gamma_{11}+\gamma)\xi} \\ &\quad \leq 0 \end{aligned}$$
provided that
$$ \bigl[d_{2}\gamma^{2}-c\gamma+r_{2}(1-a_{21}) \bigr]pe^{\gamma\xi} +2r_{2}a_{21}e^{(\gamma_{11}+\gamma_{21})\xi}\leq0 $$
(4.13)
and
$$ \bigl[d_{2}\gamma^{2}-c\gamma+r_{2}(1-a_{21}) \bigr] +2r_{2}a_{21}e^{\gamma_{11}\xi}\leq0. $$
(4.14)
Clearly, (4.13) is true if
$$p>1-\frac{2r_{2}a_{21}}{d_{2}\gamma^{2}-c\gamma+r_{2}(1-a_{21})}:=p_{1} (>1) $$
and (4.14) is true if
$$p> \biggl(\frac{2r_{2}a_{21}}{-(d_{2}\gamma^{2}-c\gamma+r_{2}(1-a_{21}))} \biggr)^{\frac{\gamma}{\gamma_{11}}}+1:=p_{2}>1. $$
-
(3)
(i) If \(\overline{\varphi}_{31}(\xi )=a_{31}+a_{32}-1< e^{\gamma_{31}\xi}+pa_{32}e^{\gamma\xi}\), then \(\overline{\varphi}_{2}(\xi)\leq1\) so that
$$\begin{aligned} & d_{3}\overline{\varphi}''_{3}( \xi)-c\overline{\varphi}'_{3}(\xi)+r_{3}\overline {\varphi}_{3}(\xi)\bigl[a_{31}-1 +a_{32}\overline{ \varphi}_{2}(\xi)-a_{31}\underline{\varphi}_{1}(\xi )-\overline{\varphi}_{3}(\xi)\bigr] \\ &\quad \leq d_{3}\overline{\varphi}''_{3}( \xi)-c\overline{\varphi}'_{3}(\xi)+r_{3}\overline {\varphi}_{3}(\xi)\bigl[a_{31}-1 +a_{32}-\overline{ \varphi}_{3}(\xi)\bigr] \\ &\quad = 0. \end{aligned}$$
(ii) If \(\overline{\varphi}_{3}(\xi)=e^{\gamma_{31}\xi}+pa_{32}e^{\gamma \xi}< a_{31}+a_{32}-1\), then \(\overline{\varphi}_{2}(\xi)\leq e^{\gamma_{21}\xi}+pe^{\gamma\xi}\) so that
$$\begin{aligned} & d_{3}\overline{\varphi}''_{3}( \xi)-c\overline{\varphi}'_{3}(\xi) +r_{3} \overline{\varphi}_{3}(\xi)\bigl[a_{31}-1+a_{32} \overline{\varphi}_{2}(\xi)- a_{31}\underline{ \varphi}_{1}(\xi)-\overline{\varphi}_{3}(\xi)\bigr] \\ &\quad \leq d_{3}\overline{\varphi}''_{3}( \xi)-c\overline{\varphi}'_{3}(\xi) +r_{3} \overline{\varphi}_{3}(\xi)\bigl[a_{31}-1+a_{32} \overline{\varphi}_{2}(\xi )-\overline{\varphi}_{3}(\xi)\bigr] \\ &\quad \leq \bigl[d_{3}\gamma^{2}_{3}-c \gamma_{31}+r_{3}(a_{31}-1)\bigr]e^{\gamma_{31}\xi} + \bigl[d_{3}\gamma^{2}-c\gamma+r_{3}(a_{31}-1) \bigr]pa_{32}e^{\gamma\xi} \\ &\qquad {}+ r_{3}\bigl(e^{\gamma_{31}\xi}+pa_{32}e^{\gamma\xi} \bigr)\bigl[a_{32}\bigl(e^{\gamma _{21}\xi}+pe^{\gamma\xi}\bigr)- \bigl(e^{\gamma_{31}\xi}+pa_{32}e^{\gamma\xi}\bigr)\bigr] \\ &\quad \leq \bigl[d_{3}\gamma^{2}-c\gamma+r_{3}(a_{31}-1) \bigr]pa_{32}e^{\gamma\xi }+r_{3}a_{32}e^{(\gamma_{21}+\gamma_{31})\xi} +r_{3}{a_{32}}^{2}pe^{(\gamma_{21}+\gamma)\xi} \\ &\quad \leq 0 \end{aligned}$$
provided that
$$ \bigl[d_{3}\gamma^{2}-c\gamma+r_{3}(a_{31}-1) \bigr]pe^{\gamma\xi}+2r_{3}e^{(\gamma _{21}+\gamma_{31})\xi}\leq0 $$
(4.15)
and
$$ \bigl[d_{3}\gamma^{2}-c\gamma+r_{3}(a_{31}-1) \bigr]+2r_{3}a_{32}e^{\gamma_{21}\xi}\leq0. $$
(4.16)
For (4.15), since \(\gamma< \gamma_{21}+\gamma_{31}\), we have
$$\begin{aligned} & \bigl[d_{3}\gamma^{2}-c\gamma+r_{3}(a_{31}-1) \bigr]pe^{\gamma\xi}+2r_{3}e^{(\gamma _{21}+\gamma_{31})\xi} \\ &\quad\leq e^{\gamma\xi}\bigl\{ \bigl[d_{3}\gamma^{2}-c \gamma+r_{3}(a_{31}-1)\bigr]p+2r_{3}\bigr\} , \end{aligned}$$
which is true if
$$p\geq\frac{-2r_{3}}{d_{3}\gamma^{2}-c\gamma+r_{3}(a_{31}-1)}+1:=p_{3}. $$
Since \(e^{\gamma_{31}\xi}+pa_{32}e^{\gamma\xi}< a_{31}+a_{32}-1\), then
$$pe^{\gamma\xi}< \frac{a_{31}+a_{32}-1}{a_{32}} $$
so that
$$\xi< \frac{1}{\gamma}\ln\frac{a_{31}+a_{32}-1}{pa_{32}} $$
and
$$e^{\gamma_{21}\xi}< \biggl( \frac{a_{31}+a_{32}-1}{pa_{32}} \biggr)^{\frac{\gamma_{21}}{\gamma}}. $$
Clearly, (4.16) holds if
$$\biggl( \frac{a_{31}+a_{32}-1}{pa_{32}} \biggr)^{\frac{\gamma _{21}}{\gamma}} < \frac{d_{3}\gamma^{2}-c\gamma+r_{3}(a_{31}-1)}{-2r_{3}a_{32}}, $$
which is true provided that
$$p> \frac{a_{31}+a_{32}-1}{a_{32}} \biggl[ \biggl(\frac{d_{3}\gamma^{2}-c\gamma +r_{3}(a_{31}-1)}{-2r_{3}a_{32}} \biggr)^{\frac{\gamma}{\gamma_{21}+\gamma _{31}}} +1 \biggr]:=p_{4}. $$
-
(4)
If \(\underline{\varphi}_{1}(\xi)=0\), then
$$\begin{aligned} & d_{1}\underline{\varphi}''_{1}( \xi)-c\underline{\varphi}'_{1}(\xi) +r_{1} \bigl[1-\underline{\varphi}_{1}(\xi)\bigr] \bigl[a_{12} \underline{\varphi}_{2}(\xi) +a_{13}\underline{ \varphi}_{3}(\xi)-\underline{\varphi}_{1}(\xi)\bigr] \\ &\quad = r_{1}\bigl[a_{12}\underline{\varphi}_{2}( \xi)+a_{13}\underline{\varphi}_{3}(\xi)\bigr] \\ &\quad \geq 0. \end{aligned}$$
-
(5)
(i) If \(\underline{\varphi}_{2}(\xi)=0>e^{\gamma_{21}\xi }-qe^{(\gamma_{21}+\epsilon)\xi}\), then
$$d_{2}\underline{\varphi}''_{2}( \xi)-c\underline{\varphi}'_{2}(\xi) +r_{2} \underline{\varphi}_{2}(\xi)\bigl[1-a_{21}+ a_{21} \underline{\varphi}_{1}(\xi)-\underline{\varphi}_{2}(\xi )-a_{23}\overline{\varphi}_{3}(\xi)\bigr]=0. $$
(ii) If \(\underline{\varphi}_{2}(\xi)=e^{\gamma_{21}\xi}-qe^{(\gamma _{21}+\epsilon)\xi}>0\), then
$$\underline{\varphi}_{2}(\xi)< e^{\gamma_{21}\xi},\qquad\overline{\varphi }_{3}(\xi)\leq e^{\gamma_{31}\xi}+pa_{32}e^{\gamma\xi} $$
so that
$$\begin{aligned} & d_{2}\underline{\varphi}''_{2}( \xi)-c\underline{\varphi}'_{2}(\xi) +r_{2} \underline{\varphi}_{2}(\xi)\bigl[1-a_{21}+a_{21} \underline{\varphi}_{1}(\xi )-\underline{\varphi}_{2}(\xi) -a_{23}\overline{\varphi}_{3}(\xi)\bigr] \\ &\quad \geq d_{2}\underline{\varphi}''_{2}( \xi)-c\underline{\varphi}'_{2}(\xi) +r_{2} \underline{\varphi}_{2}(\xi)\bigl[1-a_{21}-\underline{ \varphi}_{2}(\xi )-a_{23}\overline{\varphi}_{3}(\xi) \bigr] \\ &\quad \geq d_{2}\bigl[\gamma^{2}_{2}e^{\gamma_{21}\xi}-q( \gamma_{21}+\epsilon)^{2}e^{(\gamma _{21}+\epsilon)\xi}\bigr] -c\bigl[ \gamma_{21}e^{\gamma_{21}\xi}-q(\gamma_{21}+ \epsilon)e^{(\gamma _{21}+\epsilon)\xi}\bigr] \\ &\qquad {}+r_{2}\bigl(e^{\gamma_{21}\xi}-qe^{(\gamma_{21}+\epsilon)\xi }\bigr) \bigl[1-a_{21}-e^{\gamma_{21}\xi}-a_{23}\bigl(e^{\gamma_{31}\xi} +pa_{32}e^{\gamma\xi}\bigr)\bigr] \\ &\quad \geq -\bigl[d_{2}(\gamma_{21}+\epsilon)^{2}-c( \gamma_{21}+\epsilon )+r_{2}(1-a_{21}) \bigr]qe^{(\gamma_{21}+\epsilon)\xi} \\ &\qquad {} -r_{2}a_{23}\bigl[e^{(\gamma_{21}+\gamma_{31})\xi}+pa_{32}e^{(\gamma+\gamma _{21})\xi} \bigr]-r_{2}e^{2\gamma_{21} \xi} \\ &\quad \geq {-}\bigl[d_{2}(\gamma_{21}+\epsilon)^{2}-c( \gamma_{21}+\epsilon )+r_{2}(1-a_{21}) \bigr]qe^{(\gamma_{21}+\epsilon)\xi} \\ &\qquad {}-r_{2}e^{(\gamma_{21}+\epsilon)\xi}\bigl[1+a_{23}(1+a_{32}p) \bigr]. \end{aligned}$$
Let
$$q\geq q_{1}=1+\frac{-r_{2}[1+a_{23}(1+a_{32}p)]}{d_{2}(\gamma_{21}+\epsilon )^{2}-c(\gamma_{21}+\epsilon)+r_{2}(1-a_{21})}, $$
then
$$d_{2}\underline{\varphi}''_{2}( \xi)-c\underline{\varphi}'_{2}(\xi) +r_{2} \underline{\varphi}_{2}(\xi)\bigl[1-a_{21} +a_{21} \underline{\varphi}_{1}(\xi)-\underline{\varphi}_{2}(\xi )-a_{23}\overline{\varphi}_{3}(\xi)\bigr]\geq0. $$
-
(6)
(i) If \(\underline{\varphi}_{3}(\xi)=0>e^{\gamma_{31}\xi}-qe^{(\gamma _{31}+\epsilon)\xi}\), then
$$d_{3}\underline{\varphi}''_{3}( \xi)-c\underline{\varphi}'_{3}(\xi) +r_{3} \underline{\varphi}_{3}(\xi)\bigl[a_{31}-1 +a_{32} \underline{\varphi}_{2}(\xi)-a_{31}\overline{ \varphi}_{1}(\xi )-\underline{\varphi}_{3}(\xi)\bigr]=0. $$
(ii) If \(\underline{\varphi}_{3}(\xi)=e^{\gamma_{31}\xi}-qe^{(\gamma _{31}+\epsilon)\xi}>0\), then
$$\overline{\varphi}_{1}(\xi)\leq e^{\gamma_{11}\xi}+pe^{\gamma\xi},\qquad \underline{\varphi}_{3}(\xi)< e^{\gamma_{31}\xi} $$
so that
$$\begin{aligned} & d_{3}\underline{\varphi}''_{3}( \xi)-c\underline{\varphi}'_{3}(\xi) +r_{3} \underline{\varphi}_{3}(\xi)\bigl[a_{31}-1+ a_{32} \underline{\varphi}_{2}(\xi)-a_{31}\overline{ \varphi}_{1}(\xi )-\underline{\varphi}_{3}(\xi)\bigr] \\ &\quad \geq d_{3}\underline{\varphi}''_{3}( \xi)-c\underline{\varphi}'_{3}(\xi) +r_{3} \underline{\varphi}_{3}(\xi)\bigl[a_{31}-1-a_{31} \overline{\varphi}_{1}(\xi )-\underline{\varphi}_{3}(\xi) \bigr] \\ &\quad \geq d_{3}\bigl[\gamma_{31}^{2}e^{\gamma_{31}\xi}-q( \gamma_{31}+\epsilon)^{2}e^{(\gamma _{31}+\epsilon)\xi}\bigr] -c\bigl[ \gamma_{31}e^{\gamma_{31}\xi}-q(\gamma_{31}+ \epsilon)e^{(\gamma _{31}+\epsilon)\xi}\bigr] \\ &\qquad {}+ r_{3}\bigl(e^{\gamma_{31}\xi}-qe^{(\gamma_{31}+\epsilon)\xi }\bigr) \bigl[a_{31}-1-a_{31}\bigl(e^{\gamma_{11}\xi}+ pe^{\gamma\xi} \bigr)-e^{\gamma_{31}\xi}\bigr] \\ &\quad \geq -\bigl[d_{3}(\gamma_{31}+\epsilon)^{2}-c( \gamma_{31}+\epsilon )+r_{3}(a_{31}-1) \bigr]qe^{(\gamma_{31}+\epsilon)\xi} \\ &\qquad {} -r_{3}a_{31}\bigl[e^{(\gamma_{31}+\gamma_{11})\xi}+pe^{(\gamma_{31}+\gamma )\xi} \bigr]-r_{3}e^{2\gamma_{31}\xi} \\ &\quad \geq -\bigl[d_{3}(\gamma_{31}+\epsilon)^{2}-c( \gamma_{31}+\epsilon )+r_{3}(a_{31}-1) \bigr]qe^{(\gamma_{31}+\epsilon)\xi} \\ &\qquad {}-r_{3}e^{(\gamma_{31}+\epsilon)\xi}\bigl[1+a_{31}(1+p) \bigr]. \end{aligned}$$
Let
$$q\geq q_{2}=1+\frac{-r_{3}[1+a_{31}(1+p)]}{d_{3}(\gamma_{31}+\epsilon )^{2}-c(\gamma_{31}+\epsilon) +r_{3}(a_{31}-1)}, $$
then
$$d_{3}\underline{\varphi}''_{3}( \xi)-c\underline{\varphi}'_{3}(\xi) +r_{3} \underline{\varphi}_{3}(\xi)\bigl[a_{31}-1+a_{32} \underline{\varphi}_{2}(\xi) -a_{31}\overline{ \varphi}_{1}(\xi)-\underline{\varphi}_{3}(\xi)\bigr]\geq0. $$
By what we have done, we first fix \(p=p_{1}+p_{2}+p_{3}+p_{4}\), then let \(q=q_{1}+q_{2}\), which completes the proof. □
Summarizing the above, we have the following conclusions.
Theorem 4.6
Assume that (4.10) holds. If
\(c>c^{*}\)
is such that (4.11)–(4.12) are true, then (4.3) has a nonconstant positive solution.
About the traveling wave solution, we also give the following remark.
Remark 4.7
By direct calculations in P, we see that \(\varphi_{i}'(\xi),\varphi _{i}''(\xi), i=1,2,3\), are uniformly bounded.