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Traveling wave solution for a reaction-diffusion competitive-cooperative system with delays
Boundary Value Problems volume 2016, Article number: 46 (2016)
Abstract
This paper investigates the existence of traveling wave solution to a three species reaction-diffusion system with delays, which includes competitive relationship, cooperative relationship and predator-prey relationship. By using the method of upper-lower solutions, the cross iteration method and Schauder’s fixed point theorem, the existence of a traveling wave solution is obtained.
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:
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
considered by Huang and Zou [2]. When \(u_{2}=0\), system (1) is reduced to the two species predator-prey system
studied by Zhang and Li [17]. When \(u_{3}=0\) system (1) is reduced to the two species competing system
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
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:
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:
with asymptotic boundary conditions
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
Without loss of generality, we can assume
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:
-
(A1)
\(f_{i}(0, 0, 0)=f_{i}(k_{1}, k_{2}, k_{3})=0\) for \(i=1, 2, 3\);
-
(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
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
and
Let
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
Define
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:
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):
-
(P1)
\((0,0,0)\leq(\underline{\phi}(t),\underline{\varphi }(t),\underline{\psi}(t))\leq(\overline{\phi }(t), \overline{\varphi}(t), \overline{\psi}(t))\leq(M_{1}, M_{2},M_{3})\), \(t \in R\).
-
(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
where
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
Proof
From (8), a direct calculation shows that
□
From the definitions of \(H_{1}\), \(H_{2}\), and \(H_{3}\) in (11), system (6) can be rewritten as
We define
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
It is easy to see that \(F_{i}(\phi,\varphi,\psi)\) (\(i=1,2,3\)) satisfy
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
Lemma 4
Assume that (8) holds, then
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
we know that
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
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:
Substituting \(s=x+ct\) into (1) and denoting the variable s still by t, then the corresponding wave profile equations are
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 \(\phi_{1}(s), \phi_{2}(s), \varphi_{1}(s), \varphi_{2}(s), \psi_{1}(s), \psi_{2}(s) \in C([-\tau, 0], R)\),
-
(i)
\(0\leq\phi_{2}(s)\leq\phi_{1}(s)\leq M_{1}\), \(0\leq\varphi_{2}(s)\leq\varphi_{1}(s)\leq M_{2}\), \(0\leq\psi_{2}(s)\leq\psi_{1}(s)\leq M_{3}\), \(s\in[-\tau, 0]\);
-
(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
Let
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
For \(f_{c2}\), we have
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
and
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
There exist \(\lambda_{i}> 0\) (\(i=1,3,5\)) so that
We find that there exist \(\varepsilon_{i}> 0\) (\(i=0,1,2,3,4,5,6\)) satisfying
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:
where \(\lambda>0\) is a constant to be chosen later and
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
Obviously,
It is easy to see that \(I_{1}(0)<0\) and there exists \(\lambda^{\ast}_{1}>0\), such that
for all \(\lambda \in (0,\lambda^{\ast}_{1})\).
If \(t\leq t_{1}\), \(\overline{\phi}(t)=e^{\lambda_{1} t}\), we have
If \(t_{1}< t\leq t_{1}+c\tau_{11}\), then we have
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
Therefore, there exists a \(\lambda^{\ast}_{2}\), such that for all \(\lambda \in (0,\lambda^{\ast}_{2})\), we have
From the above argument, we see that
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
Obviously,
It is easy to see that \(I_{3}(0)<0\) and there exists \(\lambda^{\ast}_{3}>0\), such that
for all \(\lambda \in (0,\lambda^{\ast}_{3})\).
If \(t\leq t_{3}\), \(\overline{\varphi}(t)=e^{\lambda_{3} t}\), we have
If \(t_{3}< t\leq t_{3}+c\tau_{22}\), then we have
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
Therefore, there exists a \(\lambda^{\ast}_{4}\), such that for all \(\lambda \in (0,\lambda^{\ast}_{4})\)
From the above argument, we see that
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
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 \(t\leq t_{2}\),
If \(t> t_{2}+c\tau_{11}\),
Obviously,
\(a_{11}\varepsilon_{2}-a_{12}\varepsilon_{3}-a_{13}\varepsilon _{5}>\varepsilon_{0}\) implies that \(I_{5}(0)>0\) and there exists \(\lambda^{\ast}_{4}>0\) such that
for all \(\lambda \in (0,\lambda^{\ast}_{5})\).
If \(t_{2}< t\leq t_{2}+c\tau_{11}\),
It is easy to see that \(I_{6}>I_{5}>0\) and
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
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
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 \((k_{1},k_{2},k_{3})\).
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Acknowledgements
The authors express their sincere thanks to the anonymous reviewers for their valuable suggestions and corrections for improving the quality of the paper. This work is supported by the Natural Science Foundation of China (Grant No. 11471146), and partially supported by PAPD of Jiangsu Higher Education Institutions, postgraduate training project of Jiangsu Province (KYLX15_1465) and Jiangsu Normal University.
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Du, Z., Xu, D. Traveling wave solution for a reaction-diffusion competitive-cooperative system with delays. Bound Value Probl 2016, 46 (2016). https://doi.org/10.1186/s13661-016-0556-0
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DOI: https://doi.org/10.1186/s13661-016-0556-0