Abstract
This paper is concerned with the minimal wave speed in a nonlocal dispersal predator–prey system with delays. We define a threshold. By presenting the existence and nonexistence of traveling wave solutions, we confirm that the threshold is the minimal wave speed, which completes the known results.
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1 Introduction
Spatial propagation dynamics of parabolic type systems has been widely investigated in the literature. In the past decades, some important results were established for monotone semiflows; see [1–6] and a survey paper by Zhao [7]. In particular, there are some important thresholds that have been widely and intensively studied, and one is the minimal wave speed of traveling wave solutions, which plays an important role modeling biological processes and chemical kinetic [8, 9]. Here, the minimal wave speed implies the existence (nonexistence) of a desired traveling wave solution if the wave speed is not less (is less) than the threshold.
It is well known that energy transfer is one basic law in nature and one typical model on the topic is the predator–prey system, and the spatial distribution of individuals is also important to understand the evolutionary process [10–13]. Since the work of Dunbar [14–16], much attention has been paid to traveling wave solutions of reaction–diffusion systems with predator–prey nonlinearities to model the transmission of energy. However, the dynamics of predator–prey systems is a very field of research since they do not generate monotone semiflows, and there are many open problems on the minimal wave speed of traveling wave solutions.
In this paper, we shall investigate the following nonmonotone system:
in which \(x\in \mathbb{R}\), \(t>0\), \((u_{1},u_{2})\in \mathbb{R}^{2}\), \(r_{1}\), \(r_{2}\), \(d_{1}\) and \(d_{2}\) are positive constants, \(F_{1}\) and \(F_{2}\) are defined by
hereafter, \(a_{1}> 0\), \(a_{2}> 0\), \(b_{1}\ge 0\), \(b_{2} \ge 0\), \(c_{1}\ge 0\), \(c _{2}\ge 0\), \(\tau > 0\) are constants such that
Moreover, \([J_{1}*u_{1}](x,t)\) and \([J_{2}*u_{2}](x,t)\) formulate the spatial dispersal of individuals (see Bates [17], Fife [18] and Hopf [19] for the backgrounds and applications of dispersal models) and are illustrated by
where \(J_{1}\), \(J_{2}\) are probability kernel functions formulating the random dispersal of individuals and satisfy the following assumptions:
-
(J1)
\(J_{i}\) is nonnegative and continuous for each \(i=1,2\);
-
(J2)
for any \(\lambda \in \mathbb{R}\), \(\int_{\mathbb{R}}J_{i}(y)e ^{\lambda y}\,dy < \infty\), \(i=1,2\);
-
(J3)
\(\int_{\mathbb{R}}J_{i}(y)\,dy=1\), \(J_{i}(y)=J_{i}(-y)\), \(y \in \mathbb{R}\), \(i=1,2\).
Clearly, (1.1) is a predator–prey system and does not generate monotone semiflows. In Yu and Yuan [20], Zhang et al. [21], if \(a_{1}=a_{2}=0\) with small delay or \(b_{1}=b_{2}=0\), the authors obtained a threshold. If the wave speed is larger than the threshold, they proved the existence of traveling wave solutions, which formulates that both the predator and the prey invade a new habitat. But the question remains open of the existence or nonexistence of traveling wave solution if the wave speed is not larger than the threshold. Our main purpose of this paper is to answer the question.
The rest of this paper is organized as follows. In Sect. 2, we recall some known results. Section 3 is concerned with the existence of nonconstant traveling wave solutions. In Sect. 4, the asymptotic behavior and nonexistence of traveling wave solutions are presented. Finally, we give a discussion of the methods and results in this paper.
2 Preliminaries
In this part, we shall give some preliminaries. Since \(a_{1}>0\), \(a_{2}>0\) are positive constants, we assume that \(a_{1}=a_{2}=1\) due to the scaling recipe. Let
be a traveling wave solution of (1.1). Then \((\phi_{1} (\xi),\phi _{2} (\xi))\) and c satisfy
with
and
Similar to [20, 22], we shall focus on the positive \((\phi_{1},\phi_{2})\) satisfying
where \((k_{1},k_{2})\) is the unique spatial homogeneous steady state of (1.1) and
provided that
When the scalar equation is concerned, Jin and Zhao [23] studied a periodic equation with dispersal. Their results remain true for the following equation with constant coefficients:
where J satisfies (J1)–(J3), \(d >0\) and \(r>0\) are constants, and the initial value \(\chi (x)\) is uniformly continuous and bounded. By [23], Theorem 2.3, we have the following comparison principle of (2.4).
Lemma 2.1
Assume that \(0\le \chi (x)\le 1\). Then (2.4) admits a solution for all \(x\in \mathbb{R}\), \(t>0\). If \(w(x,0)\) is uniformly continuous and bounded, and \(w(x,t)\) satisfies
then
For \(\lambda >0\), define
Then \(c'>0\) holds. Moreover, it also admits the following property [23].
Lemma 2.2
Assume that \(\chi (x)>0\). Then, for any \(c< c'\), we have
If \(\chi (x)\) has nonempty compact support, then
For \(\lambda >0\), \(c>0\), we further define \(c^{\ast } =\max \{c_{1}^{ \ast },c_{2}^{\ast }\}\) with
and
By the convexity, we have the following conclusion.
Lemma 2.3
Assume that \(c^{\ast }\), \(\Theta_{1}(\lambda,c)\), \(\Theta_{2}(\lambda,c)\) are defined as the above.
-
(1)
\(c_{i}^{\ast }>0\) holds and \(\Theta_{i}(\lambda,c)=0\) has two distinct positive roots \(\lambda_{i}^{c}<\lambda_{i+2}^{c}\) for any \(c>c^{\ast }\) and each \(i=1,2\). Moreover, for each \(i=1,2\), and \(c>c_{i}^{\ast }\), if \(\lambda_{i}\in (\lambda_{i}^{c},\lambda_{i+2} ^{c})\), then \(\Theta_{i}(\lambda_{i},c)<0\).
-
(2)
If \(c\in (0,c_{i}^{\ast })\), then \(\Theta_{i}(\lambda,c)>0\) for any \(\lambda >0\) and \(i=1,2\).
-
(3)
If \(c=c_{i}^{\ast }\), then \(\Theta_{i}(\lambda,c^{\ast }) \geq 0\) for any \(\lambda >0\) and \(\Theta_{i}(\lambda,c^{\ast })=0\) has a unique positive root \(\lambda_{i}^{{\ast }}\), where \(i=1,2\).
For convenience, we use the following notation:
for any positive bounded continuous functions \(\phi_{1}(\xi)\), \(\psi _{1}(\xi)\), \(\phi_{2}(\xi)\), \(\xi \in \mathbb{R}\).
Similar to Pan [24], Theorem 3.2, we can prove the following conclusions.
Lemma 2.4
Assume that \(\underline{\phi}_{1}(\xi)\), \(\overline{\phi}_{1}(\xi)\), \(\underline{\phi}_{2}(\xi)\), \(\overline{\phi}_{2}(\xi)\) are continuous functions satisfying
-
(A1)
\(0\le \underline{\phi}_{1}(\xi)\le \overline{\phi}_{1}( \xi)\le 1\), \(0\le \underline{\phi}_{2}(\xi)\le \overline{\phi}_{2}( \xi)\le 1+c_{2}\), \(\xi \in \mathbb{R}\);
-
(A2)
there exists a set E containing finite points of \(\mathbb{R}\) such that they are differentiable and their derivatives are bounded if \(\xi \in \mathbb{R}\backslash E\);
-
(A3)
they satisfies the following inequalities:
$$\begin{aligned}& d_{1}[J_{1}\ast \overline{\phi}_{1}](\xi)-c\overline{\phi}_{1}^{ \prime }(\xi)+r_{1} \overline{\phi}_{1}(\xi)H_{1}(\overline{\phi} _{1},\underline{\phi}_{1},\underline{\phi}_{2}) (\xi)\leq 0, \end{aligned}$$(2.5)$$\begin{aligned}& d_{1}[J_{1}\ast \underline{\phi}_{1}](\xi)-c\underline{\phi}_{1} ^{\prime }(\xi)+r_{1} \underline{\phi}_{1}(\xi)H_{1}(\underline{ \phi }_{1},\overline{\phi}_{1},\overline{\phi}_{2}) (\xi) \geq 0, \end{aligned}$$(2.6)$$\begin{aligned}& d_{2}[J_{2}\ast \overline{\phi}_{2}](\xi)-c \overline{\phi}_{2}^{ \prime }(\xi)+r_{2}\overline{\phi}_{2}(\xi)H_{2}(\overline{\phi} _{1}, \underline{\phi}_{2},\overline{\phi}_{2}) (\xi) \leq 0, \end{aligned}$$(2.7)$$\begin{aligned}& d_{2}[J_{2}\ast \underline{\phi}_{2}](\xi)-c \underline{\phi}_{2} ^{\prime }(\xi)+r_{2}\underline{ \phi }_{2}(\xi)H_{2}(\underline{ \phi }_{1}, \overline{\phi}_{2},\underline{\phi}_{2}) (\xi) \geq 0, \end{aligned}$$(2.8)for \(\xi \in \mathbb{R}\backslash E\).
Then (2.1) has a positive solution \((\phi_{1}(\xi),\phi_{2}( \xi))\) such that
Remark 2.5
Here, \((\overline{\phi}_{1}(\xi), \overline{\phi}_{2}(\xi))\), \((\underline{\phi}_{1}(\xi), \underline{\phi}_{2}(\xi))\) are a pair of generalized upper and lower solutions of (2.1). Therefore, the existence of traveling wave solutions is deduced to the existence of generalized upper and lower solutions, of which the recipe has been earlier utilized in delayed reaction–diffusion systems by Ma [25] and Wu and Zou [26] for quasimonotone systems, and by Huang and Zou [27] for predator–prey systems. When the dispersal models are involved, we also refer to [20, 21, 28–31].
3 Existence of traveling wave solutions
In this section, we shall present the existence of traveling wave solutions for any \(c\ge c^{*}\). When the wave speed is large, there exists a positive traveling wave solution.
Theorem 3.1
If \(c>c^{*}\), then (2.1) has a positive solution \((\phi_{1}( \xi),\phi_{2}(\xi))\) such that
and
Proof
We shall prove it by Lemma 2.4, and first construct generalized upper and lower solutions. For convenience, we denote \(\lambda_{i} ^{c}\) by \(\lambda_{i}\) for simplicity, and we prove the result for any fixed \(c>c^{*}\).
Define continuous functions
and
where
and \(p>1\), \(q>1\) are constants, of which the definitions will be clarified later. We now show these functions satisfy (2.5)–(2.8) if they are differentiable.
If \(\overline{\phi}_{1}(\xi)=1< e^{\lambda_{1}\xi }\), then \(H_{1}(\overline{ \phi }_{1},\underline{\phi}_{1},\underline{\phi}_{2})(\xi)\leq 0 \) such that (2.5) is clear. Otherwise, \(\overline{\phi}_{1}( \xi)=e^{\lambda_{1}\xi }<1\) implies that
which implies what we wanted.
If \(\overline{\phi}_{2}(\xi)=1+c_{2}< e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }\), then \(H_{2}(\overline{\phi}_{1},\underline{\phi} _{2},\overline{\phi}_{2})(\xi)\leq 0\) such that (2.7) is clear. Otherwise, \(\overline{\phi}_{2}(\xi)=e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }<1+c_{2}\) such that
and
Note that
then there exists \(p_{1}>1+c_{2}\) such that \(p= p_{1}\) leads to
since \(\lambda_{1}+\lambda_{2}-\eta \lambda_{2}>0\), \(\xi <0\) and \(\Theta_{2}(\eta \lambda_{2},c)<0\) is a constant.
When \(\underline{\phi}_{1}(\xi)=0>e^{\lambda_{1}\xi }-qe^{\eta \lambda_{1}\xi }\), then \(H_{1}(\underline{\phi}_{1},\overline{\phi} _{1},\overline{\phi}_{2})(\xi)=0\) such that (2.6) is clear. Otherwise, \(\underline{\phi}_{1}(\xi)=e^{\lambda_{1}\xi }-qe^{ \eta \lambda_{1}\xi }>0\). Firstly, let \(q>q_{1}>1\) such that \(e^{\lambda_{1}\xi }-q_{1}e^{\eta \lambda_{1}\xi }>0\) implies \(\xi <0\) and
which is admissible once p is fixed. Therefore, the monotonicity and \(q>q_{1}\) indicate
By what we have done, (2.6) is true once
Let
then (3.2) holds since \(\xi <0\) and
The verification of (2.6) is finished.
We now consider (2.8), which is clear if \(\underline{\phi}_{2}( \xi)=0>e^{\lambda_{2}\xi }-qe^{\eta \lambda_{2}\xi }\). If \(\underline{ \phi }_{2}(\xi)=e^{\lambda_{2}\xi }-qe^{\eta \lambda_{2}\xi }>0\), we first select \(q_{3}\geq q_{2}\) implies
for any \(q\geq q_{3}\), which is admissible for fixed \(p=p_{1}\). Then
Therefore, if
then (2.8) holds since
Summarizing what we have done, it suffices to verify that (3.1) is true. We now show \(\phi_{1}(\xi)>0\), \(\xi \in \mathbb{R}\). If \(\phi_{1}(\xi_{0})=0\), then it arrives the minimal and so \(\phi_{1}'( \xi_{0})=0\), which further implies that
Therefore, \(\phi_{1}(\xi)=0\) on an interval. Repeating the process, we see that \(\phi_{1}(\xi)=0\), \(\xi \in \mathbb{R}\). A contradiction occurs since \(\underline{\phi}_{1}(\xi)>0\) if −ξ is large. Similarly, we can verify (3.1). The proof is complete. □
Theorem 3.2
Assume that \(c^{\ast }=c_{1}^{\ast }>c_{2}^{\ast }\). Further suppose that \(k_{1}(y)\) admits compact support. Then (2.1) with \(c=c^{*}\) has a positive solution \((\phi_{1}(\xi),\phi_{2}(\xi))\) such that
and
Proof
By Lemma 2.3, \(\Theta_{1}(\lambda,c^{\ast })\) arrives at its minimum when \(\lambda =\lambda_{1}^{\ast }\), and so
Let \(S>0\) be a constant such that \(k_{1}(y)=0\), \(\vert y\vert >S\). Moreover, let \(\eta >1\) such that
Consider the continuous function \(-L\xi e^{\lambda_{1}^{\ast }\xi }\), \(\xi <0\), where \(L>0\) is a constant. Clearly, if \(L>1\) is large, then
where \(\xi_{2}\), \(\xi_{1}\) with \(\xi_{2}-\xi_{1}>0\) are two roots of \(-L\xi e^{\lambda_{1}^{\ast }\xi }=1\). Moreover, let \(q>L\) be a constant clarified later, then there exists \(\xi_{3}=-q^{2}/L^{2}<-1\) such that
By the above constants, define the continuous functions
and
where \(p>1\), \(q>1\) are constants, of which the definition will be further illustrated later. We now show these functions satisfy (2.5)–(2.8) if they are differentiable.
If \(\overline{\phi}_{1}(\xi)=1\), then \(H_{1}(\overline{\phi}_{1},\underline{ \phi }_{1},\underline{\phi}_{2})(\xi)\leq 0\) such that (2.5) is clear. Otherwise, \(\overline{\phi}_{1}(\xi)=-L\xi e^{\lambda_{1} ^{\ast }\xi }<1\) implies that
and (3.3) indicates that
which implies what we wanted.
If \(\overline{\phi}_{2}(\xi)=1+c_{2}< e^{\lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }\), then \(H_{2}(\overline{\phi}_{1},\underline{\phi} _{2},\overline{\phi}_{2})(\xi)\leq 0\) such that (2.7) is clear. Otherwise, let \(p_{2}>0\) such that \(\overline{\phi}_{2}(\xi)=e^{ \lambda_{2}\xi }+pe^{\eta \lambda_{2}\xi }<1+c_{2}\) with \(p\geq p_{2}\) implies that
which is evident by simple limit analysis. Thus, the monotonicity implies
and
Note that
then there exists \(p_{3}>p_{2}+1+c_{2}\) such that \(p\ge p_{3}\) leads to
since \(\lambda_{1}^{\ast }/2+\lambda_{2}-\eta \lambda_{2}>0\), \(\xi <0\) and \(\Theta_{2}(\eta \lambda_{2},c^{\ast })<0\) is a constant. Now, we fix it by \(p=p_{3}\).
When \(\underline{\phi}_{1}(\xi)=0\) with \(\xi <\xi_{3}\), then \(H_{1}(\underline{\phi}_{1},\overline{\phi}_{1},\overline{\phi} _{2})(\xi)=0\) such that (2.6) is clear. Otherwise, if \(\xi \ge \xi_{3}\), then \(\underline{\phi}_{1}(\xi)= ( -L\xi -q\sqrt{- \xi } ) e^{\lambda_{1}^{\ast }\xi }>0\). Firstly, let \(q>q_{1}>1\) such that \(-L\xi -q\sqrt{-\xi }>0\) implies \(\xi <0\) and
for some \(\theta \in {}[ \frac{2}{3},1)\) with \(\theta \lambda_{1} ^{\ast }+\lambda_{2}>\lambda_{1}^{\ast }\), which is admissible once p is fixed. Therefore, \(q>q_{1}\) indicates
Moreover, (3.3) leads to
By what we have done, (2.6) is true if
or
We first analyze the left of the above inequality
Let
then (3.2) holds since \(\xi <0\) and
The verification of (2.7) is finished.
We now consider (2.8), which is clear if \(\underline{\phi}_{2}( \xi)=0>e^{\lambda_{2}\xi }-qe^{\eta \lambda_{2}\xi }\). If \(\underline{ \phi }_{2}(\xi)=e^{\lambda_{2}\xi }-qe^{\eta \lambda_{2}\xi }>0\), we first select \(q_{3}\geq q_{2}\) such that \(\underline{\phi}_{2}( \xi)>0\) implies
for any \(q\geq q_{3}\), which is admissible for fixed \(p=p_{1}\). Then
Therefore, if
then (2.8) holds since
By Lemma 2.4 and a discussion similar to (3.1), we complete the proof. □
Theorem 3.3
If \(c^{\ast }=c_{2}^{\ast }>c_{1}^{\ast }\). Further suppose that \(k_{2}(y)\) admits compact support. Then (2.1) with \(c=c^{\ast }\) has a positive solution \((\phi_{1}(\xi),\phi_{2}(\xi))\) such that
and
Proof
Under the assumption, we see that
by Lemma 2.3. Let \(S>0\) be a constant such that \(k_{2}(y)=0\), \(\vert y\vert >S\). Select a constant \(\eta >1\) such that
Let \(L>1\) be large enough such that
has two real roots \(\xi_{5}< \xi_{6}\) and \(\xi_{6}-\xi_{5}>2S\).
We now define
and
where \(\xi_{3}=L^{2}/q^{2}\) and \(\xi_{4}<\xi_{5}\) such that \(\overline{\phi}_{2}(\xi)\) is continuous.
For \(\overline{\phi}_{1}(\xi)\), the verification is similar to that in Theorem 3.1 and we omit it here. If \(\overline{\phi}_{2}( \xi)=1+c_{2}\), then \(H_{2}(\overline{\phi}_{1},\underline{\phi} _{2},\overline{\phi}_{2})(\xi)\leq 0\) such that (2.7) is clear. Otherwise, let \(p_{2}>0\) such that
Thus,
and
if
When \(\underline{\phi}_{1}(\xi)=0\) with \(\xi <\xi_{3}\), then \(H_{1}(\underline{\phi}_{1},\overline{\phi}_{1},\overline{\phi} _{2})(\xi)=0\) such that (2.6) is clear. Otherwise, if \(\xi \geq \xi_{3}\), then \(\underline{\phi}_{1}(\xi)=e^{\lambda_{1} \xi }-qe^{\eta \lambda_{1}\xi }>0\). Firstly, let \(q>q_{1}>1\) such that \(e^{\lambda_{1}\xi }-qe^{\eta \lambda_{1}\xi }>0\) implies \(\xi <0\) and
which is admissible once p is fixed. Therefore, \(q>q_{1}\) indicates
Moreover, (3.3) leads to
provided that
Let \(q_{3}\ge q_{2}\) such that \(q>q_{3}\) indicates
and \(q>q_{3}\), \(( -L\xi -q\sqrt{-\xi } ) >0\), imply
and so
By direct calculations, we see
if
Fix \(q=q_{5}\), we complete the proof by Lemma 2.4 and a discussion similar to (3.1). □
Theorem 3.4
Assume that \(c_{1}^{*}=c_{2}^{*}\). Further suppose that \(k_{1}\), \(k_{2}\) have compact supports. Then (2.1) with \(c=c^{\ast }\) has a positive solution \((\phi_{1}(\xi),\phi_{2}(\xi))\) such that
and
Proof
Using the notation in Theorems 3.2–3.3, we define
and
where \(p,q>1\) are large enough, \(\xi_{1}\), \(\xi_{2}\), \(\xi_{3}\), \(\xi_{4}\) are similar to above. Then we can obtain a pair of upper and lower solutions. Since the verification is similar to those in Theorems 3.2–3.3, we omit it here. □
4 Asymptotic behavior and nonexistence of traveling wave solutions
In the previous section, we obtain the existence of nonconstant traveling wave solutions of (1.1). In this part, we shall first consider the behavior if \(\xi \to \infty \) by the idea of contracting rectangle [32] in Lin and Ruan [33]. For \(s\in [0,1]\), define the continuous functions
and
with \(\epsilon \in (0,1)\) such that
Then they satisfy
-
(C1)
\(1-a_{1}(s)-b_{1}b_{1}(s)-c_{1}b_{2}(s) > 0\),
-
(C2)
\(1-a_{2}(s)-b_{2}b_{2}(s)+c_{2}a_{1}(s) > 0\),
-
(C3)
\(1-b_{1}(s)-b_{1}a_{1}(s)-c_{1}a_{2}(s) < 0\),
-
(C4)
\(1-b_{2}(s)-b_{2}a_{2}(s)+c_{2}b_{1}(s) < 0\),
for any \(s\in (0,1)\), we now verify them [34]. In (C1), we have
(C2) is true since
On (C3), we have
Finally, (C4) is true since
Remark 4.1
In Pan [34], we proved the stability of positive steady state by (C1)–(C4) of the corresponding kinetic system. Moreover, Faria [35] gave some sharp conditions on the general Lotka–Volterra systems with delays.
Theorem 4.2
Assume that \(c\ge c^{*}\). Further suppose that \((\phi_{1}(\xi),\phi _{2}(\xi))\) is a solution of (2.1) and satisfies
If
then
Proof
We first verify that
By (4.1), we see that
for any \(\xi \in \mathbb{R}\). Then \(u_{1}(x,t)=\phi_{1}(x+ct)\) satisfies
for \(x\in \mathbb{R}\), \(t>0\). By Lemmas 2.1 and 2.2, we have
and so
by the definition of traveling wave solutions.
Similarly, we have
for \(x\in \mathbb{R}\), \(t>0\). Then Lemmas 2.1 and 2.2 imply that
and so
Define
Then there exists \(s^{\prime }\in (0,1]\) such that
Define \(s=\sup s^{\prime }\). If \(s=1\), then the result is true. Otherwise, \(s<1\) and at least one of the following is true:
If \(a_{1}(s)=\phi_{1}^{-}\), then there exists \(\{\xi_{m}\}_{m=1}^{ \infty }\) such that
and
By (C1), we see that
which implies a contradiction by the definition of \(\phi_{1}(\xi)\), \(\phi_{2}(\xi)\).
By a similar discussion of
we complete the proof. □
We now present the nonexistence of (2.1) with (2.2) if \(c< c^{*}\).
Theorem 4.3
If \(c< c^{*}\), then there is not a positive solution of (2.1) with (2.2).
Proof
Were the statement false, then, for some \(c'\in (0,c^{*})\), there is a positive solution \((\phi_{1}(\xi),\phi_{2}(\xi))\) of (2.1) with (2.2). Firstly, it is easy to confirm that
If \(c^{*}=c_{1}^{*}\), then there exists \(\epsilon >0\) such that
Let \(\xi^{\prime }\in \mathbb{R}\) such that
then
Define \(\inf_{x\geq \xi^{\prime }}\phi_{1}(\xi)=\underline{\underline{ \phi_{1}}}\), then \(\underline{\underline{\phi_{1}}}>0\) by the positivity and limit behavior. Let \(M\geq 1\) such that
then
Therefore, \(\phi_{1}(\xi)=\phi_{1}(x+c^{\prime }t)=u_{1}(x,t)\) satisfies
By Lemma 2.1, we see that if
then
which also implies that \(x+c^{\prime }t\rightarrow -\infty \), \(t\rightarrow \infty \) and
a contradiction occurs.
Similarly, we can prove the result if \(c^{*}=c^{*}_{2}\). The proof is complete. □
5 Conclusion and discussion
In this paper, we firstly show the existence and nonexistence of traveling wave solutions for all positive wave speed, and thus obtain the minimal wave speed. In [20, 21], the authors studied the existence of traveling wave solutions when \(c>c^{*}\), and the traveling wave solutions decay exponentially. In this paper, if \(c=c^{*}\), these traveling wave solutions do not decay exponentially, the asymptotic behavior coincides with the conclusions in [36, 37] when \(b_{1}=b_{2}=c_{1}=c_{2}\). That is, for the minimal wave speed, the corresponding traveling wave solutions may have different properties. Moreover, there are also some results on the minimal wave speed of nonmonotone coupled systems with time delay, which was proved by constructing upper and lower solutions, part of recent results can be found in Fu [38], Lin [39] and Yang and Li [40].
In mathematical biology, the spreading speed is also an important threshold [41]. For monotone systems, see Liang and Zhao [3], Lui [4, 42], Weinberger [5], Weinberger et al. [6]. Recently, Pan [43] estimated the invasion speed of the predator in a predator–prey system, which equals the minimal invasion wave speed in Lin [44]. It is a challenging question to estimate the spreading speeds of (1.1), of which the corresponding undelayed system with classical Laplacian diffusion were studied by Lin [45], Pan [46], Wang and Zhang [47], Wang and Zhao [48].
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Funding
The first author was partially supported by Scientific Research Project of High Education of Gansu Province of China (2016B-080) and Lanzhou City University (LZCU-QN20). The second author was partially supported by NSF of China (11461040, 11471149). The third author was partially supported by Natural Science Foundation of Jiangsu Province (BK20151288).
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Li, XS., Pan, S. & Shi, HB. Minimal wave speed in a dispersal predator–prey system with delays. Bound Value Probl 2018, 49 (2018). https://doi.org/10.1186/s13661-018-0966-2
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DOI: https://doi.org/10.1186/s13661-018-0966-2