Abstract
This paper deals with the minimal wave speed of delayed lattice dynamical systems. We obtain the minimal wave speed by presenting the existence and nonexistence of traveling wave solutions, which completes the earlier results. In particular, when the wave speed equals the minimal wave speed, traveling wave solutions do not exponentially decay.
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1 Introduction
Lattice dynamical systems are spatially discrete evolutionary models, which could more reasonably reflect many important facts of natural phenomena than the continuous cases. For example, these systems successfully characterized the propagation failure of excitable cells [17]. In literature, there are some results on general theory of lattice dynamical systems, see Chow [10], Mallet-Paret [26–28]. In particular, one important topic in these works is the traveling wave solution. We refer to [1, 3–9, 13, 14, 31, 33, 40, 42–44] for some models and results on wave propagation of lattice dynamical systems.
Because time delay is universal in natural phenomena, much attention has been paid to traveling wave solutions of lattice dynamical systems with delayed effect, see [15, 16, 18, 21, 23–25, 32, 35–39, 41]. When the propagation dynamics are concerned, there are some important thresholds modeling crucial features, and one is the minimal wave speed of traveling wave solution in the sense that wave speed larger (smaller) than the threshold or equivalent to the threshold implies the existence (nonexistence) of traveling wave solutions, see some results in the above works.
It should be noted that the comparison principle appealing to monotone semiflows plays an important role, and some results are sharp in the works mentioned above. However, when the noncooperative systems are concerned, there are some open problems. In this paper, we consider the following lattice dynamical system [20]:
with
where \(n\in\mathbb{Z}\), \(t>0\), \(d_{1}\), \(d_{2}\), \(r_{1}\), \(r_{2}\) are positive and \(b_{1}\), \(b_{2}\), \(\tau_{1}\), \(\tau_{2}\) are nonnegative.
In Lin and Li [20], if \(b_{1},b_{2}\in[0,1)\), then (1.1) has a positive equilibrium \(K=(k_{1},k_{2})\), where
They proved the existence of traveling wave solutions connecting \((0,0)\) with \((k_{1},k_{2})\) when the wave speed is larger than a threshold \(c^{*}\), which will be clarified in the subsequent section. But it remains open on the existence/nonexistence of traveling wave solutions when the wave speed is not large. To answer this question when \(b_{1},b_{2}\in[0,1)\) is the main purpose of this paper.
In this paper, we shall confirm the existence or nonexistence of traveling wave solutions with smaller wave speed, which will complete the conclusions in Lin and Li [20]. More precisely, by Schauder’s fixed point theorem, we obtain a sufficient condition on the existence of nontrivial traveling wave solutions. We then confirm the existence of traveling wave solutions by constructing upper and lower solutions if the wave speed is \(c^{*}\). To obtain the asymptotic behavior of traveling wave solutions, we use the theory of asymptotic spreading. Moreover, we also confirm the nonexistence of traveling wave solutions if the wave speed is smaller than \(c^{*}\), which is investigated by constructing auxiliary equations and utilizing the theory of asymptotic spreading.
In Sect. 2, we shall give some preliminaries, which implies that the existence of traveling wave solutions can be obtained by the existence of proper upper and lower solutions. In Sect. 3, we give our conclusions including the existence and nonexistence of traveling wave solutions.
2 Preliminaries
In this paper, we use the standard partial ordering in \(\mathbb{R}^{2}\). That is, for \(u=(u_{1},u_{2})\) and \(v=(v_{1},v_{2})\), we denote \(u\leq v\) if \(u_{i}\leq v_{i}\), \(i=1,2\), and \(u< v\) if \(u\leq v\) but \(u\neq v\). Let \(C(\mathbb{R},\mathbb{R}^{2})\) be a set of bounded and uniform continuous functions from \(\mathbb{R}\) to \(\mathbb{R}^{2}\). Define
then \(C(\mathbb{R},\mathbb{R}^{2})\) is a Banach space with supremum norm \(\| \cdot\|\).
A traveling wave solution of (1.1) is a special translation invariant solution of the form
where \(\phi,\psi\in C^{1}(\mathbb{R},\mathbb{R})\) are the so-called wave profiles propagating through the one-dimensional spatial lattice at a constant velocity \(c>0\). Thus, ϕ, ψ, and c satisfy the following mixed functional differential equations:
with
Because we are interested in the invasion process of two competitive invaders [20], then ϕ, ψ satisfy the following asymptotic boundary conditions:
For \(\lambda>0\), \(c>0\), we further define
and
By the convexity, we have the following conclusion.
Lemma 2.1
Assume that \(c^{\ast}\), \(\Lambda_{1}(\lambda,c)\), \(\Lambda _{2}(\lambda,c)\) are defined as the above.
-
(1)
\(c^{\ast}_{i}>0\) holds and \(\Lambda_{i}(\lambda,c)=0\) has two distinct positive roots \(\lambda_{i}^{c}<\lambda_{i+2}^{c}\) for any \(c>c^{\ast}_{i}\) and each \(i=1,2\). Moreover, if \(c>c^{\ast}_{i}\) and \(\lambda_{i}\in(\lambda_{i}^{c},\lambda _{i+2}^{c})\), then \(\Lambda _{i}(\lambda_{i},c)<0\), \(i=1,2\).
-
(2)
If \(c\in(0,c_{i}^{\ast})\), then \(\Lambda_{i}(\lambda,c)>0\) for any \(\lambda>0\) and \(i=1,2\).
-
(3)
If \(c=c_{i}^{\ast}\), then \(\Lambda_{i}(\lambda,c^{\ast })\geq0 \) for any \(\lambda>0\) and \(\Lambda_{i}(\lambda,c^{\ast})=0\) has a unique positive root \(\lambda_{i}^{{\ast}}\), where \(i=1,2\).
On the existence of (2.1), we have the following conclusion.
Lemma 2.2
If \((\underline{\phi}(\xi),\underline{\psi}(\xi)), (\overline{\phi}(\xi),\overline{\psi}(\xi)) \in C(\mathbb{R},\mathbb {R}^{2})\) satisfy
Moreover, except several points, they are differentiable such that
Then (2.1) with \(c=c^{*}\) has a positive solution \((\phi(\xi), \psi (\xi))\) such that
Proof
Let \(\beta>0\) be a constant such that
are monotone in
Define
Equip \(C(\mathbb{R},\mathbb{R}^{2})\) with the norm \(|\cdot|_{\mu}\) defined by
and define
Then we obtain a Banach space \((B_{\mu}(\mathbb{R},\mathbb{R}^{2}), |\cdot|_{\mu})\). Let Γ be a subset of \(B_{\mu}(\mathbb{R},\mathbb{R}^{2})\) such that \((\phi(\xi),\psi (\xi))\in\Gamma\) implies
Then Γ is bounded and closed with respect to the norm \(|\cdot|_{\mu}\), and it is a nonempty and convex subset of \(B_{\mu}(\mathbb{R},\mathbb{R}^{2})\).
For \((\phi(\xi),\psi(\xi))\in\Gamma\), we define \(F=(F_{1},F_{2})\) by
Then, by direct calculations, we can prove that \(F: \Gamma\to\Gamma\) and the mapping is complete continuous in the sense of \(|\cdot|_{\mu}\), which is similar to that in Huang et al. [16]. Due to Schauder’s fixed point theorem, the proof is complete. □
We also consider the following initial value problem:
where \(r>0\), \(d>0\) and
By Ma et al. [23] and Weng et al. [35], we have the following conclusions.
Lemma 2.3
If \(0\le\psi(n)\le1\), \(n\in\mathbb{Z}\), then (2.7) has a solution \(u_{n}(t)\) for all \(n\in\mathbb{Z}\), \(t>0\). If \({w}_{n}(t)\), \(n\in\mathbb{Z}\), \(t>0\), satisfies
then \({w}_{n}(t) \ge(\le)\ u_{n}(t)\) for all \(n\in\mathbb{Z}\), \(t>0\). In particular, \(w_{n}(x)\) is called an upper (a lower) solution of (2.7).
Lemma 2.4
Define \(c_{1}=:\inf_{\lambda>0}\frac{D(e^{\lambda}+e^{-\lambda}-2)+ h(0)}{\lambda}>0\). If \(\psi(n)\ge0\), \(n\in\mathbb{Z}\) such that \(\psi _{n}(0)> 0\) for some \(n\in\mathbb{Z}\), then
for any given \(c< c_{1}\).
3 Main results
Our main conclusion of this paper is given as follows.
Theorem 3.1
If \(c< c^{*}\), then (2.1) does not admit a positive solution satisfying (2.2). If \(c=c^{*}\), then (2.1) with \(c=c^{*}\) has a strict positive solution satisfying (2.2) and
-
(1)
\(\lim_{\xi\to-\infty}\frac{\phi(\xi)}{-\xi e^{\lambda_{1}^{\ast} \xi }}\), \(\lim_{\xi\to-\infty}\frac{\psi(\xi)}{e^{\lambda_{2}^{c^{*} } \xi}} \) are positive if \(c=c^{*}= c_{1}^{*}> c_{2}^{*}\);
-
(2)
\(\lim_{\xi\to-\infty}\frac{\phi(\xi)}{e^{\lambda_{1}^{c^{*} } \xi}}\), \(\lim_{\xi\to-\infty}\frac{\psi(\xi)}{-\xi e^{\lambda_{2}^{* } \xi}}\) are positive if \(c=c^{*}= c_{2}^{*}> c_{1}^{*}\);
-
(3)
\(\lim_{\xi\to-\infty}\frac{\phi(\xi)}{-\xi e^{\lambda_{1}^{\ast} \xi }}\), \(\lim_{\xi\to-\infty}\frac{\psi(\xi)}{-\xi e^{\lambda_{2}^{* } \xi }}\) are positive if \(c=c^{*}= c_{2}^{*}= c_{1}^{*}\).
The above result will be proved by several lemmas, the first one is the following.
Lemma 3.2
Assume that \(c_{1}^{*} >c_{2}^{*}\). Then (2.1) with \(c=c^{*}= c_{1}^{*}> c_{2}^{*}\) has a positive solution \((\phi(\xi),\psi(\xi))\) and \(\lim_{\xi\to-\infty}\frac{\phi(\xi)}{-\xi e^{\lambda_{1}^{\ast} \xi }}\), \(\lim_{\xi\to-\infty}\frac{\psi(\xi)}{e^{\lambda_{2}^{c^{*} } \xi}} \) are positive.
Proof
Note that \(\Lambda_{1}(\lambda,c^{\ast})\geq0\) and \(\Lambda _{1}(\lambda ,c^{\ast})\) arrives at its minimal when \(\lambda=\lambda_{1}^{\ast }\), then \(\frac{\partial\Lambda_{1}(\lambda,c^{\ast})}{\partial\lambda}\vert_{\lambda =\lambda_{1}^{\ast}}=0\) or
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 \(\xi_{1}\), \(\xi_{1}^{\prime}\) with \(\xi_{1}^{\prime}-\xi _{1}>0\) are two roots of \(-L\xi e^{\lambda_{1}^{\ast}\xi }=1\). We now fix L and define
Moreover, let \(q_{1}> L\) such that
At the same time, there exists \(q_{2}>q_{1}\) such that
Further select
and
where \(\eta\in(1,2)\) is a fixed constant such that
Now, we define
By the above constants, define
and
We now show that these continuous functions satisfy (2.3)–(2.6) if they are differentiable.
(1) Equation (2.3) is clear when \(\xi> \xi_{1}\). If \(\xi<\xi_{1}\), then \(\overline{\phi}(\xi)=-L\xi e^{\lambda _{1}^{\ast}\xi}\) such that
and
Thus (2.3) is true if
which is evident by \(\Lambda_{1}\). This completes the verification of (2.3).
(2) If \(\xi>0\) such that \(\overline{\phi}(\xi)=1\), then (2.4) is clear. When \(\xi<0\), it suffices to show
which is clear by the definition of \(\lambda_{2}^{c^{*}}\). We obtain (2.4) when \(\xi\neq0\).
(3) On \(\underline{\phi}(\xi)\), we shall prove (2.5) when \(\underline{\phi}(\xi)\) is differentiable, and it is clear if \(\xi>\xi _{2}\). If \(\xi<\xi_{2}\) and \(\underline{\phi}(\xi )= ( -L\xi-q\sqrt{-\xi} ) e^{\lambda_{1}^{\ast}\xi}\), then
Since \(\lambda^{\prime}=\min \{ \lambda_{1}^{\ast}/2,\lambda _{2}^{c^{\ast}} \} \), then
and so
Then it suffices to verify
or
By the properties of \(\Lambda_{1}(\lambda,c)\), the above is true if
or
Since
then (2.5) is true if
Note that \(\xi_{2}<-1\), then
implies what we wanted.
(4) When \(e^{\lambda_{2}^{c^{\ast}}\xi}-qe^{\eta\lambda _{2}^{c^{\ast}}\xi}<0\), (2.6) is clear. If \(\underline{\psi }(\xi)>0\), then
Then it suffices to verify that
which is equivalent to
Clearly, the above is true if
Summarizing what we have done, we obtain (2.3)–(2.6) except several points. From Lemma 2.2, the proof is complete. □
Similar to the proof of Lemma 3.2, we have the following result.
Lemma 3.3
Assume that \(c_{1}^{*} < c_{2}^{*}\). Then (2.1) with \(c=c^{*}= c_{2}^{*}\) has a positive solution \((\phi(\xi),\psi(\xi))\) such that \(\lim_{\xi\to-\infty}\frac{\phi(\xi)}{e^{\lambda_{1}^{c^{*} } \xi}}\), \(\lim_{\xi\to-\infty}\frac{\psi(\xi)}{-\xi e^{\lambda_{2}^{* } \xi}} \) are positive.
Further, combining the recipes in Lemmas 3.2–3.3, we obtain the following conclusion.
Lemma 3.4
Assume that \(c_{1}^{*} =c_{2}^{*}\). Then (2.1) with \(c=c^{*}\) has a positive solution \((\phi(\xi),\psi(\xi))\) such that \(\lim_{\xi\to-\infty} \frac{\phi(\xi)}{-\xi e^{\lambda_{1}^{* } \xi }}\), \(\lim_{\xi\to-\infty}\frac{\psi(\xi)}{-\xi e^{\lambda_{2}^{* } \xi}} \) are positive.
Lemma 3.5
Assume that \((\phi(\xi),\psi(\xi))\) is given by one of Lemmas 3.2–3.3. Then it is strictly positive and satisfies (2.2).
Proof
Assume that \(\phi(\xi_{0})=0\) for some \(\xi_{0}\in\mathbb{R}\). Then
and so
by the definition of F, which implies a contradiction since \(\phi (\xi) \ge\underline{\phi}(\xi)>0\) if −ξ is large. By a similar discussion on \(\psi(\xi)\), we see that
By the definition of traveling wave solutions, \(\phi(\xi)=u_{n}(t)\) satisfies
which implies that
Similarly, we obtain
Define
then
and
Applying the dominated convergence theorem in F, the monotonicity implies
which indicates
and so
by \(b_{1},b_{2}\in{}[0,1)\). The proof is complete. □
By what we have done, we have proven the existence of traveling wave solutions. We now consider the nonexistence of traveling wave solutions.
Lemma 3.6
If \(c< c^{*}\), then (2.1) does not admit a positive solution satisfying (2.2).
Proof
Without loss of generality, we assume that \(c^{*}=c_{1}^{*}\). Were the statement false, then for some fixed \(c< c^{*}\), (2.1) has a positive solution \((\phi(\xi),\psi(\xi))\) satisfying (2.2). It is evident that
Select \(\epsilon>0\) such that
Let \(\xi'\) such that
where \(\xi'\) is admissible since
Define
then \(\Phi>0\) by the positivity of \(\phi(\xi)\) and (2.2).
Therefore, \(\phi(\xi)\) satisfies
with
By the definition of \(\phi(\xi)=u_{n}(t)\), we see that
From Lemma 2.4, we have
for
Let \(-2n=(c+c')t\), then \(n\to-\infty\) implies
and \(\lim_{n\to\infty}u_{n}(t)=\lim_{\xi\to-\infty} \phi(\xi)=0\), a contradiction occurs between the above and (3.3). The proof is complete. □
4 Conclusion and discussion
Minimal wave speed of traveling wave solution of evolutionary systems is very visual in characterizing some natural phenomena, e.g., modeling the diffusion of epidemic [11, 12]. In Lin and Li [20], the authors proved the existence of positive solutions of (2.1)–(2.2) if the wave speed is larger than \(c^{*}\). In this paper, we obtain the existence and nonexistence of (2.1)–(2.2) if \(c\le c^{*}\), which implies that \(c^{*}\) is the minimal wave speed and completes the earlier results. In particular, when the wave speed equals the minimal wave speed, traveling wave solutions do not exponentially decay, which is different from that in Lin and Li [20].
Besides the minimal wave speed, spreading speed [2] is also an important threshold. In some monotone systems, it has been proven that the spreading speed equals the minimal wave speed [18, 22, 34]. Even for the predator-prey system, a similar conclusion is obtained [19, 30]. On nonnomotone lattice differential equations, we also obtain a result in Pan [29]. But for coupled systems, it seems to be more difficult, and we shall consider the spreading speed of such a competitive system in forthcoming papers.
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Supported by NSF of China (11461040, 11471149), Natural Science Foundation of Jiangsu Province (BK20151288).
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Pan, S., Shi, HB. Minimal wave speed of competitive lattice dynamical systems with delays. Adv Differ Equ 2018, 279 (2018). https://doi.org/10.1186/s13662-018-1745-1
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DOI: https://doi.org/10.1186/s13662-018-1745-1