Discrete Dynamics in Nature and Society
Volume 2019 (2019), Article ID 1950767, 8 pages
https://doi.org/10.1155/2019/1950767

Research Article

Minimal Wave Speed in a Delayed Lattice Dynamical System with Competitive Nonlinearity

Fuzhen Wu

Department of Basic, Zhejiang University of Water Resources and Electric Power Hangzhou, Zhejiang 310018, China

Correspondence should be addressed to Fuzhen Wu; fuzhwu@yeah.net

Received 8 April 2019; Accepted 15 May 2019; Published 2 June 2019

Academic Editor: Luca Pancioni

Copyright © 2019 Fuzhen Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper deals with the minimal wave speed of delayed lattice dynamical systems without monotonicity in the sense of standard partial ordering in . By constructing upper and lower solutions appealing to the exponential ordering, we prove the existence of traveling wave solutions if the wave speed is not smaller than some threshold. The nonexistence of traveling wave solutions is obtained when the wave speed is smaller than the threshold. Therefore, we confirm the threshold is the minimal wave speed, which completes the known results.

1. Introduction

Propagation thresholds including minimal wave speed of traveling wave solutions and asymptotic speed of spreading in population models have attracted much attention; see [1, 2] for their biological backgrounds. For monotone systems, some sharp results were established [3, 4]. If a noncooperative system admits comparison principle in the sense of standard partial ordering in , there are also some results on propagation thresholds, e.g., predator-prey systems [5–7] and competitive system [8, 9]. For some delayed models, the dynamics may be very plentiful [10, 11], and the propagation modes may be complex; see some by [12–17] and recent papers [18–20] and references cited therein.

In a very recent paper, Pan and Shi [21] studied the minimal wave speed of the following lattice dynamical systems with time delay

with

in which , are positive constants and are nonnegative constants. By showing the existence or nonexistence of traveling wave solutions with small wave speed, they completed the conclusions in Lin and Li [22, Example 5.1] if

Although (1) does not satisfy the quasimonotonicity [23, 24], it admits a comparison principle appealing to competitive systems in the sense of standard partial ordering in . In fact, Lin and Li [22, Example 5.7] also studied the following coupled system:

where , are nonnegative. Evidently, if , then the comparison principle in the sense of standard partial ordering in does not work. Lin and Li [22] studied the existence of traveling wave solutions of (3) if the wave speed is larger than some threshold clarified later. The purpose of this paper is to confirm the existence or nonexistence of traveling wave solutions if the wave speed is positive when

In this paper, a traveling wave solution of (3) is still a special solution taking the following form:

where denote the wave profiles and reflects the wave speed. Therefore, , and must satisfy

with

Because (4), (3) have a positive steady state defined by

Similar to that in [21, 22], we want to study the existence or nonexistence of (6) satisfying

which may formulate the coinvasion-coexistence of two competitors.

In this paper, we establish the existence of traveling wave solutions by an abstract result in Lin and Li [22, Theorem 4.5] with the help of exponential ordering [25], which will be finished by constructing proper upper and lower solutions if the wave speed is large. Then the nonexistence of traveling wave solutions is proved by the theory of asymptotic spreading if the wave speed is small. It should be noted that our discussion includes all positive wave speed, so we obtain a minimal wave speed and complete the conclusion in [22].

2. Main Result

In this paper, we shall prove the following result.

Theorem 1. Assume that are small enough. Then (6) and (9) have a strict positive solution satisfying

if and only if , where and

We now prove the above conclusion. And we use the standard partial ordering in That is, if and , then

and

If , we define

Lemma 2. Assume that , are defined as the above.

(1) holds.(2)For with any fixed , there exist such that and implies (3)For with any fixed , .(4)For with , holds and has a unique positive root .

When the wave speed is small, we have the following conclusion.

Lemma 3. If , then (6) with (9) does not have a positive solution.

Proof. The proof is similar to that in Pan and Shi [21]. We prove the result if Assume that for some fixed , (6) has a positive solution satisfying (9) such that

Select such that

Let such that

where is admissible since

Define

and then .
By what we have done, satisfies

with

By the definition of , we see that

According to the theory of asymptotic spreading, we have

with

For any , select with , and then implies

such that , and a contradiction occurs between the above and (23). The proof is complete.

Lin and Li [22, Theorem 4.5] proved the following abstract conclusion on the existence of (6) with (9).

Lemma 4. Assume that is fixed. Further suppose that continuous functions

such that

(1),(2),(3),(4)there exists such that

 are nondecreasing for

Moreover, except for several points, they are differentiable such that

If are small enough, then (6) with (9) has a positive solution such that

Remark 5. The continuous functions satisfying (28)-(31) are a pair of upper and lower solutions of (6). In Lin and Li [22], there are also some sufficient conditions on the bounds of and At least if are small, there exists a bounded , and we do not focus on precise conditions on in this paper. Moreover, the existence of (6) and (9) has been confirmed if ; we refer to Lin and we refer to Lin and Li [22, Example 5.2]. Therefore, it suffices to consider

Lemma 6. If and are small enough, then (6) with (9) has a positive solution.

Proof. We now prove the conclusion by constructing upper and lower solutions. Firstly, we show the definition of potential upper and lower solutions and show the logistic sequence and the admissible of several parameters in upper and lower solutions; then we shall verify the necessary inequalities.
When , by selecting parameters, we define positive continuous functions

and

where all the parameters are clarified later. We now show the key logistic sequence on the selection and verification of parameters and functions as follows.

(P1)Fix a constant such that

 where with are two roots of

By the selection of , we see that such that and

and

Thus (28) is true if

which is evident by This completes the verification of (28) when .

(P2)Let be a constant such that

 Denote Fix

 such that implies

 which is admissible by simple limit analysis as

We now verify (30) if By the definition, we see that

By the definition of , we see

and so

Then it suffices to verify

or

By the properties of , the above is true if

or

Since

(30) is true if

Note that , and then

implies what we wanted.

(P3)Fix a constant such that

 Let

 such that implies

 which is admissible by the limit as

We now prove (31) if such that . In fact, we have

Then it suffices to verify that

which is equivalent to

Clearly, the above is true if

(P4)Let

 For any given , define

 Then there exists such that

(P5)Fix small enough, which will be clarified later.

With the above parameters, we now illustrate the other parameters in upper and lower solutions.

(X1)Let be the smallest root of

(X2)Let be the smallest root of

(X3)Let be the larger root of

(X4)Let be the larger root of

We now complete the verification of (28)-(31). On (28), since , the verification of has been given by (P1). If is differentiable, the conclusion is also true if is small. If , then such that

and

Let be small such that , and then (28) is true if

Clearly, (68) holds if , and then it is true if is small. In particular, we may fix such that

for all
If but , we see that varies in a bounded interval, of which the length is . Then it suffices to verify

which holds by the continuity if is small. We complete the proof of (28).
Similarly, we can confirm that (29)-(31) are true if are small and is small. The proof is complete if .
If , the proof can be finished similar to that for . When , define

and

in which all the parameters are similar to that in the case of . By combining the above analysis with Pan and Shi [21], the upper and lower solutions can be verified. We complete the proof.

3. Conclusion and Discussion

To obtain the existence and asymptotic behavior of traveling wave solutions of nonmonotone systems is not easy; at least the oscillation of traveling wave solutions has been observed in some delayed equations without quasimonotonicity [26]. In Li et al. [27], the authors constructed upper and lower solutions to confirm the existence and asymptotic behavior of traveling wave solutions of delayed competitive systems. Furthermore, Lin and Li [22] also gave some similar conclusions for lattice dynamical systems without quasimonotonicity.

However, to construct upper and lower solutions in [22, 27], several parameters are needed. What is more, the logistic sequence of selection of these parameters is very important. Although Lin and Li [22] showed the existence of desirable upper and lower solutions, the logistic sequence was not given. For large wave speed, Lin et al. [28] presented the sequence of a competitive reaction-diffusion system with time delay. In this paper, we show the sequence of parameters for minimal wave speed of model (3), and the sequence can be utilized to illustrate that in [21].

Moreover, for the size of time delay, it should be further investigated to show the existence of traveling wave solutions satisfying desirable asymptotic behavior. On this topic, very likely the recipe in Kwong and Ou [15] is a good motivation. In the future, we shall try to fix the bounds of time delays such that (6) and (9) have a strict positive solution.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

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