Some Qualitative Properties of Traveling Wave Fronts of Nonlocal Diffusive Competition-Cooperation Systems of Three Species with Delays

Yao, Meiping; Qiao, Pengzhi; Wang, Yang

doi:https://doi.org/10.1155/2020/6909567

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 6909567 | https://doi.org/10.1155/2020/6909567

Some Qualitative Properties of Traveling Wave Fronts of Nonlocal Diffusive Competition-Cooperation Systems of Three Species with Delays

Meiping Yao,¹Pengzhi Qiao,¹and Yang Wang¹

Academic Editor: Mojtaba Ahmadieh Khanesar

Received06 Jul 2020

Accepted29 Aug 2020

Published18 Sept 2020

Abstract

This paper is concerned with nonlocal diffusion systems of three species with delays. By modified version of Ikehara’s theorem, we prove that the traveling wave fronts of such system decay exponentially at negative infinity, and one component of such solutions also decays exponentially at positive infinity. In order to obtain more information of the asymptotic behavior of such solutions at positive infinity, for the special kernels, we discuss the asymptotic behavior of such solutions of such system without delays, via the stable manifold theorem. In addition, by using the sliding method, the strict monotonicity and uniqueness of traveling wave fronts are also obtained.

1. Introduction

In this paper, we are concerned with the following nonlocal diffusive competition-cooperation systems of three species with delays:where , denote the diffusive kernel functions of three species, respectively, and represent the density of three species. We assume that , , , , and , .

By directly computing, (1) has eight equilibria as follows, which are also the equilibria of the corresponding ordinary differential system of (1):wherewhere , , are non-negative equilibria. Moreover, by directly computing, gives . Thus, when , is the unique positive equilibrium if and only if

On the other hand, for (1) without diffusion and delays, by direct calculation, is unstable, and is stable, if

Furthermore, if (5) and , then the third component of is negative, which implies that is not in the first octant.

The existence and other properties of traveling wave solutions are important research fields in diffusive equations including reaction diffusion equations and nonlocal diffusion equation. For (1), the vector value functionis called a traveling wave solution connecting and , if it satisfies

Moreover, if , , are monotone in , then we call it the traveling wave front.

If the diffusive kernel functions , where is the Dirac delta function and , , under some assumptions on the parameters, Hung in [9] firstly proved the existence of traveling wave fronts of (1) connecting with and gave the exact solutions. Then, in [16], when the wave speed is large, Ma et al. invested the nonlinear stability of such traveling wave fronts of (1) via the weighted energy method and comparison principle. In addition, Liu and Weng obtained the relevant conclusions of the asymptotic speeds of (1) by using the method of super-sub solutions and comparison principle in [12]. Very recently, in [21], Tian and Zhao proved the existence and the global stability of bistable traveling wave fronts of (1) with finite or infinite delays by using the theory of monotone semiflows, the squeezing technique, and the dynamical systems approach. When delays are taken into consideration and holds, in [10], Huang and Weng proved the existence of traveling wave fronts of (1) connecting with if (4) exists and connecting with if (5) exists.

Besides the aforementioned papers, for the research on the existence of traveling wave solutions, we can refer to [2–4, 7, 8, 13, 15, 17, 19–21, 23, 24]. Specially, Li et al. in [13] introduced the nonlocal diffusion and time delays into the classical Lotka–Volterra reaction diffusion system and proved the existence, asymptotic behavior, and uniqueness of the traveling wave front of this system connecting the equilibria on the two axes. From the above statements, as we know, there is no conclusion about the existence, asymptotic behavior, and other properties of the traveling wave solutions of (1), thus, inspired by [13], we mainly consider the asymptotic behavior, strict monotonicity, and uniqueness of the traveling wave fronts of (1), connecting and . Firstly, we make a variable substitution . Thus, (1) is changed into the following cooperative system, by dropping the tildes for simplicity:

Correspondingly, system (9) also has eight corresponding equilibria as follows:

From the above statements, the traveling wave fronts of (1) connecting with are equivalent to the traveling wave fronts of (9) connecting with . And, the boundary conditions (7) and (8) are correspondingly changed intowhere

In the sequel, the traveling wave front of (9) as we mention is the nondecreasing solution to (11) and (12). We give the basic assumptions of this paper to end this section. Firstly, we always assume that (5) holds. Secondly, the kernels , , satisfy (P1) (P2) For every ,

In Section 2, we introduce some notations and the main results. In Section 3, we discuss the asymptotic behavior of the nondecreasing solutions to (11) and (12). In Section 4, we give the strict monotonicity and uniqueness of the nondecreasing solutions to (11) and (12).

2. Preliminaries and Main Results

In this section, we will discuss the eigenvalue problems and introduce the main results in this paper. First of all, let

By some simple computations, we have the following lemma.

Lemma 1. Assume that (P1) and (P2) hold. Then, there exist , , such that and , . For , the equation has two positive roots , with , while for , for . Similarly, for , the equation has two positive roots , with , while for , for .

Then, for any , we will discuss the root of .

Lemma 2. Assume that (P1) and (P2) hold and with small , has a unique positive bounded root for , and for . Moreover, is a unique root with of .

Proof. LetNote thatand , . Thus, is increasing in . Then, we remark that there are and , such that and on ; otherwise, by the symmetry of , which contradicts (P1). Therefore, by (P2),which implies has a unique zero. Hence, in , there is only such thatand , for . Thus, by further noting , and , there is only such that .
If , then let be the corresponding root, which in fact satisfies . Suppose , for . Since , , then . Hence, we havewhich is a contradiction. Thus, . The rest of the proof is similar to the proof of Lemma 2.8 in [13], so we omit it here.
Let , and . Now, we give the following the exponential decay rates of at the minus infinity.

Theorem 1. Assume that (P1), (P2), and (5) hold. For , assume that is any nondecreasing solution of (11) and (12) for either or sufficiently small positive , . Then,(i)There exist such that(i)There exist such that

Case 1. When and ,

Case 2. When .
If , thenwhere , .
If , thenwhere .
If , thenwhere
If , thenwhere .

Case 3. When , by choosing different , the conclusions are the same as Case 2.
Moreover, when ,Next, we investigate the exponential decay rates of at the plus infinity. Similarly, let

Lemma 3. Assume that (P1), (P2), and (5) hold. Then, has a unique negative root for .

Proof. Sincefor ,and from (5), then obviously has a unique negative root .
Then, we introduce the following theorem.

Theorem 2. Assume that (P1), (P2), and (5) hold. For , assume that is any nondecreasing solution of (11) and (12). Then, there exists such that

We cannot use the method offered in [13] to obtain the asymptotic behavior similar to Theorem 1. In order to obtain more detailed results of at the plus infinity, we consider a special case, , , and . Moreover, from Lemma 8 in Section 3, we conclude that are the negative roots of

Then, with the methods in [5, 18, 22] and the stable manifold theorem, we give the following asymptotic behavior of at the plus infinity.

Theorem 3. Assume that (P1), (P2), and (5) hold. For , we assume that is any nondecreasing solution of (11) and (12), , , andwhere is the Dirac delta function. In addition, letThen, when , has the following asymptotic behavior:
(i) When ,where .(ii) When ,where , , .
(iii) When ,where , .
Moreover, suppose(iv) When ,where , .(v) When ,where , , , and

We give a remark at once to show (38) is reasonable.

Remark 1. Choose , , ; then, . Thus, the left side of the equation (38) is equal to , while the right side of the equation (38) is , which are not equal for .
Finally, we introduce conclusions on the strict monotonicity and the uniqueness.

Theorem 4. Assume that (P1), (P2), and (5) hold, and , . For , if is any nondecreasing solution of (11) and (12), then it is strictly monotone.

Theorem 5. Under the assumptions of Theorem 3, let and be two nondecreasing solutions of (11) and (12), with the same speed and the same decay rate at . Then, they are unique (up to a translation).

Remark 2. When , , and , the existence of nondecreasing solutions of (11) and (12) had been discussed in [9]. From Theorems 1–5, we further investigate the asymptotic behavior, strict monotonicity, and uniqueness of such solutions in this paper.

Remark 3. The proof of existence of traveling wave fronts is standard. For example, suppose , and (P1), (P2), (5) hold. Since system (11) is the quasimonotone system or weak quasimonotone system , with the methods in the [11–13, 17, 19, 23] and references therein, the existence of traveling wave fronts to (9) can be proved.

3. Asymptotic Behavior of Traveling Wave Solutions

In this section and next section, we always assume or , , small enough, and assume (P1), (P2), and (5) hold; then, we investigate the asymptotic behavior, strict monotonicity, and uniqueness of nondecreasing solutions to (11) and (12).

To study the asymptotic behavior, we introduce the following result given in [24].

Proposition 1. Let be a constant and be a continuous function having finite limits at infinity . Let be a measurable function satisfyingThen, is uniformly continuous and bounded. In addition, exist and are real roots of the characteristic equation

Firstly, we give some properties of the solutions to (11) and (12).

Lemma 4. Suppose (P1), (P2), and (5) hold, and for , is any nondecreasing solution of (11)-(12). Then,Moreover, if , , then

Proof. From the assumptions, we remark that is nondecreasing and . Thus, let be the right-most point of ; then, attains the minimum at , implying . It follows from the first equation of (11) that . By (P1), there exist , such that for . Sincefollowing from , . Then, we obtain for , which contradicts the choice of . Thus, in . Similarly, we can prove , .
When , similarly, suppose there exists a point such that , implying . By recalling and , , we deduceThus, . On the other hand,which is a contradiction. Thus, . Similarly, when , .
When , we also assume there exists a point such that , and thus . Also by noting , , , we haveTherefore,which is a contradiction, and thus .
By Lemmas 1 and 4 and Proposition 1, it is easy to verify the following theorem.

Theorem 6. Suppose (P1), (P2), and (5) hold, and for , is any nondecreasing solution of (11) and (12). Then,where , , are described as Lemma 1.

Define the bilateral Laplace transformwhere is a continuous function. The following lemma is derived from (12) and Theorem 6.

Lemma 5. Suppose (P1) and (P2) hold, and for , is any nondecreasing solution of (11) and (12). Then,

Then, we discuss the convergence of via the following lemma.

Lemma 6. Suppose (P1), (P2), and (5) hold, and for , suppose is any nondecreasing solution of (11) and (12). Then,where .

Proof. We first search such that , . From Lemma 2, we can conclude that has two roots 0 and for , andfor . Since implies thatfor , then there exists a small such thatfor . For , we take small enough such that for .
Multiplying the third equation of (11) by with and integrating from to givesFor the left side of (59), obviously,By recalling the convergence in Lemma 5, from (56)–(61), similar to [13], we deducewhich implies, for any ,We conclude that there exists such that for .
We now prove that . Multiplying the third equation of (11) by with and integrating from to givesWe claim that . Otherwise, if , then . Taking in (64), the left side of (64) equals 0 by , while the right side of (64) is always negative by , which is a contradiction. It also follows easily from (64) that if or if . Furthermore, if , it also follows that exists.
In order to study the asymptotic behavior, we introduce the following modified version of Ikehara’s theorem [1], which has been applied early in [6].

Lemma 7 (Ikehara’s theorem). Let be a positive nondecreasing function on and define . Assume that can be written as , where , and is analytic in the strip ; then,

With the aid of the above lemmas, we will finish the proof of Theorem 1.

Proof of Theorem 1. The proof is motivated by [13–15]. From Lemmas 5 and 6, are well defined for with , , and is well defined for with , respectively. It follows from (11) thatfor with , , andfor with , where , are defined in Theorem 1 andBy Lemmas 2, 5, and 6 and directly calculating, we conclude the following facts.(1) is a unique root with of , , and is a unique root with of .(2) are analytic in the strip , , and is analytic in the strip .From (66), we definewhere when and when .Then, from (1) and (2), we deduce that are analytic in the strip , . Thus, from Lemma 7,which implies (i) in Theorem 1 holds if . In the following, we will prove that , , in two cases.
For , ; since and is a simple root of , then we have at . For , if we suppose , then from (69), exist, which contradicts Lemma 5. Thus, , . Therefore, from (70), we arrive atFor , in this case, there are three subcases, , , and . The first two subcases are equivalent by changing the roles of and . Thus, without loss of generality, we consider . In this subcase, is a simple root of ; repeating the above proofs of with , , take , and we directly haveSince is a double root of , then at . When , we can take such thatWhile , must be a simple root of ; otherwise, exists by (69), which contradicts Lemma 5. Thus, we can take such thatWhen , repeating the above proofs gives , , where , .
Now we prove (ii) in Theorem 1. From (67), we definewhere is the same as above. By using a similar argument as (i), is analytic in the strip . Hence, also from Lemma 7,which implies (ii) in Theorem 1 holds if . In the following, we will prove that in three cases.

Case 1. . In this case, andIf , thenimplying in , which contradicts the results in Lemma 4.

Case 2. . In this case, , and from Lemma 2, we note that . Thus, from (78), in order to prove , it is sufficient for us to proveFirst of all, when , in this case , and there are four subcases:For subcase (i), , and is a simple root of and ; then, by repeating the process to prove with , we easily obtainFor subcase (iii), , and is a simple root of , . By similar process, we deduceThe proofs of subcases (ii) and (iv) are similar.
Secondly, we consider . Similarly, we only consider and . For , we can get the corresponding conclusions by similar steps, and we omit the process here.
When , there are also four subcases.For subcases (i) and (ii), . Since is a simple root of , similar to the proof of subcase (i), when , we getFor subcase (iii), , and is a double root of , and thusat . We can take such thatwhen . On the other hand, ; then, must be a simple root of . Thus, we can take such thatFor subcase (iv), and is a double root of , while it is a simple root of . Since , we can take such thatwhen . When , we can take such thatWhen , there are three subcases:Repeating the above proofs, one can easily obtain in each subcase.

Case 3. . In this case, the proof is similar.
The following corollary is obvious by Theorem 1.

Corollary 1. If is described as Theorem 1, then .

Now, we investigate the exponential decay rates of at the plus infinity. For convenience, let , , and ; substituting into (11), we havesatisfyingwhere

Let . Since by Lemma 4, Theorem 2 is based on and Proposition 1.

Finally, from Lemma 6, the convergence of is determined both by and , while for , we cannot obtain the convergence of and only by . Thus, in order to investigate the asymptotic behavior of the nondecreasing solution , we enforce the assumptions and adopt the methods in [5, 18, 22]. Now, we suppose , , and , where is the Dirac delta function. With these assumptions, system (11) is transformed into the following system:which is equivalent to (by taking )

Linearizing the above system at the equilibrium point giveswhereand , , . Letand then the characteristic equation of matrix isnamely,or

Since (5) holds, by directly computing, (103) has only a negative root

Although it is hard to get exact expression of the roots of (104), we can judge the real and positive or negative properties of them in the following lemma, which is sufficient for us to discuss the asymptotic behavior.

Lemma 8. For , (104) has four different real roots, and among the four real roots, two are positive roots while the others are negative roots.

Proof. Letwhere . AndFirstly, when , by simple calculation, the functionhas the following four zero points:When and , then has four different roots, two positive and two negative. Moreover, we notice that the image of is strictly decreasing as is increasing, and , , . Thus, we deduce that has two different positive roots and two different negative roots.
For and , has one positive root (the algebraic multiplicity is 2) and two different negative roots. Repeating the above process, we can prove that has two different positive roots and two different negative roots. When and or and , the same result still holds.

From Lemma 8, (104) has two different negative roots, denoted by and separately. Without loss of generality, we assume .

Now, we give the proof of Theorem 3 to finish this section.

Proof of Theorem 3. We firstly supposewhich is equivalent to , , and which are three different negative real roots of . Then, there are three corresponding eigenfunctions:whereFrom the proofs of Lemma 8, it is easy to inferand thus , . Therefore, when , (99) has the following solutions:where , , are arbitrary constants. Thus, by the stable manifold theorem, we haveSimilar to [18], we finally determine the positive or negative of , . Since , then .
When , we deduce that . Thus,which implies . Moreover, by noting , , (115) can be simplified intoWhen , the denominators of are positive, , and , . Since the signs of molecules of may not be positive, , and . If , we suppose ; then, for large . We suppose ; similarly, for large . Thus, , which is a contradiction. Hence, . Thus, (115) can be simplified intoWhen , the proof is similar.
In the following, we assume that (110) does not hold, namely, or , and we want to find two independent vectors of at first. By directly computing, is a vector of ; when , , with , and when , , with .
Then, we suppose another linearly independent vector is , which satisfies and , since . We can solve to find . Thus, in fact, satisfieswhich is equivalent toTake into the last formula above, and we getFrom (103), for any , the coefficient of equals zero. If , then , for some . Thus, we can deduce thatTaking , givesFrom the first equation, we getSubstituting it into the last one givesThus,From the first equation, we haveWhen , let be the negative root ofSince , then . Moreover, by recalling (113), , which impliesThus,Then, we will prove , which is equivalent toSince , , , thenThus,which is equivalent to (131).
When , from (38),Since , , thenIf , then . If , then from , , we havewhich impliesSuppose ; by recalling , from the above proof, we deduce thatwhich is a contradiction, and thus .
Therefore, when , , (99) has the following solutions:where , , are arbitrary constants and .By the stable manifold theorem, we havefor some , .
Moreover, when , , (99) has the following solutions:where , , are arbitrary constants and .By the stable manifold theorem, we havefor some , , .

4. Other Properties

In this section, we adopt the sliding method to prove the strict monotonicity and the uniqueness. Similar to [13], we first give the strong comparison principle.

Lemma 9. Assume that (P1), (P2), and (5) hold. For , let and be two nondecreasing solutions of (11) and (12). Then, either , , and in or , , and in .

Finally, we finish the proof of Theorem 4.

Proof of Theorem 4. Let be a nondecreasing solution of (11) and (12); then, , . By Theorem 6 and Corollary 1, for a large enough , we have , , . We need to prove for , . Let be the left-most point such that ; then, is the minimum of , implying . Differentiating the first equation of (11) givesThus, we conclude , , which contradicts the definition of . Therefore, . With similar process, in .
In the end, the proof of Theorem 5 is completely similar to [14, 15], so we omit it here.

Data Availability

No data were used in this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

Meiping Yao was partially supported by the NSFC (11501339). Yang Wang was partially supported by the NSFC (11901366), the Natural Science Foundation of Shanxi Province (201801D221008), and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2019L0021).

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Copyright © 2020 Meiping Yao et al. 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.

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