Some Qualitative Properties of Traveling Wave Fronts of Nonlocal Diffusive Competition-Cooperation Systems of Three Species with Delays
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.
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  firstly proved the existence of traveling wave fronts of (1) connecting with and gave the exact solutions. Then, in , 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 . Very recently, in , 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 , 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  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 , 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 , 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.
We cannot use the method offered in  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
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.
Remark 2. When , , and , the existence of nondecreasing solutions of (11) and (12) had been discussed in . 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 .
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
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.
Then, we discuss the convergence of via the following lemma.
Proof. We first search such that , . From Lemma 2, we can conclude that has two roots 0 and for , andfor . Since