SomeQualitative Properties of TravelingWave Fronts ofNonlocal Diffusive Competition-Cooperation Systems of Three Species with Delays

*is 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.


Introduction
In this paper, we are concerned with the following nonlocal diffusive competition-cooperation systems of three species with delays: where (J i * u i )(x, t) � R J i (x − y)u i (y, t)dy, i � 1, 2, 3, J i denote the diffusive kernel functions of three species, respectively, and u i (x, t) represent the density of three species.
On the other hand, for (1) without diffusion and delays, by direct calculation, e 3 is unstable, and e 4 is stable, if 1 − a 31 k 1 − a 32 k 2 < 0.
Furthermore, if (5) and D > 0, then the third component of e 7 is negative, which implies that e 7 is not in the first octant.
e 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 function is called a traveling wave solution connecting e 3 and e 4 , if it satisfies Moreover, if ϕ i (ξ), i � 1, 2, 3, are monotone in ξ ∈ R, then we call it the traveling wave front.
If the diffusive kernel functions J i (x) � δ(x) − δ ′′ (x), i � 1, 2, 3, where δ is the Dirac delta function and τ ij � 0, i, j � 1, 2, 3, under some assumptions on the parameters, Hung in [9] firstly proved the existence of traveling wave fronts of (1) connecting e 3 with e 4 and gave the exact solutions. en, in [16], when the wave speed c 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 D, D 1 , D 2 > 0 holds, in [10], Huang and Weng proved the existence of traveling wave fronts of (1) connecting e 3 with e 7 if (4) exists and connecting e 3 with e 4 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 e 3 and e 4 . Firstly, we make a variable substitution us, (1) is changed into the following cooperative system, by dropping the tildes for simplicity: Correspondingly, system (9) also has eight corresponding equilibria as follows: , 0, a 31 1 − a 13 1 − a 13 a 31 , , a 32 1 − a 23 1 − a 23 a 32 , From the above statements, the traveling wave fronts of (1) connecting e 3 with e 4 are equivalent to the traveling wave fronts of (9) connecting E 3 with E 4 . And, the boundary conditions (7) and (8) are correspondingly changed into Complexity 3 where 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 J i , i � 1, 2, 3, satisfy 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).

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.

Complexity
We give a remark at once to show (38) is reasonable. , while the right side of the equation (38) is (−c/2) + �������� (c/2) 2 + 2, which are not equal for c > 0. Finally, we introduce conclusions on the strict monotonicity and the uniqueness.

Theorem 5.
Under the assumptions of eorem 3, let φ 1 (ξ) and φ 2 (ξ) be two nondecreasing solutions of (11) and (12), with the same speed c ≥ c * and the same decay rate at ξ � ± ∞. en, they are unique (up to a translation).
Define the bilateral Laplace transform where ϕ: R ⟶ R is a continuous function. e following lemma is derived from (12) and eorem 6.
With the aid of the above lemmas, we will finish the proof of eorem 1.
en, from (1) and (2), we deduce that H i (λ) are analytic in the strip 0 < Reλ ≤ Λ i , i � 1, 2. us, from Lemma 7, which implies (i) in eorem 1 holds if H i (Λ i ) ≠ 0. In the following, we will prove that H i (Λ i ) ≠ 0, i � 1, 2, in two cases. For c > c * , i � 1, 2; since ] � 0 and Λ i is a simple root of For c � c * , in this case, there are three subcases, . e first two subcases are equivalent by changing the roles of φ 1 and φ 2 .