Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application

In this paper, we provide a general approach to study the asymptotic behavior of traveling wave solutions for a three-component system with nonlocal dispersal. Then as an important application, we establish a new type of entire solutions which behave as two traveling wave solutions coming from both sides of \begin{document} $x$ \end{document} -axis for a three-species Lotka-Volterra competition system.

1. Introduction. In this paper, we are interested in the asymptotic behavior of traveling wave solutions for the following three-component nonlocal dispersal system: ∂v(x,t) ∂t ∂w(x,t) ∂t where u(x, t), v(x, t) and w(x, t) measure the density of population at location x ∈ R and time t ∈ R, respectively. The reaction field (f 1 , f 2 , f 3 ) describes the population birth, death as well as the competition between individuals and so on. J i * ϑ is the standard spatial convolution on R, that is, (J i * ϑ)(x, t) = R J i (x − z)ϑ(z, t)dz, where i = 1, 2, 3. Then J i * ϑ − ϑ (i = 1, 2, 3) is called the nonlocal dispersal and represents transportation due to long range dispersion mechanisms. The nonlocal dispersal problem has been proposed in many practical fields, such as population biology [11], phase transition [1,7,8] and network model [10]. Here, the convolution kernels J i (i = 1, 2, 3) can be understood as the probability density and satisfy (J): J i ∈ C 1 (R), J i (−x) = J i (x) ≥ 0, R J i (x)dx = 1 and has compact support, i = 1, 2, 3. For convenience, we further assume J i (x) = 0 if |x| > 1, i = 1, 2, 3.
Here, we only deal with the asymptotic behavior of traveling wave solutions when ξ goes to −∞, then it is not necessary to impose the boundary condition at +∞. As we all know, the investigation of asymptotic behavior is essential and important in several aspects of traveling wave solutions, such as the stability of traveling wave solutions, monotonicity and uniqueness of wave profiles as well as the estimation of wave speed and existence of front-like entire solutions (see, e.g., [19-21, 24, 35, 36]). Although there are a lot of work devoted to asymptotic behavior, most of them are focused on the scalar equations (see, e.g., [2,3,6,9,30,36]) or two-component systems (see, e.g., [14,24,27]). Notice that the research of asymptotic behavior for some two-component systems largely depends on the particular form of nonlinear term. To the best of our knowledge, there are only a few papers concerning the asymptotic behavior of traveling wave solutions for some Ncomponent systems [15,33] (N ≥ 3). This is the motivation of our current paper. In particular, we will provide a general approach to tackle the asymptotic behavior of traveling wave solutions for a three-component nonlocal dispersal system (1). Some ideas come from Wu [33] for a class of three-component lattice dynamical system.
The way we treat other cases is a whole lot like the above three cases studied.
Theorem 1.2. Assume that (J), (H1) and (H2-2) hold. Then (4) holds. Also, let V 2 = ψ * , V 3 = ϕ * . Then for j = 2, 3, V j satisfy the following dichotomy: Theorem 1.3. Assume that (J), (H1) and (H2-3) hold. Then (4) holds. Moreover, the following statements are valid: In the following, we demonstrate an application of the above results to construct front-like entire solutions for a three-species Lotka-Volterra competition system with nonlocal dispersal: where u = u(x, t), v = v(x, t) and w = w(x, t) represent the population density of three competing species at location x ∈ R and time t ∈ R, respectively. b ij > 0 stands for the competition coefficient of species j to species i, r i > 0 is the growth rate of species i and J i is the migration rate of species i. Besides, the carrying capacity of each species is assumed to be 1 by some suitable time and space scaling. Further, we assume species v and w have different preference for food resource, that is, there is no competition between species v and w. However, species u can compete with both species v and w. In (10), we give an additional hypothesis on the competition coefficients b 12 , b 13 , b 21 and b 31 : , which indicates that the competition ability of species u is stronger than the species v and w.
Under (A), it is easy to verify that the corresponding kinetic system (10) has an unstable equilibrium (0, 1, 1) and a stable equilibrium (1, 0, 0). This indicates that we can model the population invasion process between the invader u and the residents v and w. Mathematically, this dynamics could be characterized by a traveling wave solution or an entire solution (see definition below). According to the previous definition, a solution (u(x, t), v(x, t), w(x, t)) of (10) is called a traveling wave solution connecting (0, 1, 1) and (1, 0, 0) with speed c, if (u(x, t), v(x, t), w(x, t)) = φ(ξ), ψ(ξ), ϕ(ξ) , ξ = x + ct, for some function φ, ψ, ϕ satisfying The following theorem gives the existence of solutions of (11). Although the traveling wave solution is a key object characterizing the dynamics of evolution equations, it is not enough to understand the whole dynamics. In fact, traveling wave solutions are only special examples of the so-called entire solutions which are defined for whole space and all time. The study of entire solutions is crucial and significant. In a nutshell, entire solutions provide some new spread and invasion ways of the epidemic and species [17,24,27] and can help us for the mathematical understanding of transient dynamics and the structures of global attractor [26]. Accurately speaking, the research of two-front entire solutions can be tracked back to the works of Hamel-Nadirashvili [19,20] and Yagisita [35]. Since then, the investigation of entire solutions for scalar equations or two-component systems with or without delay has been widely concerned (see, e.g., [4,5,13,18,20,23,26,27,29,31,34]), and (see, e.g., [16,17,22,24,28,32,37]) for nonlocal dispersal equations or lattice differential equations. In this paper, we further study this type of entire solutions for a three-component system with nonlocal dispersal.
We end this part with the last theorem about the entire solutions of system (10). Unfortunately, we still need the following technical condition on the solution c, φ, ψ, ϕ of system (11): 1− ϕ(ξ) ≥ η for any ξ ≤ 0 and some η > 0, which has appeared in some papers (see, e.g., [17,24,27,33]). Thankfully, we can verify that (M) is valid under some proper conditions (see Remarks 3 and 4).
and lim Remark 1. We point out that the techniques and theories developed for system (1) under the assumption that J i (x) = 0, if |x| > 1, i = 1, 2, 3 can be parallel extended to the following situation: 1]. Moreover, all the analysis can be repeated if we replace J i by J iLi , respectively. Therefore, in our paper, we take L i = 1 (i = 1, 2, 3) directly just for convenience.
The remaining part of this paper is organised as follows. In Section 2, we establish the asymptotic behavior of traveling wave solutions for system (1) and complete the proof of Theorems 1.1-1.3. In Section 3, we first obtain the existence of traveling wave solutions of system (10), then we apply the results established in Section 2 to construct front-like entire solutions.
2. Asymptotic behavior of traveling wave solutions. In this section, our aim is to study the asymptotic behavior of traveling wave solutions of system (1). For convenience, we denote (φ * , ψ * , ϕ * ) by (φ, ψ, ϕ) just in the current section. Now we give some preparing works.
Consider the characteristic equations as follows: Then we have the following observation.
We now introduce a lemma coming from [36,Proposition 3.7], which plays an important role in proving Theorem 1.1.
where J is a kernel function and satisfies (J). Then Z is uniformly continuous and bounded. Moreover, z ± = Z(±∞) exists and are real roots of the characteristic equation aω = R J(y)e −ωy dy + B(±∞). Based on Lemmas 2.1 and 2.2, we can easily obtain the asymptotic behavior of φ near −∞.

Asymptotic behavior of solutions under (H2-1).
In this subsection, we always assume that (J), (H1) and (H2-1) hold. In this case, similar to Lemma 2.3, the following result holds.
Notice that if condition (22) is removed, (23) may be incorrect. Here, we give another estimate of ϕ ϕ in a bounded interval.
where µ is given in Lemma 2.5.
Proof. By virtue of (19) and the choice of µ, the conclusion is obvious and we omit the detail here.
To continue our work, we need the following result [33, Lemma 2.6].
Based on Lemma 2.7, the following result can be derived by a parallel discussion to that of [33, Lemma 2.7] and the details are omitted here.
where P is given by (18).
exists and is finite, where P is given by (18).
Proof. If P (·) is eventually monotone for ξ < 0, then we have the assertion by Lemma 2.8. Next, we continue our discussion for P (·) is not eventually monotone for ξ < 0, that is, P oscillates as ξ → −∞. that 0 ≤ m ≤ M < ∞. Moreover, we can choose a sequence {x n } (resp., {y n }) of local maximal (resp., minimal) points of P such that x n → −∞ (resp., y n → −∞) and P (x n ) → M (resp., P (y n ) → m) as n → ∞. If M = 0, then lim ξ→−∞ P (ξ) = 0 and the conclusion is valid. So we only consider the case M > 0. We divide our discussion into two cases. To start with, we assume Λ 2 (c) ≥ Λ 1 (c) without loss of generality.
Case I. When inf ξ≤0 P (ξ) > 0, then m > 0. By the choice of {x n } and Lemma 2.7 Let That is, for any fixed z ∈ [−1, 1]. Further, for any given ε > 0 and any fixed z ∈ [−1, 1], according to the choice of {x n } and (20), we can get that In view of (27), (28) and Lemma 2.3, taking n → ∞ in (21), we have By the arbitrariness of ε, we further have On the other hand, replacing {x n } by {y n } and using m > 0, we can obtain that It then follows from (29) and (30) that M = m, that is, the proposition is true.
P (ξ). Motivated by [14] and [33], we can define {z n } ∞ n=0 to be the sequence of local minimal points of P in (−∞, 0) such that z n < z n−1 , P (z n ) < P (z n−1 ) for all n ∈ N, and P (z) ≥ P (z n−1 ) for any minimal point z of P in (z n , z n−1 ) (if it exists). Notice that this sequence is well defined due to the fact that inf ξ≤0 P (ξ) = 0 and P oscillates as n → −∞. It is clear that z n ↓ −∞ and P (z n ) ↓ 0 as n → ∞. We now give a claim: for any α ∈ [0, 2] and n ∈ N, there holds In fact, if not, then there exists α 0 ∈ (0, 2] and n 0 ∈ N such that P (z n0 . Thus the claim is true. According to P (z n ) = 0, (19) and Lemma 2.7 (ii), we have lim n→∞ ϕ (zn) ϕ(zn) = Λ 1 (c). Furthermore, we claim that there exists α 1 ∈ (0, 1] and N ∈ N, such that for any n ≥ N, If not, then P (z n ) ≤ P (z n − z) for any z ∈ (0, 1] and n ∈ N. By (20), (21) and (31), letting n → ∞, by Lemma 2.3 and (28), we have which contradicts to (29) since M > 0, so (32) holds. Moreover, by the definition of z n and (32), if z n+1 + α 1 < z n − α 1 , we have which implies that any local maximal point of P near −∞ must fall in [z n − α 1 , z n + α 1 ] for enough large n ∈ N. For convenience, we assume x n ∈ [z n − α 1 , z n + α 1 ] for enough large n (if necessary we take a sub-sequence), where x n is defined as above. Note that Since x n ∈ [z n − α 1 , z n + α 1 ] for enough large n ∈ N. By Lemma 2.3 and (28), we have for some K 2 > 0 and for all large n ∈ N. Hence If we can prove sup n ϕ(zn) ϕ(xn) < +∞, then m = 0 can't occur when M > 0. Therefore, we next want to show sup n ϕ(zn) ϕ(xn) < +∞. Specifically, we complete the proof in two ways: c > 0 and c < 0.
We are ready to prove Theorem 1.1.
Proof of Theorem 1.1. By Lemmas 2.3 and 2.4, we have (4) and (5). Note that P (−∞) exists and is finite by Proposition 1, where P is given by (18).
3. Application. In this section, we focus on a three-species Lotka-Volterra competition system (10). We first study the existence of traveling wave solutions and then obtain the asymptotic behaviour of them by using the abstract results in Section 2. Finally, we construct a class of front-like entire solutions.
In system (40), denote It is easy to check that f = (f 1 , f 2 , f 3 ) satisfies (H1) and (H2-2), so we have the following corollary by virtue of Theorem 1.2.
According to Corollary 1, we give some remarks about the assumption (M).
So far, we have obtained the asymptotic behavior of the solutions for (40) at −∞. Next, we want to investigate the behavior of wave tails near +∞. To start with, we study the following characteristic equations: By (A) and Lemma 2.1, we can verify that for each c < 0, κ j (c, λ) = 0 has one real root λ j (c) > 0, j = 2, 3 and κ 1 (c, λ) = 0 admits a real root λ 1 (c) > 0.
By the earlier rules, we have for any ξ ∈ R, which together with (44)-(46) lead to the following results.
Corollary 2. Assume that (J) and (A) hold, and let (c, φ, ψ, ϕ) be a solution of (40). Then we have Moreover, there exists K 3 > 0 such that 3.3. Existence of entire solutions of system (10). This subsection is devoted to the existence of front-like entire solutions for (10). In particular, the limiting argument of a sequence of Cauchy problems combining comparison principle and upper/lower-solutions method are used in solving this issue.
We just study the entire solutions of (48), since (48) and (10) are equivalent. Consider the following initial value problem: where x ∈ R, t > 0.

3.3.2.
Upper-lower solutions. As a consequence of Corollaries 1 and 2, we have following lemma which is essential in constructing lower/upper-solutions of (48). and where u = φ, ψ, ϕ.
Proof. The proof is similar to that of Lemma 3.6 and then we omit the details here.