Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal

This paper is concerned with the traveling waves for a three-species competitive system with nonlocal dispersal. It has been shown by Dong, Li and Wang (DCDS 37 (2017) 6291-6318) that there exists a minimal wave speed such that a traveling wave exists if and only if the wave speed is above this minimal wave speed. In this paper, we first investigate the asymptotic behavior of traveling waves at negative infinity by a modified version of Ikehara's Theorem. Then we prove the uniqueness of traveling waves by applying the stronger comparison principle and the sliding method. Finally, we establish the exponential stability of traveling waves with large speeds by the weighted-energy method and the comparison principle, when the initial perturbation around the traveling wavefront decays exponentially as x → -∞, but can be arbitrarily large in other locations.

1. Introduction. In this paper, we consider the following three species competition system with nonlocal dispersal where d i > 0, r i > 0, b ij > 0. Here u, v, w are the population densities of species 1, 2, 3, respectively, b ij is the competition coefficient of species j to species i, r i is the growth rate of species i and d i is the diffusion coefficient of species i. Also, we have taken the scales of species so that the carrying capacity of each species is normalized to be 1. J i * χ − χ models nonlocal dispersal processes [4,10,16], and 1512 GUO-BAO ZHANG, FANG-DI DONG AND WAN-TONG LI (J1): J i ∈ C(R), J i (x) = J i (−x) ≥ 0, x ∈ R, and R J i (x)dx = 1, i = 1, 2, 3; (J2): For every λ > 0, R J i (x)e −λx dx < ∞, i = 1, 2, 3. The system (1) describes the relation that species u competes with both species v and w, while species v and species w have no competition with each other. In this paper, we give a hypothesis on the competition coefficients b 12 , b 13 , b 21 and b 31 : (H1): b 21 , b 31 > 1, b 12 + b 13 < 1, which means that the competition ability of the species u is stronger than the species v and w.
In biology, it is very meaningful to study the population invasion between the residents v, w and the invader u. Since the species v, w are weak competitors to the species u, it is expected that the species u shall win the competition eventually. This dynamics of the invasion process can be mathematically characterized by a traveling wave solution connecting the equilibria (0, 1, 1) and (1, 0, 0).
As we all know, the traveling waves for nonlocal dispersal equations and systems has been extensively studied, see [3,9,[13][14][15][20][21][22]30,31,36,39,41,43]. In a recent paper, Dong, Li and Wang [8] have established the existence and asymptotic behavior of traveling waves of (1), i.e., solutions of (2) with (3). Based on the asymptotic behavior of traveling waves, they further studied a new type of entire solutions which behave as two traveling waves coming from both sides of x-axis. It is very natural and interesting to investigate the uniqueness of the traveling waves (up to translation) and their asymptotic stabilities. This will be the main purpose of the present paper. The uniqueness of traveling waves for various evolution equations has been established, for example, see [1-3, 5-7, 11, 19] and the references therein. In the past few years, many researchers employed the strong comparison principle and the sliding method to investigate the uniqueness of traveling waves for two-component systems [11,13,[17][18][19]. In particular, Guo and Wu [11] considered a two-component non-delayed lattice dynamical system arising in competition models, and proved the uniqueness of traveling wave solutions connecting two half-positive equilibria by the sliding method. Recently, Li et al. [19] further used the sliding method to prove the uniqueness of traveling wave fronts for a two-component nonlocal dispersal competitive system with time delay. Motivated by [11,13,[17][18][19], in this paper, we apply the sliding method to prove the uniqueness of traveling waves for the three-component nonlocal dispersal competitive system (1). We should point out that a key step in proving uniqueness of traveling waves by the sliding method is to establish rather precise exponential decay rate of wave profiles when x → −∞. Although Dong, Li and Wang [8] have obtained some results on the asymptotic behavior of traveling waves of (1) at ±∞, it is not precise enough for the uniqueness of wave profiles. Inspired by Carr and Chmaj [3] for integro-differential equation without delay, we use a modified version of Ikehara's Theorem to derive more precise asymptotically exponential tails of wave profiles, see also [11,19].
In addition to the uniqueness of traveling waves, the stability of traveling waves is an extremely important subject. In the past decades, there have been extensively investigations on the stability of traveling waves, see [5,14,15,20,[23][24][25][26][27][28][29][32][33][34][35]40,42]. The main methods are the (technical) weighted energy method [15,20,23,27,28,40], the sub-and supersolutions method and squeezing technique [5,32], and the combination of the comparison principle and the weighted energy method [25,26,29,42]. In 2014, Lv and Wang [24] used the third method to prove the stability of traveling waves for a delayed two-component Lotka-Volterra cooperative system with nonlocal dispersal. More recently, Yu et al. [35] took the same method to study the stability of traveling waves for a two-component Lotka-Volterra competitive system with nonlocal dispersal. Encouraged by the work of [24,33,35], it is very natural to expect that we can still apply the third method to prove the stability of traveling wave for our three-component Lotka-Volterra competition system (1). It is well known that when the component of the Lotka-Volterra competitive system is greater than or equal to three, it is hard to transform the competitive system into a cooperative system any more. Fortunately, since the nonlinearity of our system (1) has a special structure, and we consider the traveling waves connecting the equilibria (0, 1, 1) and (1, 0, 0), by a simple transformation u 1 = u, u 2 = 1 − v and u 3 = 1 − w, the system (1) can be reduced into a cooperative system, see (5). Thus, the comparison principle works for the transformed system of (1). Therefore, in this paper, we take the weighted energy method together with the comparison principle to establish the stability of traveling waves of (1). We should point out that by this method, our stability result only holds for waves with lager speed due to the presence of the nonlocal dispersal term.
The rest of this paper is organized as follows. In Section 2, we give the main results of this paper. In Section 3, we are devoted to proving the monotonicity and uniqueness of traveling waves. In Section 4, the stability of traveling waves with large speeds is obtained.
In [8], Dong et al. have established the existence of traveling waves of (5).
Remark 1. From Proposition 1, we can see that c min is the minimal wave speed. By Remark 2 in [8], c min ≥ c * . Note that c min = c * means that the minimal speed is linearly determined. From the random diffusion system and lattice dynamical system [12], we conjecture c min = c * holds under some conditions on the coefficients and Kernel functions J i , i = 1, 2, 3, of (1).
In order to study the uniqueness, we shall need more precise information on the wave tails. Inspired by the approach developed by Carr and Chmaj [3] for integrodifferential equation, we can establish the exact asymptotic behavior of traveling waves at negative infinity. Then the uniqueness of traveling waves of (5) with speed c ≥ c min can be obtained by applying the sliding method. Theorem 2.4 (Uniqueness of traveling waves). Assume that (H1) holds, and J i satisfies (J1) and is compactly supported, i = 1, 2, 3, and The wave profile of (5) is unique up to translation for a given wave speed c ≥ c min .

Remark 2.
The assumption in Theorems 2.3 and 2.4 that J i is compactly supported, i = 1, 2, 3, is given, due to the application of the asymptotic behaviors in Lemmas 2.1 and 2.2. We should point out that the conclusions in Theorems 2.3 and 2.4 also hold, if the assumption that J i is compactly supported, i = 1, 2, 3, is replaced by (J2). This is because that we can directly use the Ikehara's Theorem to show the asymptotic behavior of traveling waves, and then prove the monotonicity and uniqueness of traveling waves, see [19,37,38].
To obtain the stability of traveling wavefronts of (5), we need the following technical assumption: (H2): Define three functions on η as follows: By the assumption (H2), we can see that Then by the continuity of M i (η) with respect to η, there exists η 0 > 0 such that M i (η 0 ) > 0 , i = 1, 2, 3. Furthermore, define where (ϕ 1 (ξ), ϕ 2 (ξ), ϕ 3 (ξ)) is a traveling wavefront of (5). It is easy to see that which imply that there exists a number ξ 0 > 0 large enough such that Notation. Throughout the paper, C > 0 denotes a generic constant, while C i > 0(i = 0, 1, 2 · ··) represents a special constant. Letting I be an interval, especially I = R, L 2 (I) is the space of the square integrable function on I, and H k (I)(k ≥ 0) is the Sobolev space of the L 2 -function f (x) defined on I whose derivatives d i dx i f, i = 1, · · ·, k, also belong to L 2 (I). L 2 w (I) represents the weighted L 2 -space with the weight w(x) > 0 and its norm is defined by Letting T > 0 and B space, we denote by For above η 0 and ξ 0 , we define a weight function w(ξ) by Let Now, we state our stability theorem. hold. For any given traveling wave (ϕ 1 (x + ct), ϕ 2 (x + ct), ϕ 3 (x + ct)) of (5) with the wave speed c > max{c min ,c}, wherẽ if the initial data satisfy and the initial perturbations satisfy , then the nonnegative solution of the Cauchy problem (5) and (6) uniquely exists and satisfies where C and µ are some positive constants.
3. Monotonicity and uniqueness of traveling waves. In this section, we adopt the strong comparison principle and the sliding method to prove the monotonicity and uniqueness (up to a translation) of traveling waves of (1). We first give the strong comparison principle.
In order to study the uniqueness, we further investigate the asymptotic behavior of traveling waves of (5) at negative infinity by using a modified version of Ikehara's Theorem.

GUO-BAO ZHANG, FANG-DI DONG AND WAN-TONG LI
In the sequel, we let such that the following statements hold.
Remark 3. We should point out that when c min = c * , the statements (i), (ii) and (iii) hold. When c min > c * , the statements (i) with c > c * and (ii) only hold.
By taking the limit ε → 0, we can obtain the corresponding energy estimate for the original solution U i (ξ, t). For the sake of simplicity, below we formally use U i (ξ, t) to establish the desired energy estimates.
for some positive constant C.
By Lemma 4.9, we obtain the following results. for some positive constant C and t > 0.