MULTIDIMENSIONAL STABILITY OF PLANAR TRAVELING WAVES FOR THE DELAYED NONLOCAL DISPERSAL COMPETITIVE LOTKA-VOLTERRA SYSTEM

. In this paper, we consider the multidimensional stability of planar traveling waves for the nonlocal dispersal competitive Lotka-Volterra system with time delay in n –dimensional space. More precisely, we prove that all planar traveling waves with speed c > c ∗ are exponentially stable in L ∞ ( R n ) in the form of t − n 2 α e − ε τ σt for some constants σ > 0 and ε τ ∈ (0 , 1), where ε τ = ε ( τ ) is a decreasing function refer to the time delay τ > 0. It is also realized that, the eﬀect of time delay essentially causes the decay rate of the solution slowly down. While, for the planar traveling waves with speed c = c ∗ , we show that they are algebraically stable in the form of t − n 2 α . The adopted approach of proofs here is Fourier transform and the weighted energy method with a suitably selected weighted function.


(Communicated by Igor Kukavica)
Abstract. In this paper, we consider the multidimensional stability of planar traveling waves for the nonlocal dispersal competitive Lotka-Volterra system with time delay in n-dimensional space. More precisely, we prove that all planar traveling waves with speed c > c * are exponentially stable in L ∞ (R n ) in the form of t − n 2α e −ετ σt for some constants σ > 0 and ετ ∈ (0, 1), where ετ = ε(τ ) is a decreasing function refer to the time delay τ > 0. It is also realized that, the effect of time delay essentially causes the decay rate of the solution slowly down. While, for the planar traveling waves with speed c = c * , we show that they are algebraically stable in the form of t − n 2α . The adopted approach of proofs here is Fourier transform and the weighted energy method with a suitably selected weighted function.
1. Introduction. The theory of traveling wave solutions of reaction-diffusion equations has been attached much attention since the seminal works of Fisher [7] and Kolmogorove [16], due to its significant nature in biology, chemistry, epidemiology and physics (see, [3,7,8,29,32,34,38,39,40,41,43] ). Among the basic problems in the theory of traveling wave solutions, the stability of traveling wave solutions, which is one of the central questions in the study of traveling waves, is a very challenge question. Recently, a great interest has been drawn to the study of the multidimensional stability of traveling wave solutions. Xin [36] first considered the following bistable reaction-diffusion equation, ∂u(t, x) ∂t = ∆u(x, t) + f (u(x, t)), x ∈ R n , t > 0, (1.1) where f (u) = u(1 − u)(u − θ) for some θ ∈ (0, 1/2). In fact, he obtained the multidimensional stability of planar traveling waves of (1.1) via an application of linear semigroup theory. He showed that if the perturbation of a planar traveling wave is small enough in H m (R n ) ∩ L 1 (R n )(m ≥ n + 1, n ≥ 4), then the solution of the initial value problem converges to the planar traveling wave in H m (R n ) as t goes to infinity with rate O(t − n−1 4 ). Levermore and Xin [17] further investigated the same problem by using the maximum principle and spectral theory. They proved that the planar traveling waves of (1.1) are stable in L 2 loc (R n ) for n ≥ 2. Matano et al. [24] obtained that planar traveling waves of (1.1) are asymptotically stable under almost periodic perturbation or under any possibly large initial perturbations which decay at space infinity. Furthermore, they also found a special solution that oscillates permanently between two planar traveling waves, which implies that planar traveling waves are not asymptotically stable under more general perturbations. Matano and Nara [23] extended the results in [24] and obtained that the planar traveling waves are asymptotically stable in L ∞ (R n ) under spatially ergodic perturbations, which include quasi-periodic and almost periodic ones as special cases. We can study more works of the multidimensional stability of traveling waves by referring to [1,2,14,23,24,30,31,35,42] and references therein for more details.
Mei and Wang [26] considered the following Fisher-KPP type reaction-diffusion equation x−y))dy, x ∈ R n , t > 0, (1.2) where D > 0 denotes the diffusion rate and d(u), b(u) are nonnegative nonlinear functions. They obtained that all noncritical planar traveling waves are exponentially stable and critical planar traveling waves are algebraically stable in the form t − n 2 by using weighted energy method and Fourier transform. Huang et al. [10] extended the results in [26] to the nonlocal diffusion equations. Chern et al. [4] studied the stability of non-monotone critical traveling waves for reaction-diffusion equations with time-delay. The adopted method is the technical weighted-energy method with some new flavors to handle the critical oscillatory waves. We can refer to [11,19,27] and the references therein for more results on the study of the stability of oscillatory traveling waves.
Very recently, Faye [6] extended the local diffusion equation (1.1) to the following nonlocal diffusion equation and investigated the multidimensional stability of planar traveling waves by using semigroup estimates, where J(x) is the kernel function and f is a smooth function with bistable type. They showed that if the traveling wave is spectrally stable in one-dimensional space, then it is stable in n-dimensional space under some special perturbations of planar traveling waves. Although the multidimensional stability of planar traveling waves for scalar reaction-diffusion equation has been studied, little attention has been paid to systems especially with time delay in higher dimensional space. In this paper, we study the multidimensional stability of planar traveling waves of the following competitive Lotka-Volterra system with nonlocal diffusion, x − y)dy, i = 1, 2.
In system (1.4), d i > 0 denotes the nonlocal diffusion rate of the species i, r i > 0 denotes the intrinsic rate of natural increase of species i and b i denotes the competitive rate of the inter-species (i = 1, 2). The nonlocal kernel function J satisfies The assumption (H1) is natural from a modeling point of view and (H2) is required to ensure the existence of the planar traveling waves to the system (1.4). By assumption (H3), as F[J](η) − 1 ∼ −K |η| 2 with α = 1 for η → 0, the operator u → J * u − u approaches the local operator K∆ R n . Thus, we recover the classical local dispersal competitive system. Remark that we have F[J](η) − 1 ∼ −1 for |η| → +∞ such that u → J * u − u is a bounded operator, which is a very different feature from the local operator (Laplacian operator). If we take Thus, the assumptions (H1)-(H3) can be ensured. Now, we first examine the competitive Lotka-Volterra system (1.4) without the nonlocal diffusion and time delay response term, which is reduced to (1.5) The system (1.5) has four constant equilibria: (0, 0), (0, 1), (1, 0) and coexistence equilibrium ( 1−b1 1−b1b2 , 1−b2 1−b1b2 ) with the condition b 1 b 2 = 1. By a phase diagram (see [28] for more details), we list the following asymptotic behavior of the solution as t → +∞, In this paper, we only treat cases (i) and (ii). Since (i) and (ii) are similar by exchanging the role of u and v, in the sequel we always assume that 0 A planar traveling wave of system (1.4) is a special solution in the form of (u 1 , u 2 )(t, x) = (φ 1 , φ 2 )(x · ν + ct) (where ν ∈ R n is a fixed unit vector) connecting the two equilibria of (1.4). In one-dimensional space, existence and stability of traveling waves of system (1.4) have been discussed in [9,12,18,20]. However, to the best of our knowledge, there is no any results for multidimensional stability of planar traveling waves of competitive system (1.4) in high dimensional space. Here, the main purpose of the present paper is to investigative the multidimensional stability of planar traveling waves of system (1.4). We prove that all planar traveling waves with speed c > c * (here, c * is not the minimal speed and the definition of c * can be found in Remark 2.1) are exponentially stable in L ∞ (R n ) in the form of t − n 2α e −ετ σt by using the weighted energy method and Fourier transform, where σ > 0 and ε(τ ) ∈ (0, 1) is a decreasing function for τ > 0. Furthermore, the effect of time delay can essentially make the decay rate of the solution slow down. While, for c = c * , the planar traveling waves are algebraically stable like t − n 2α in L ∞ (R n ). The rest of this paper is organized as follows. In Section 2, we introduce some necessary notations and present the main results of the existence and stability of planar traveling waves. In Section 3, we introduce the time delay ODE system and give the sufficient condition for stability of the trivial solution. In Section 4, we mainly prove the multidimensional stability of planar traveling waves, including the case of c = c * . In Section 5, we obtain the exact planar traveling waves in some special cases and further give some numerical simulations to illustrate the main results.
2. Preliminaries and main results. First, we introduce some necessary notations throughout this paper. C > 0 denotes a generic constant and C i (i = 0, 1, 2, · · · ) represents a specific constant. Let · and · ∞ denote 1−norm and ∞−norm of the matrix (or vector), respectively. Let Ω be a domain, typically Ω = R n .Let α = (α 1 , α 2 , · · · , α n ) be a multi-index with nonnegative integers α i ≥ 0(i = 1, 2, · · · , n). The derivatives for function f (x) are denoted as is the Sobolev space in which the function f (x) is defined on Ω and its weak derivatives ∂ α f (x)(|α| ≤ k) also belong to L p (Ω), and in particular, we denote W k,2 (Ω) as H k (Ω). Further, L p w (Ω) denotes the weighted L p space with a weighted function w(x) > 0. Its norm is defined by Fourier transform is defined as and the inverse Fourier transform is given by In order to apply the comparison principle, we transform the competitive system (1.4) into a cooperative system by changing the variablesũ 1 = u 1 ,ũ 2 = 1−u 2 . Thus, we obtain the following system by dropping the tildes for the sake of convenience, (2.1) for x ∈ R n , t > 0, with the initial condition By the properties of the monotone semiflows [5], we have the following comparison principal.
If we look for a planar traveling wave (u 1 (t, x), u 2 (t, x)) = (φ 1 (ζ), φ 2 (ζ)) (ζ = ν · x + ct, where ν ∈ R n is a fixed unit vector, here we set ν = e 1 = (1, 0, · · · , 0) for simplicity) of the system (2.1) connecting E − and E + , then φ i has to satisfy the following system on the line To obtain the existence of planar traveling waves, we consider the following function By the properties of function ∆ 1 (λ, c) (see [21, Lemma 2.2] for more details), we have the following lemma.
To obtain the multidimensional stability of the planar traveling waves, we consider the following function where q = e −cλτ max 1, r2b2 r1b1 .
Li et al [18] obtained the existence, uniqueness and asymptotic behaviour of the solution of the system (2.3) by using upper-lower solutions, Schauder's fixed point theorem and the sliding method.
Let c ≥ c * and (φ 1 (x · e 1 + ct), φ 2 (x · e 1 + ct)) be the planar traveling wave of (2.1) with the speed c connecting E − and E + . Now, we define a weighted function as where ζ 0 is a very large constant and λ * is defined in Remark 2.1.
Here, we present the main results of this paper.
Theorem 2.2 (Stability). Assume that conditions in Theorem 2.1 hold. For any given planar traveling wave (φ 1 (x · e 1 + ct), φ 2 (x · e 1 + ct)) of the system (2.1) with the speed c ≥ c * connecting E − and E + , if the initial data satisfies and the initial perturbation then nonnegative solution of the Cauchy problem (2.1) and (2.2) uniquely exists and satisfies w (R n ))(i = 1, 2) and the assumption (H3) holds, we have for some constant σ > 0, where ε τ = ε(τ ) ∈ (0, 1) is a decreasing function for τ > 0, and Remark 2.2. The proof of Theorem 2.2 is mainly motivated by [26,37], but there is some difference. In one-dimensional space, Yu et al. [37] considered the system 1.4 without time delay and proved the stability of traveling waves with speed c > max{c * , 1 η0 max{c 1 , c 2 }} (see [37, Theorem 2.1] for more details) with decay rate e −µt . Here, we not only give the much more accurate decay rate (t − n 2 e −ετ σt ) of the stability of planar traveling waves with speed c > c * in high dimensional space, but also obtain the algebraic stability of planar traveling waves with speed c = c * .

Remark 2.3.
To overcome the effect of the nonlocal diffusion, we make the con- or is a function with compact supported set.
3. Linearized delay differential system. In this section, we will derive the solution formulas for the linearized delay differential system and their decay rates, which will play a key role in the stability proof in Section 4. Now let us consider the following delay differential system, d dt where A, B ∈ C N ×N and τ > 0 denotes a time delay. In [15], Khusainov and Ivanov presented the solution formula of (3.1) in the case of space dimension N = 1 and A, B ∈ R. Similarly, we can obtain the solution formula of system (3.1) in general case of space dimension N ≥ 2 and A, B ∈ C N ×N .
, then the solution of system (3.1) can be shown as where B 1 = Be −Aτ and e B1t τ is the so-called delayed exponential function in the form Now, we are going to give a sufficient condition of the global stability for the trivial solution of the linear delay system (3.1).
From Lemmas 4.1 and 4.2, we can obtain the decay rates for U i (t, ξ)(i = 1, 2) in L ∞ (R n ).
Next, we give the proof of our main result Theorem 2.2. Proof of Theorem 2.2. For c ≥ c * , let ξ = x + ct · e 1 and x) ≤ 1, for x ∈ R n , t > 0, i = 1, 2, by the squeeze argument, we have for t > 0, i = 1, 2.
5. Exact planar traveling waves and numerical simulations. In this section, we first show that exact planar traveling waves of the local dispersal competitive system corresponding to (1.4) can be given explicitly under some certain restrictions. We apply the approach developed in [13] to obtain the exact planar traveling waves. Furthermore, we give some numerical simulations to illustrate our main result. If we take the nonlocal kernel function J(x) = n i=1 δ 1 (x i ) + δ n (x) (δ 1 (·) and δ n (·) are the one-dimensional and n-dimensional Dirac functions) and the time delay τ = 0, then (1.4) reduces to the following Laplacian diffusion system (A2):
Now we present the exact planar traveling wave given in Theorem 5.1.
In system (1.4), we take r 1 = r 2 = 1, d 1 = d 2 = 1, b 1 = 1 2 , b 2 = 3 2 , τ = 0.5 and J(x) = 1 (4π) n/2 e −| x 2 | 2 , x ∈ R n (here, we assume n = 1 for simplicity). Then, the system (1.4) with above coefficients has two steady states (0, 1) and (1, 0). From Theorem 2.1, system (1.4) admits a planar traveling wave with speed c > c * = 1.5505. Furthermore, if the initial data satisfies (2.6) and the initial perturbation satisfies (2.7), the planar traveling wave with speed c > c * = 2.9955 is exponential stable in L ∞ (R n ) and the planar traveling wave solution with speed c = c * is algebraic stable in L ∞ (R n ). It is not hard to verify that the initial conditions in Theorem 2.2 can be ensure for such initial data (5.11). With the help of the software MATLAB, we can obtain the numerical solution of (2.1) (see the Figures 2-3). It follows from the Figures 2-3 that the solution of system (1.4) will eventually converge to the equilibrium (1, 0), which implies that the species u 1 will survive and species u 2 will die out. After a large time ( here, the time t ≥ 4 is enough ), the solution of system (1.4) behaves exactly as a stable planar traveling wave in the sense of stability of no change of wave's shape. From the Figure 2-3, we can obtain that the solution of system (1.4) travels from the positive direction of the axis x to the negative direction. An ecological phenomenon related to these results can be explained in the following manner. In the beginning, system (1.4) with species u 1 being absent only have one native species u 2 . Then the exotic species u 1 competing with u 2 invades this system in the way described by system (1.4). Now a natural question arises, namely, can the two species coexist? According to our result, the native species u 2 will eventually become extinct while the invading species u 1 will survive. It also can be seen from Theorems 2.2 that this process is stable if the initial perturbation is proper. Neumann boundary conditions (5.9) and initial data (5.11). From (a) to (f), the solution u1(t, x) plots at times t = 0, 1, 2, 10, 30, 50 and behaves as a stable monotone increasing traveling wave ( no change of the waves's shape after a large time in the sense of stability) and travels from right to left. Neumann boundary conditions (5.9) and initial data (5.11). From (a) to (f), the solution u2(t, x) plots at times t = 0, 1, 2, 10, 30, 50 and behaves as a stable monotone increasing traveling wave ( no change of the waves's shape after a large time in the sense of stability) and travels from right to left.