Stability of Planar Traveling Waves for a Class of Lotka–Volterra Competition Systems with Time Delay and Nonlocal Reaction Term

In this paper, we consider the multidimensional stability of planar traveling waves for a class of Lotka–Volterra competition systems with time delay and nonlocal reaction term in n–dimensional space. It is proved that, all planar traveling waves with speed c>c∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c>c^{*}$$\end{document} are exponentially stable. We get accurate decay rate t-n2e-ϵτσt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^{-\frac{n}{2}} \textrm{e}^{-\epsilon _{\tau } \sigma t}$$\end{document}, where constant σ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma >0$$\end{document} and ϵτ=ϵ(τ)∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _{\tau }=\epsilon (\tau )\in (0,1)$$\end{document} is a decreasing function for the time delay τ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau >0$$\end{document}. It is indicated that time delay essentially reduces the decay rate. While, for the planar traveling waves with speed c=c∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=c^{*}$$\end{document}, we prove that they are algebraically stable with delay rate t-n2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^{-\frac{n}{2}}$$\end{document}. The proof is carried out by applying the comparison principle, weighted energy and Fourier transform, which plays a crucial role in transforming the competition system to a linear delayed differential system.


Introduction
Recently, nonlocal reaction-diffusion equations have been getting attention for their accuracy in describing numerous biological models or other models. The authors of [6] proposed a new method to derive an essential integral kernel in convolution type from any given reaction-diffusion network. In [11], the authors dealed with traveling wave phenomena of a degenerate reaction-diffusion equation with the nonlocal effect. In addition, there are many related works for the existence and stability of traveling waves for nonlocal reaction-diffusion equations. Such as [19] for a class of twospecies chemotaxis models with Lotka-Volterra competitive kinetics. In [35] for an age-structured nonlocal dispersal model derived from an epidemic model with vertical infection. In [37] for a class of nonlocal dispersal SIR epidemic models with nonlinear incidence.
We consider the following Lotka-Volterra competitive system with time delay and nonlocal reaction term in n-dimensional space for x ∈ R n , t > 0, where u(x, t) and v(x, t) denote the population densities of two competitive species at location x and time t. d 1 and d 2 are the diffusion rates of two competitive species. The parameters r i , b i (i = 1, 2) and τ are all positive constants, where r i denotes the intrinsic rate of natural increase of species i, b i denotes the competitive rate of the inter-species, and τ > 0 is time delay. The kernel functions g i : R n → R, i = 1, 2 are probability functions of the random dispersal of individuals. g 1 * v and g 2 * u are defined as These two nonlocal terms aim to account for the fact that, at previous time t − τ , the individuals move and reach x from all other positions y in space. Throughout this paper, we need the following assumptions about the kernel functions g 1 , g 2 .
Especially, we can derive the following common Lotka-Volterra model (1.2) by letting g 1 (x) = g 2 (x) = δ(x) and τ = 0 in system (1.1), where δ(·) is Dirac's delta function, Here are some consequences about the existence and stabilities of traveling wave solution for system (1.2). Morita et al. [21] proved the existence of an entire solution. Yu and Yuan [42] investigated the existence of traveling wave solution. Zhao and Ruan [45] obtained the existence, uniqueness and asymptotic stability of time periodic traveling waves for system (1.2). Due to the influences of the time delay and spatial diffusion in the real world, Lv and Wang [20] considered the following delay system for x ∈ R n , t > 0. They studied the existence of traveling wave solution connecting (0, 1) to (1, 0) for system (1.3) by using the upper-lower solution technique and the monotone iteration. For other related researches of reaction diffusion equations with delay, we refer to [4,12,13,20,24,29,38].
In the theory of traveling wave solutions, the stability of them is an extremely important and challenge object. For the past few years, a great interest has been drawn to the study of the multidimensional stability of traveling wave solutions. Here we give a brief introduction about it's development: Levermore and Xin [18] first considered the following bistable reaction-diffusion equation, 1 2 ). They proved that the planar traveling wave solutions of (1.4) are stable in L 2 loc (R n ) with a small initial perturbation by using the maximum principle and energy methods. Xin [39] studied the multidimensional stability of planar traveling wave solutions of (1.4) by means of an application of linear semigroup theory. Matano et al. [28] obtained the asymptotically stability of the planar traveling waves of (1.4) under any-possibly large-initial perturbations that decay to zero at space infinity as well as under almost periodic perturbations. Furthermore, they also found a special solution that oscillated permanently between two planar waves, which implied that planar waves were not asymptotically stable under more general perturbations. We can refer to [1,5,9,15,27,28,33,36,39,44] for more details about the multidimensional stability of traveling waves.
Mei and Wang [31] considered the following Fisher-KPP type reaction-diffusion equation, 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 . Huang et al. [14] extended the results in [31] to the nonlocal diffusion equations. Although the multidimensional stability of planar traveling waves for scalar reactiondiffusion equations has been studied, little attention has been paid to the systems especially those with the nonlocal reaction term and time delay in higher dimensional space.
Motivated by the above analysis, in this paper, we are devoted to investigating the multidimensional stability of planar wave (φ(x · e 1 + ct), ϕ(x · e 1 + ct)) of the system (1.1) connecting the equilibria (0, 1) and (1, 0). Firstly, we examine the Lotka-Volterra competition system (1.1) without the diffusion, time delay and nonlocal response term, which is reduced to (1.6) The system (1.6) has four equilibrium points (0, 1), (1, 0), (0, 0) and coexistence equilibrium We list the following asymptotic behavior of the solution for the system (1.6) equipped with strictly positive initial conditions as t → +∞ (see [25] for more details), Since cases (i) and (ii) are similar by exchanging the role of u and v, we only consider the case (i) in this paper for the sake of convenience. Here we suppose the parameters b 1 , b 2 satisfy This paper is organized as follows. In Sect. 2, some preparatory results are given, and we state the results about the existence and stability of the planar traveling waves for system (1.1). In Sect. 3, we prove the multidimensional stability of the planar traveling waves using the comparison principle, weighted energy and Fourier transform. In Sect. 4, we give the numerical simulation of the planar traveling waves for (1.1).

Preliminaries and Main Results
Throughout this paper, we introduce some necessary notations at first. C > 0 denotes a generic constant and C i > 0(i = 1, 2, · · · ) represents a specific constant. Let · and · ∞ denote 1−norm and ∞−norm of the matrix (or vector), respectively. α = (α 1 , α 2 , · · · , α n ) denotes a multi-index with non-negative integers α i ≥ 0(i = 1, 2, · · · , n), and |α| = α 1 + α 2 + · · · + α n . The derivatives for multi-dimensional Let be a domain, typically = R n . L p ( ) is the Lebesgue space of the integrable functions defined on . W k, p ( )(k ≥ 0, p ≥ 1) is the Sobolev space of the L p functions f (x) defined on whose derivatives ∂ α x f (|α| ≤ k) also belong to L p ( ), and particularly we represent W k,2 ( ) as H k ( ). Further, L p w ( ) denotes the weighted L p space with a weight function w(x) > 0, defined as For an n-D function f (x), its Fourier transform is defined as and its inverse Fourier transform is given by For the convenience of proving and using the comparison principle, we do the transform u = u, v = 1−v. Then the competitive system (1.1) becomes the following cooperative system with monotonicity by dropping the overline for convenience, The equilibria (0, 1) and (1, 0) of system (1.1) are corresponding to the equilibria (0, 0) and (1, 1) of system (2.1).
By the properties of the monotone semiflow [8] or using the similar method in [22, Lemma 2.3], we have the following comparison principle.
Here, the type of order we are using is lexicographic law, that is If we look for a planar traveling wave solution (u(x, t), v(x, t)) = (φ(x · e 1 + ct), ϕ(x · e 1 + ct)) of the monotone system (2.1) connecting the two equilibria (0, 0) and (1, 1), then the planar traveling wave solution has to satisfy the following nonlinear system That is, To obtain the existence and stability of planar traveling wave solutions, we need the following assumption and we consider the following function By direct calculations, we can obtain In view of the above properties of the function (λ, c), we have the following lemma.

Lemma 2.2 Under the conditions (A1) and (A2)
, there exist λ * > 0 and c * > 0 such that Furthermore, Yu and Zhao [43] obtained the existence of planar traveling waves of the system (2.1) by using the cross-iteration technique and Schauder's fixed point theorem. Let c ≥ c * and (φ(x · e 1 + ct), ϕ(x · e 1 + ct)) be the planar traveling wave of system (2.1) with the speed c connecting E − and E + . We define a weight function as where ζ 0 is a large enough constant and λ * is defined in Lemma 2.2. Now, we present the main results of this paper.

6)
and the initial perturbation t)) of the Cauchy problem (2.1) uniquely exists and satisfies for some constant σ > 0, where τ = (τ ) is a decreasing function for τ > 0, and

Remark 2.2
The idea of the proof of Theorem 2.2 is from [31,40], but there exists difference. In [40] for the system (1.1) without time delay, the authors proved the stability (see [40, Theorem 2.1] for more details) of traveling waves with speed c > max{c * , 1 η 0 max{c 1 , c 2 }} in one-dimensional space with decay rate e −μt , where μ > 0 is some constant. Here, we not only give still more accurate decay rate (t − n 2 e − τ σ t ) 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 * with delay rate t − n 2 . It is also indicated that time delay essentially reduces the decay rate.

Proof of Stability
Before proving the stability of the planar traveling wave solutions, let's introduce the essential formula for the solution of linear delayed differential system firstly.
As for linear delayed differential system where A, B ∈ C p× p , τ > 0 is the delay. Khusainov and Ivanov [16] had given the solution of the system (3.1) in case of p = 1 and A, B ∈ R. Similarity, we can get the solution in case of p ≥ 2.
, then the solution to the system (3.1) can be represented as where B 1 = Be Aτ and e B 1 t τ is the so-called delayed exponential function in the form Ma et al. gave a sufficient condition of the global stability for trivial solution of the linear delay system (3.1) (see [23] for the proof).
We can obtain the global existence and uniqueness of the solution for Cauchy problem (2.1) and (2.2) by the standard energy method and continuity extension method (see [30]) or the theory of abstract functional differential equations in [26]. Therefore we present the following proposition omitting the proof.

Proposition 3.1 (Global existence and uniqueness). Assume that the initial data satisfies
For any given planar traveling wave (φ(x · e 1 + ct), ϕ(x · e 1 + ct)) of (2.1) with c ≥ c * connecting the equilibria E − and E + , if the initial perturbation satisfies then nonnegative solution of the Cauchy problem (2.1) and (2.2) uniquely exists and satisfies Secondly, we mainly concentrate on proving the stability of all planar traveling waves to (2.1) with a specific convergence rate. Here, for the sake of convenience, we take g 1 (x) = g 2 (x) = g(x), x ∈ R n and assume that the conditions in Theorem 2.2 hold throughout this section.
For any c ≥ c * , we define the functions Obviously, the initial data (U ± 0 (x, s), V ± 0 (x, s)) are piecewise continuous and has a poor regularity, which may also cause the deficiency of regularity for the corresponding solutions. In order to overcome such a shortcoming, instead of these initial data, we choose smooth functions ( U ± 0 (x, s), V ± 0 (x, s)) as the new initial data and ( U ± 0 (x, s), V ± 0 (x, s)) satisfy t)) be the corresponding solution of (2.1) with the initial data for x ∈ R n , t > 0 and s ∈ [−τ, 0].
By the comparison principle in Lemma 2.1, we have and for (x, t) ∈ R n × R + . Next, we will complete the proof of Theorem 2.2 in three steps.
Based on Lemmas 3.2 and 3.3, we immediately obtain the following lemma.
According to the definition of U 1 (ξ, t) and V 1 (ξ, t), we have the following convergence of the solution (U + (x, t), V + (x, t)).

Proof of Theorem 2.2 Since
for x ∈ R n , t > 0, then by the Lemmas 3.5 and 3.6 and using the squeeze argument, we immediately obtain that, for any c > c * , there exists a positive number σ such that and for c = c * , where 0 < σ < min{σ 1 , δ 1 / τ }, constants σ 1 and δ 1 are given in (3.27) and (3.43), respectively. The proof of Theorem 2.2 is completed.

Numerical Simulations of the Planar Traveling Waves
In this section, we do the numerical simulation of the planar traveling waves of Lotka-Volterra competition system (1.1). We consider the system (1.1) satisfying the following initial value condition and the Neumann boundary value condition (4.1) and where ⊆ R n has smooth boundary, ∂ ∂n denotes the exterior normal derivative on ∂ . Homogeneous Neumann boundary value condition implies individual motion cannot cross the boundary ∂ . In system (1.1), we let b 1 = 1/2, b 2 = 3/2, d 1 = d 2 = r 1 = r 2 = 1, g 1 (x) = e − x 2 2 , x ∈ R n (here, we assume n = 1 for simplicity). Then, system (1.1) has two stable equilibrium points (0, 1) and (1, 0). According to the Theorem 2.1, system (1.1) has a planar wave. Furtherly, when the initial value satisfies (2.6) and the initial perturbation satisfies (2.7), the planar wave with the speed c > c * = 2.4495 is exponentially stable, and the planar wave with the speed c = c * is algebraically stable.
Here, we choose = [−50, 50], τ = 0.5, the initial function We can get it easily that the initial data (4.3) satisfy the initial conditions in Theorem 2.2.
With the help of the software MATLAB, we can obtain the numerical solution of (1.1) (see the following Figs. 1, 2). According to our results, the solution of system (1.1) will eventually converge to the equilibrium (1, 0), which corresponds to a biological phenomenon. In the beginning, system (1.1) only has one native species v with species u being absent, then the exotic species u competing with v invades this system in the way described by system (1.1). It follows from the Figs. 1 and 2 that the invading species u will survive and the native species v will eventually die out. From the Figs. 1 and 2, we can also find the solution of system (1.1) travels from the positive direction of the axis x to the negative direction. After a large time (here, the time t ≥ 4 is enough), the solution of system (1.1) behaves exactly as a stable planar traveling wave in the sense that the shape of the wave does not change. It can also be seen from Theorem 2.2 that this process is stable if the initial perturbation is proper.