Stability of Traveling Wave Fronts for a Three Species Predator-Prey Model with Nonlocal Dispersals

In this paper, we consider a predator-prey model with nonlocal dispersals of two cooperative preys and one predator. We prove that the traveling wave fronts with the relatively large wave speed are exponentially stable as perturbation in some exponentially weighted spaces, when the dierence between initial data and traveling wave fronts decay exponentially at negative innity, but in other locations, the initial data can be very large. e adopted method is to use the weighted energy method and the squeezing technique with some new avors to handle the nonlocal dispersals.


Introduction
In this paper, we investigate the stability of traveling wave fronts for a three species predator-prey model: with the initial condition u(x, 0) u 0 (x, 0), v(x, 0) v 0 (x, 0), z(x, 0) z 0 (x, 0), where r i , i 1, 2, 3, a 1 , a 2 , h, and k are positive constants, u(x, t) and v(x, t) denote the densities of two cooperative preys at time t and location x, respectively, z(x, t) denotes the density of the predator at time t and location x, r 1 and r 2 are the intrinsic growth rates of u(x, t) and v(x, t), respectively, r 3 is the death rate of z(x, t), h and k are interspeci c cooperative coe cients between two preys, r 1 a 1 and r 2 a 2 are the predation rates, and r 3 a 1 and r 3 a 2 are the conversion rates.J i : R ⟶ R (i 1, 2, 3) are probability functions of the random dispersal of individuals and satisfy the following assumptions: (H 1 )J i (x) � J i (− x) ≥ 0, x ∈ R,  R J i (x)dx � 1 and  R J i (y)e λy dy < ∞, λ > 0 is the eigenvalue of the characteristic equation of model ( 1) (H 2 )0 < hk < 1 Wu [1] investigated the spreading speed for a predatorprey model with one predator and two preys: with the initial condition where u(x, t) and v(x, t) denote the densities of two competitive preys at time t and location x, respectively and z(x, t) denotes the density of the predator at time t and location x. e parameter d i , i � 1, 2, 3, is the diffusion coefficients of u, v, and z, respectively, the remaining parameters are the same as model (1).Under certain conditions, the author characterized the asymptotic spreading speed by the parameters of model (3).e interaction of three species eological systems has been studied before (see [2][3][4]).
Yu and Pei [5] studied the stability of traveling wave fronts for the cooperative system with nonlocal dispersals: with the initial condition u 1 (x, 0) � u 10 (x, 0), e authors adopt the weighted energy method and the squeezing technique to prove the stability of the traveling wave fronts.e weighted energy method for treating timedelayed reaction diffusion equations was firstly introduced by Mei et al. [6].
en, by combining the squeezing argument, it was developed for proving the global stability of wavefronts by [7][8][9].For the nonlocal model using a single integrodifferential equation, the existence, uniqueness, and stability of traveling waves have been widely studied in [10][11][12][13][14].For the multicomponent nonlocal systems, the existence of traveling waves was also investigated in [15,16], while the stability of traveling waves is less investigated and can only be found in [5,17].Many researchers are widely focused on the complex dynamics of biological systems such as stochastic delay population system [18] and many researchers have studied the Lotka-Volterra time delay models with two competitive preys and one predator [19].Note that the composite population systems with stochastic effects and time delays present some complex dynamics; thus, this causes widespread researchers concern [20,21].
Inspired by works [1,5], we will study the stability of traveling wave fronts for problems (1) and (2).We consider the case when both preys are cooperative.In this case, there is the constant state: e rest of this paper is organized as follows.In Section 2, we present some preliminaries and summarize our main results.Section 3 is concerned with the proof of the main result by the technical weighted energy method and establishes the desired priori estimate, and then we get it by the squeezing technique.Finally, we make a simple summary in Section 4.
Before stating our main result, we introduce some notations.

Notations.
roughout this paper, C > 0 denotes a generic constant, while C i > 0 (i � 0, 1, 2, . ..) represents a specific constant.Let I be an interval, typically I � R. L 2 (I) is the space of square integrable functions defined on I, and H k (I) (k ≥ 0) is the Sobolev space of the L 2 functions f(x) defined on the interval I whose derivatives (d i /dx i )f (i � 1, . . ., k) also belong to L 2 (I).Further, L 2 w (I) be the weighted L 2 space with a weight function w(x) > 0, with the norm defined as H k w (I) be the weighted Sobolev space with the norm given by Let T > 0 be a number and B be a Banach space.We denote by C 0 ([0, T]; B) the space of the B-valued continuous functions on [0, T], and L 2 ([0, T]; B) as the space of the B-valued L 2 functions on [0, T].
e corresponding spaces of the B-valued functions on [0, ∞) are defined similarly.

Preliminaries and Main Result
In this section, we consider the existence of traveling wave fronts for the Cauchy problems (1) and (2) by using the comparison principle and squeezing technique.Traveling waves solution of system (1) is a special solution (u, v, z) with the form u(x, t) � ϕ 1 (x + ct) � ϕ 1 (ξ), v(x, t) � ϕ 2 (x + ct) � ϕ 2 (ξ), and z(x, t) � ϕ 3 (x + ct) � ϕ 3 (ξ) with c > 0 is the wave speed, where the wave profile In order to state our stability result, we need to give the following two propositions in [5].
In addition, define three functions on ξ as follows: where (ϕ 1 , ϕ 2 , ϕ 3 ) is a traveling wave front of (10).We can easily prove which imply that Complexity where ξ 0 is chosen to be large enough.Define a weight function by Now, we state the stability result.

Stability
In this section, the existence of the solution to problems ( 1) and ( 2) can be proved via the upper and lower solutions method (see [24]).We first examine the case when two species cooperate while remaining in the same area without diffusive movement: where all parameters are positive and have four constant equilibria (0, 0), (0, 1/a 2 ), and(1/a 1 , 0) and coexistence equilibrium Li and Lin [22] investigated the existence of traveling wave fronts connecting (0, 0) to the constant equilibrium (k 1 , k 2 ) for system (5) by using the known theory [25].Integrodifferential system ( 5) is related to the classic Laplacian diffusion system, for example, where δ is the Dirac delta (see, Medlock et al. [26]), and then (5) reduces to Because time delay often affects the evolutionary process, the system was incorporated into discrete and nonlocal delays while the existence of traveling wave fronts was obtained (see, [27][28][29]).
e existence of traveling wave solutions of system (1) is similar to them.at is, the initial data satisfies and the initial perturbation satisfies 4 Complexity and then the nonnegative solution of problems ( 1) and ( 2) exists uniquely and satisfies Here, we omit the details.Define and then we obtain erefore, it follows from the comparison principle that where ξ � x + ct, and it follows from (32) and ( 33) that is section is devoted to establishing a prior estimate, which is the core of this paper.e approach is the weighted energy method.Define where In order to get the basic estimate, we must prove that A i w (ξ, t) ≥ C > 0, (i � 1, 2, 3, 4, 5, 6) and B i w (ξ, t) ≥ C > 0, (i � 1, 2, 3, 4, 5, 6), for some constant C. We need the following key lemma.
for some positive constant C.
Combining Lemma 4 and Lemma 5 and noting that w(ξ) ≥ 1 on R, we obtain the following result.
for some positive constant C and all t > 0. (88) is completes the proof of eorem 1.

Conclusions
Inspired by the works of [1,5], in this paper, we consider a predator-prey model with nonlocal dispersals of two cooperative preys and one predator.Since the predation models of the three species add to the computational difficulty, it takes a lot of effort to the priority estimation, and the key point in proving the stability of the traveling front is devoted to establishing a prior estimate by using the weighted energy method.By the standard Sobolev embedding inequality and the squeezing technique, we prove the stability of the traveling wave solution.
In recent years, there has been great progress in modeling and analysis dynamical behavior of predator-prey population involving both time delay and spatial diffusion.In a pioneer work, some researchers have studied a scalar reaction diffusion equation with a single discrete delay by using the phase-plan technique; we can take more attention and initiate the study of traveling wave solutions to delayed reaction diffusion systems on the basis of this paper.Futhermore, we will consider the existence and stability of Complexity traveling wave fronts for three-dimensional diffusion systems with convolution delay by the comparison principle and squeezing technique in the future.Another interesting and difficulty problem is the stability of the traveling wave solution under quasi-monotone or nonquasi-monotone assumptions.We leave these issues for future research.