EXISTENCE AND STABILITY OF TRAVELING WAVES FOR LESLIE-GOWER PREDATOR-PREY SYSTEM WITH NONLOCAL DIFFUSION

This paper will mainly study the information about the existence and stability of the invasion traveling waves for the nonlocal Leslie-Gower predator-prey model. By using an invariant cone in a bounded domain with initial function being defined on and applying the Schauder’s fixed point theorem, we can obtain the existence of traveling waves. Here, the compactness of the support set of dispersal kernel is needed when passing to an unbounded domain in the proof. Then we use the weighted energy to prove that the invasion traveling waves are exponentially stable as perturbation in some exponentially as x→ −∞. Finally, by defining the bilateral Laplace transform, we can obtain the nonexistence of the traveling waves.


1.
Introduction. The dynamic relationship between predators and their prey has been one of the dominant themes in ecology due to its universal existence and importance. The interactions of prey and predator species are predicted by the behavior of some mathematical models. According to the different functional response to predation, researchers have constructed many various models. The classical Lotka-Volterra model and its modified models have been studied for the stability of equilibria and the existence of traveling waves, see [6,12,15,18,29,30,31,34] et al. The properties of the model with the Leslie-Gower functional response and its modified models have been discussed in [1,19,26,37]. For the Holling-Tanner predator-prey model, Tanner in [32] has considered the stability of the underlying ordinary differential system

HONGMEI CHENG AND RONG YUAN
where r denotes the growth rate of predator and Π(u) = mu A + u (m > 0, A > 0) denotes the functional response to predation suggested by Holling in [13]. The second equation of (1) means that the intrinsic population growth rate r affects not only the potential increase of the population but also its decrease. If v is greater than u, the population will decline, and the speed of its decline is directly proportional to the intrinsic growth rate. It seems to be a contradiction, but it is realistic since species of small body size and early maturity have high intrinsic growth rates and also have low survival rates and short lives. This equation is the same as that used by Leslie and Gower in [20]. In the last decades, the model (1) has attracted some attention of many researchers. For instance, one can refer to May [22], Hsu and Huang [14] and Murray [25]. Typical example of the functional response Π is given by Holling type functional response. Let us also mention the case where Π(u) ≡ αu with constant α > 0, that corresponds to the classical Lotka-Volterra functional response. A. Ducrot in [11] has studied the reaction-diffusion system with the Laplacian diffusion where d > 0 describes the diffusivity of prey, r denotes the growth rate of predator and Π(u) = uπ(u), π : [0, ∞) → [0, ∞) is a function of the class C 1 such that π(u) > 0 for all u ∈ (0, 1]. He has proved that the system has a generalized transition wave with some determined global mean speed of propagation. As we all know, the standard Laplacian operator corresponds to expected values for individuals moving under a Brownian process. But the movement of individuals which cannot be limited in a small area is often free and random. Recently, various integral operators have been widely used to describe the nonlocal diffusion phenomena. For example, an operator of the form K[u](x) = R k(x, y)[u(y) − u(x)]dy appears in the theory of phase transition, ecology, genetics and neurology, see [2,16,17,33]. Meanwhile, many researchers give more attention on the study of traveling waves for the nonlocal reaction diffusion equations. For instance, many authors have obtained the properties of the solution for the reaction-diffusion systems with nonlocal diffusion term, see [3,4,7,8,9,10,28,40]. They consider the models under the condition of monotone or quasi-monotone, for instance in [35,38,39]. The important condition can permit the applications of the powerful monotone dynamical systems and comparison arguments. But some models cannot satisfy the monotone condition. In [27], the authors have obtained the existence of the traveling waves of the model without monotone condition. In [5], we have considered the nonexistence of the general wave solution for the model (2) with Π(u) = αu and the fractional diffusion term ∆ α (α ∈ (0, 1)).
Inspired by these results, we consider the Leslie-Gower predator-prey model with the nonlocal diffusion term, that is where the parameters d > 0, r > 0 and 0 < β < 1. Throughout this paper, we need the below assumptions of the kernel function J.
In this work, we mainly consider the existence and stability of the invasion traveling wave solution which connects the predator free state (1, 0) with the coexistence state 1 1+β , 1 1+β of the system (3). We will obtain that there exists c * > 0 such that for c > c * , system (3) admits an invasion traveling wave solution with wave speed c; for 0 < c < c * , system (3) has no invasion traveling waves with wave speed c. Further, we can get the stability of the invasion traveling wave by the weighted energy. Due to the nonlocal diffusion effect, it is more hard to obtain the uniform boundness of solutions. To overcome the difficulties, we construct an invariant cone in a large bounded domain with initial functions being defined on, then pass to the unbounded domain by limiting argument. This paper is organized as follows. In the next section, we present the proof of the existence of the invasion traveling waves by Schauder's fixed point theorem under the assumption of the compactly supported for the kernel function J. In Section 3, we get the stability of the invasion traveling wave solution by the weighted energy. Finally, we obtain the nonexistence of the invasion traveling waves by the bilateral Laplace transform.
2. Existence of the traveling waves . In this section, we will prove the existence of the invasion traveling wave solution for the system (3). The traveling wave solution means a solution of the form (u(x + ct), v(x + ct)). Let ξ = x + ct, then (u(ξ), v(ξ)) satisfies Linearizing the second equation of the system (4) at (1, 0), we have Then we can get a characteristic equation

HONGMEI CHENG AND RONG YUAN
By easy calculations, we can obtain In view of the above properties of the function ∆(λ, c), we can get the following lemma.
In the sequel, we always assume c > c * and simply denote λ i (c) by λ i for i = 1, 2, respectively.
, and a function set and consider the following initial value problems with By the definition of the operator F, it is obvious to see According to the definition of φ(ξ) and the choosing of the constant δ, we can know that which implies that 1 − βδ is a super-solution of (9). Thus we can obtain that By the definition ofφ(ξ) and Lemma 2.3, we know that By a similar argument and using Lemmas 2.2 and 2.4, it is easy to show that This ends the proof.
Lemma 2.6. The operator F : Γ A → Γ A is completely continuous.
Proof. By the definition of the operator We first show that F is continuous. By a direct calculation, we have that and where For ∀ (φ 1 (·), ϕ 1 (·)), (φ 2 (·), ϕ 2 (·)) ∈ Γ A , we have that Combining with the continuity of the compound function, we can obtain that F is continuous from equations (12) and (13). Next we show that F is compacted, that is, we should prove that for any bounded Since (φ, ϕ) ∈ Ω, (12) and (13), we have that there exists a constant M 1 > 0 such that That is, F(Ω) is uniformly bounded. Further, according to the equations (9), (10) and the above inequality, then there exists some constant M 2 > 0 such that So we can get that F(Ω) is equicontinuous. By Arzela-Ascoli Theorem, we have that F(Ω) is precompact. Then we get that F : Γ A → Γ A is completely continuous with respect to the maximum norm. Proof. By the definition of Γ A , it is easy to see that Γ A is closed and convex. Thus, according to Lemma 2.6 and using the Schauder's fixed point theorem, there exists To obtain the existence of solutions for system (4), we need some estimates . In order to get the estimates, we make another assumption on the kernel function J.
The kernel function J is compactly supported.
3. Stability of traveling waves . In this section, we will give the proof of the stability for the invasion traveling waves. Before giving our stability result, we need to show an assumption on parameters. (J 4 ) The parameters β, r and d satisfy Define two functions on η as follows and By the assumptions (J 1 ), (J 4 ) and the continuity of f i (η) (i = 1, 2), it is easily show that there exists η 0 such that f i (η 0 ) > 0 (i = 1, 2). Next, we define two functions on ξ as follows and ṽ) is a traveling wave solution given in Theorem 2.9. It is easy to see that which implies that there exists ξ 0 large enough such that Following we define a weight function by Here we state the stability of the traveling wave solution for the system (3) with the following initial data   (24). Further, they satisfy

If the initial data satisfies
for all t > 0, where C and µ are some positive constants. Namely, (u(x, t), v(x, t)) converges to the traveling wave solution (ũ(x + ct),ṽ(x + ct)) exponentially in time t.
By the standard energy method and continuity extension method (see [23,24]) or the theory of abstract functional differential equations in [21], we can obtain the existence and uniqueness of the solution for (3) and (25) in Theorem 3.1. Here we omit the details and mainly show the proof of the stability.
First of all, we will establish some key inequalities to get the basic estimates. Define G i w (ξ, t) (i = 1, · · · , 4) as follows and
Proof. We only show the proof of G 1 w (ξ, t) ≥ C 1 for some positive constant C 1 , since the other inequalities can be proven in a similar way. By the condition, we have cη 0 > c 1 , that is When ξ ≤ ξ 0 , it is easy to show that w(ξ) = e −η0(ξ−ξ0) and w(ξ) is non-increasing. Then we can obtain For ξ > ξ 0 , it follows from the definition (24) that w(ξ) = 1 and w (ξ) w(ξ) = 0. Thus by (23), we have According to the above argument, we can easily obtain the following result. For any c > max c * , 1 η0 max{c 1 , c 2 } , there exists some positive constant C such for all ξ ∈ R, t > 0 and 0 < µ < min i=1,2,3,4 Here we begin to give the priori estimates about U (ξ, t) and V (ξ, t) in the weighted Sobolev space H 1 w (R). Lemma 3.4. Assume that (J 1 ), (J 3 ) and (J 4 ) hold and w(ξ) is defined by (24).
The following estimates can be proved by modifying the arguments in the above lemma. So we omit the details of the proof for the following lemma. For any c > max c * , 1 η0 max{c 1 , c 2 } , there exists some positive constant C such that Then we can easily have a prior estimate by Lemmas 3.4 and 3.5.
For any c > max c * , 1 η0 max{c 1 , c 2 } , there exists some positive constant C such that for all t > 0.
Proof of Theorem 3.1. According to the standard Sobolev embedding inequality H 1 (R) → C(R), and the embedding inequality By the same argument, we can also obtain Thus, using the squeezing technique, we can easily get 4. Nonexistence of traveling waves . In this section, we will give the proof of the nonexistence of traveling waves for (4) when 0 < c < c * . Proof. We prove this theorem by the way of contradiction. Here, we assume that there exists a traveling wave solution (u(ξ), v(ξ)) of system (4) satisfying the limit behavior at infinity. Since u(−∞) = 1 and v(−∞) = 0, there existξ < 0 and 1 > 0 such that u(ξ) ≥ 1 − 1 and v(ξ) ≤ 1 for any ξ ≤ξ. Therefore, we have that for any ξ ≤ξ. Letting V (ξ) = ξ −∞ v(η)dη for any ξ ∈ R and integrating two sides of inequality (44) from −∞ to ξ with ξ ≤ξ, we can obtain By Fubini theorem, we have  (46), we have that V (ξ) is integrable on (−∞, ξ] for any ξ ≤ξ. Now integrating two sides of inequality (46) from −∞ to ξ with ξ ≤ξ, we can obtain Due to the fact that yV (ξ − θy) is non-increasing for θ ∈ [0, 1] and the symmetry of the kernel function J, we have yJ(y)dy V (ξ) = cV (ξ).

HONGMEI CHENG AND RONG YUAN
Then, multiplying two sides of the second equation of (4) by e −λξ and integrating two sides on R, we can obtain In view of for λ ∈ C with 0 < Reλ < µ 0 , where ∆(λ, c) is defined by equation (5). Due to 1 − β ≤ u(ξ) ≤ 1 and (47), we can get that the right hand side of equation (48) is well defined for λ ∈ C with 0 < Reλ < 2µ 0 . By the way of recursion, we can get that the right hand side of (48) is well defined for λ ∈ C with 0 < Reλ < nµ 0 (n ≥ 2). Letting n → +∞, we can obtain that the right hand side of (48) is well defined for λ ∈ C with 0 < Reλ < ∞. Then we can know that L(λ) is well defined with Reλ > 0. But (48) can be re-written as R e −λξ ∆(λ, c) − r v(ξ) u(ξ) v(ξ)dξ = 0, it follows from the definition of ∆(λ, c) and Lemma 2.1 that ∆(λ, c) → +∞ as λ → λ 0 . This contradicts with the equation (49) and we complete our proof.