Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 33, pp. 1–19. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.33 DYNAMICS OF TRAVELING WAVES FOR PREDATOR-PREY SYSTEMS WITH ALLEE EFFECT AND TIME DELAY YANG HUA, XIAOJIE LIN, JIANG LIU, HAIXIA LU Abstract. This article aims to establish the existence of traveling waves for a predator-prey system with Beddington-DeAngelis functional response, repro- ductive Allee effect, and time delay. We investigate the existence of solutions for a system with two special delay kernels by geometric singular perturbation theory, invariant manifold theory, and Fredholm orthogonality theory. In ad- dition, we discuss the asymptotic behaviors of traveling waves with the aid of the asymptotic theory. 1. Introduction The Allee effect was originally proposed by Warder Clyde Allee [2] to characterize the correlation between the population density and per capita growth rate of the population at low densities [27]. In recent decades, it has received considerable attention and numerous related studies have been conducted [1, 19, 23, 29, 31, 34]. To investigate the impact of the reproductive Allee effect in prey growth, Dey et al. [7] considered the predator-prey system with Beddington-DeAngelis functional response and reproductive Allee effect, NT = dNNyy + aN2(b−N)− dN − sNP p+N + qP , PT = dPPyy + esNP p+N + qP −mP, (1.1) where N(T, y) and P (T, y) denote the prey and predator densities at moment T and location y, respectively, dN and dP denote the diffusion coefficients of the prey and predator populations, the positive constants a, b, and d stand for the intrinsic growth rate, threshold for positive growth, and intrinsic mortality rate, the parameters s, p, and q represent the maximum predation rate, self-saturation constant and predator mutual interference, e and m are the conversation coefficient and the per capita natural death rate of the predator population. Making the change of variables u = N b , v = sP bm , t = mT, x = √ m dN y, 2020 Mathematics Subject Classification. 92D25, 35C07, 35K57, 34D15. Key words and phrases. Predator-prey system; Allee effect; traveling waves; asymptotic behavior; geometric singular perturbation. ©2024. This work is licensed under a CC BY 4.0 license. Submitted November 29, 2023. Published April 26, 2024. 1 2 Y. HUA, X. LIN, J. LIU, H. LU EJDE-2024/33 system (1.1) becomes dimensionless system ut = uxx + σu2(1− u)− ηu− uv α+ u+ βv , vt = δvxx + γuv α+ u+ βv − v, (1.2) where α = p b , β = qm s , γ = es m , σ = ab2 m , η = d m , δ = dP dN are dimensionless param- eters. Combining the shooting method discussed in Huang [15] and the invariant manifold theory, Dey et al. [7] obtained the existence of traveling waves for system (1.2) connecting from the predator free homogeneous steady-state to the coexisting homogeneous steady-state. The time delay is a kind of common nonlinearity, and numerous biological pro- cesses involve delays [21, 24, 28, 32, 33]. In ecology, the prey cannot immediately convert food into their own energy [3, 26]. Furthermore, the predator requires time for digesting prey before further activities [13], and the reproduction of predator is not instantaneous but mediated by some time delay required for gestation. Conse- quently, it is meaningful and essential to consider the time delay in predator-prey interaction.However, Dey et al. [7] did not consider the asymptotic behavior of traveling waves for the system (1.2). There are two frequently used time delays known as local delay (1.4) and nonlocal delay (1.6). Britton [4] introduced a model for a single biological population in the form of ut = uxx + u (1 + au− (1 + a)(f ∗ u)) , (1.3) where f is a given function and f ∗ u denotes a local convolution in the spatial variable that can be written as (f ∗ u)(x, t) = ∫ t −∞ f(t− s)u(x, s)ds, (1.4) which is a spatial average weighted according to distance from the original position. The time delay (1.4) is called the local delay. The kernel f(t) is any integrable non- negative function that satisfies∫ +∞ 0 f(t)dt = 1 and tf(t) ∈ L1 ((0,∞),R) . It is notable that the normalization assumption on f ensures that the uniform non- negative steady-state solutions are unaffected by the delay. The two classical and special kernels are defined by f(t) = 1 τ e−t/τ and f(t) = t τ2 e−t/τ , (1.5) where τ > 0 is a small parameter. The former is called the local weak delay kernel and the latter the local strong delay kernel. To recognize the effect of moving time, a spatio-temporal average weighted to- ward the current time and position had been studied. Subsequently, Britton [5] provided a mathematical derivation of a modified model, which incorporates a non- local convolution in space and time, taking the form (f ∗ u)(x, t) = ∫ t −∞ ∫ +∞ −∞ f(x− y, t− s)u(y, s) dy ds. (1.6) EJDE-2024/33 TRAVELING WAVES FOR PREDATOR-PREY SYSTEMS 3 The time delay (1.6) is called the nonlocal delay. The kernel f(x, t) is any integrable non-negative function satisfying the normalization assumption∫ t −∞ ∫ +∞ −∞ f(x, t) dx dt = 1, which ensures that the uniform non-negative steady-state solutions are unaffected by the delay. The two classical and special kernels are defined by f(x, t) = 1√ 4πt e− x2 4t 1 τ e−t/τ and f(x, t) = 1√ 4πt e− x2 4t t τ2 e−t/τ , (1.7) where the small parameter τ > 0 measures the time delay. The former is called the nonlocal weak delay kernel and the latter the nonlocal strong delay kernel. Traveling waves are useful in understanding the interaction of multiple species, and many powerful study methods have been established. Among them, the geo- metric singular perturbation theory developed by Fenichel [12] is an effective method to prove the existence of traveling waves in evolution equations with small param- eters. This theory has been applied to various equations, including Keller-Segel systems [10, 25], FitzHugh-Nagumo equations [14, 20, 30], nonlinear Belousov- Zhabotinskii system[11], Liénard equations [18], Camassa-Holm equations [9, 8], etc. Motivated by the aforementioned analysis, we incorporate time delay into the system (1.2). To avoid excessive technicalities, we shall only consider the case of time delay on prey in this paper. The predator-prey system with Beddington- DeAngelis functional response, reproductive Allee effect, and time delay which we study here is given as ut = uxx + σu(f ∗ u) (1− (f ∗ u))− ηu− uv α+ u+ βv , vt = δvxx + γuv α+ u+ βv − v, (1.8) where the parameters are the same as those in system (1.2), and the term (f∗u)(x, t) denotes time delay. When we ignore the time delay, system (1.8) is reduced to system (1.2). The goal of this article is to establish the existence of traveling waves for sys- tem (1.8) with two cases of (f ∗ u)(x, t) as: local delay (1.4) and nonlocal delay (1.6), where the delay kernels are chosen as (1.5) and (1.7), respectively. Further- more, the asymptotic behavior of traveling waves for system (1.2) has also been considered. It should be pointed out that the profile system of system (1.2) is four- dimensional, while the profile system of system (1.8) with local delay kernels (1.5) is six-dimensional, or eight-dimensional with nonlocal delay kernels (1.7). Conse- quently, the shooting method employed by Dey et al. [7] does not apply to system (1.8). Here we use the geometric singular perturbation theory and Fredholm or- thogonality theory to solve this difficulty. The article is organized as follows. Section 2 gives some preliminaries and intro- duces the geometric singular perturbation theory. Section 3 and Section 4 respec- tively focus on studying the existence of traveling waves for the system (1.8) with local delay kernels (1.5) and nonlocal delay kernels (1.7) by the geometric singular perturbation theory and the Fredholm orthogonality theory. Section 5 is to explore the asymptotic behaviors of traveling waves with the method of the asymptotic theory. 4 Y. HUA, X. LIN, J. LIU, H. LU EJDE-2024/33 2. Preliminaries In this section, we provide some preliminaries and introduce the geometric sin- gular perturbation theory [16, 17]. By taking the traveling wave transformation (u, v)(x, t) = (U, V )(ξ), ξ = x+ ct, (2.1) where the constant c > 0 is the wave speed, system (1.2) is transformed into cU ′ = U ′′ + σU2(1− U)− ηU − UV α+ U + βV , cV ′ = δV ′′ + γUV α+ U + βV − V, (2.2) where ′ = d dξ . System (2.2) is equivalent to the system U ′ = X, V ′ = Y, X ′ = cX − σU2(1− U) + ηU + UV α+ U + βV , Y ′ = 1 δ [ cY − γUV α+ U + βV + V ] . (2.3) According to the analysis presented in [7], for σ > 4η, it follows that the system (2.3) has three equilibria P0(0, 0, 0, 0), P1(u1, 0, 0, 0), and P2(u2, 0, 0, 0), where u1 = σ + √ σ2 − 4ση 2σ > 0, u2 = σ − √ σ2 − 4ση 2σ > 0. Furthermore, under certain conditions, see [7] for more details, the system (2.3) admits the fourth equilibrium P∗(u∗, v∗, 0, 0), where u∗ is a positive root of the cubic equation σγβu3 − σγβu2 + (γβη + γ − 1)u− α = 0, (2.4) and v∗ = γu∗ − u∗ − α β > 0. Note that the equilibrium P1(u1, 0, 0, 0) and P∗(u∗, v∗, 0, 0) correspond to the preda- tor free homogeneous steady-state E1(u1, 0) and the coexisting homogeneous steady- state E∗(u∗, v∗) of the system (1.2). For system (2.3), Dey et al. [7] obtained the existence of the heteroclinic orbit that connects the equilibrium P1 and P∗ in the case E∗ is a stable homogeneous steady-state, which can be characterized by the following lemma. Lemma 2.1 ([7]). If c ≥ 2 √ δ( γu1 α+u1 − 1), then system (2.3) admits a heteroclinic orbit Φξ(P ) = (U(ξ), V (ξ), X(ξ), Y (ξ)) with Φξ(P ) → P1(u1, 0, 0, 0) as ξ → −∞ and Φξ(P ) → P∗(u∗, v∗, 0, 0) as ξ → +∞. EJDE-2024/33 TRAVELING WAVES FOR PREDATOR-PREY SYSTEMS 5 We now introduce the geometric singular perturbation theory, which ensures the existence of invariant manifolds under certain conditions. With the use of this approach, the higher-dimensional systems can be reduced to lower-dimensional regular perturbation systems on these manifolds. This reduction greatly simplifies the analysis of traveling waves in high-dimensional systems. Lemma 2.2 ([16, 17]). Consider the system x′(t) = f(x, y, ε), y′(t) = εg(x, y, ε), (2.5) where (x, y) ∈ Rm × Rn (m,n ≥ 1), and the parameter ε satisfies 0 < ε ≪ 1. The functions f and g are C∞ on the set V × I, where V ∈ Rm+n and I is an open interval that includes 0. If the critical manifold M0 = {(x, y) ∈ Rm × Rn : f(x, y, 0) = 0} is normally hyperbolic, i.e., the m ×m matrix (Dxf)(p, 0) of first partial derivatives with respect to the fast variables x has no eigenvalues with zero real part for all p ∈M0, then for 0 < ε≪ 1 and any 0 < r < +∞, (i) there exists a slow manifold Mε that is diffeomorphic to M0, and Mε is locally invariant under the flow of (2.5); (ii) Mε is Cr in x, y, ε and can be given as a graph Mε = {(x, y) : x = hε(y)}, for some Cr function hε(y); (iii) there exists locally stable and unstable invariable manifolds W s loc(Mε) and Wu loc(Mε) lying within O(ε) and being Cr diffeomorphic to W s loc(M0) and Wu loc(M0). 3. Traveling waves for the system with local delay This section focuses on studying the existence of traveling waves for the system (1.8) with local delay (1.4) by the geometric singular perturbation theory and the Fredholm orthogonality theory. We investigate the case of local strong delay kernel, i.e., taking f(t) = t τ2 e −t/τ , the case of the local weak delay kernel can be discussed by the same way. 3.1. Perturbation analysis with local delay. Substituting the traveling wave transformation (2.1) into system (1.8), we have cU ′ = U ′′ + σUW (1−W )− ηU − UV α+ U + βV , cV ′ = δV ′′ + γUV α+ U + βV − V, (3.1) 6 Y. HUA, X. LIN, J. LIU, H. LU EJDE-2024/33 where ′ = d dξ , and W (ξ) = ∫ t −∞ t− s τ2 e− t−s τ U(x+ cs)ds = − ∫ 0 +∞ z τ2 e− z τ U(x+ ct− cz)dz = ∫ +∞ 0 z τ2 e− z τ U(ξ − cz)dz = ∫ +∞ 0 t τ2 e−t/τU(ξ − ct)dt. (3.2) Differentiating the equation (3.2) with respect to ξ, we obtain dW dξ = ∫ +∞ 0 t τ2 e−t/τUξ(ξ − ct)dt = −1 c ∫ +∞ 0 t τ2 e−t/τUt(ξ − ct)dt = −1 c ∫ +∞ 0 t τ2 e−t/τdU = −1 c ( t τ2 e−t/τU(ξ − ct) )∣∣∣+∞ 0 + 1 c ∫ +∞ 0 U(ξ − ct)d( t τ2 e−t/τ ) = 1 c ∫ +∞ 0 U(ξ − ct)d( t τ2 e−t/τ ) = 1 cτ (∫ +∞ 0 1 τ e−t/τU(ξ − ct)dt− ∫ +∞ 0 t τ2 e−t/τU(ξ − ct)dt ) = 1 cτ (ζ −W ), (3.3) where ζ(ξ) = ∫ +∞ 0 1 τ e−t/τU(ξ − ct)dt. (3.4) Differentiating equation (3.4) with respect to ξ, we obtain dζ dξ = ∫ +∞ 0 1 τ e−t/τUξ(ξ − ct)dt = −1 c ∫ +∞ 0 1 τ e−t/τUt(ξ − ct)dt = −1 c ∫ +∞ 0 1 τ e−t/τdU = − 1 cτ e−t/τU(ξ − ct) ∣∣∣+∞ 0 + 1 c ∫ +∞ 0 Ud( 1 τ e−t/τ ) = 1 cτ U(ξ)− 1 cτ ∫ +∞ 0 1 τ e−t/τU(ξ − ct)dt = 1 cτ (U − ζ). (3.5) EJDE-2024/33 TRAVELING WAVES FOR PREDATOR-PREY SYSTEMS 7 By combining equations (3.3) and (3.5), system (3.1) can be reformulated as cU ′ = U ′′ + σUW (1−W )− ηU − UV α+ U + βV , cV ′ = δV ′′ + γUV α+ U + βV − V, cτW ′ = ζ −W, cτζ ′ = U − ζ, (3.6) which can be further rewritten as a system of first order ODEs in R6 U ′ = X, V ′ = Y, X ′ = cX − σUW (1−W ) + ηU + UV α+ U + βV , Y ′ = 1 δ [ cY − γUV α+ U + βV + V ] , cτW ′ = ζ −W, cτζ ′ = U − ζ, (3.7) where the small parameter τ denotes the time delay in the system (1.8). For small τ > 0, system (3.7) is a slow system in which (W, ζ) are the fast variables and (U, V,X, Y ) are the slow variables. Let τ = 0, then the flow of the system (3.7) is constrained to the critical manifold C0 = {(U, V,X, Y,W, ζ) ∈ R6 :W = ζ = U}. (3.8) Now we study the normal hyperbolicity of the critical manifold C0. For a point P ∈ C0, the matrix of the first partial derivatives with respect to the fast variables (W, ζ) is J1 = ( −1/c 1/c 0 −1/c ) . (3.9) Thus according to Lemma 2.2, it follows that the critical manifold C0 is normally hyperbolic. It indicates that there exists a slow manifold Cτ , O(τ) close and dif- feomorphic to C0 for 0 < τ ≪ 1, which can be expressed as Cτ = { (U, V,X, Y,W, ζ) ∈ R6 :W = U +ϖ1(U, V,X, Y )τ +O(τ2), ζ = U +ϖ2(U, V,X, Y )τ +O(τ2) } , where ϖi(U, V,X, Y ) (i = 1, 2) are two smooth functions defined on a compact domain. Substituting W = U +ϖ1(U, V,X, Y )τ +O(τ2), ζ = U +ϖ2(U, V,X, Y )τ +O(τ2), into the system (3.7), we have cτ(U ′ +ϖ′ 1τ) = cτX +O(τ2) = (ϖ2 −ϖ1)τ +O(τ2), cτ(U ′ +ϖ′ 2τ) = cτX +O(τ2) = −ϖ2τ +O(τ2). 8 Y. HUA, X. LIN, J. LIU, H. LU EJDE-2024/33 Comparing the coefficients of τ , it follows that ϖ1(U, V,X, Y ) = −2cX, and ϖ2(U, V,X, Y ) = −cX. Then system (3.7) restricted to Cτ can be reformulated as U ′ = X, V ′ = Y, X ′ = cX − σU(U − 2cXτ)(1− U + 2cXτ) + ηU + UV α+ U + βV +O(τ2), Y ′ = 1 δ [ cY − γUV α+ U + βV + V ] , (3.10) which is a regular perturbation of the system (2.3). It is evident that the system (3.10) is simplified to the system (2.3) when τ = 0. 3.2. Analysis by the Fredholm orthogonality theory. In this section, we shall prove that the system (3.10) admits a heteroclinic orbit connecting the equilibrium P1(u1, 0, 0, 0) to P∗(u∗, v∗, 0, 0) for 0 < τ ≪ 1. Let (U0, V0, X0, Y0)(ξ) be the heteroclinic orbit of system (2.3) obtained in Lemma 2.1, connecting the equilibrium P1(u1, 0, 0, 0) to P∗(u∗, v∗, 0, 0). To solve the system (3.10) for 0 < τ ≪ 1, we set U = U0 + U1τ +O(τ2), V = V0 + V1τ +O(τ2), X = X0 +X1τ +O(τ2), Y = Y0 + Y1τ +O(τ2). (3.11) Substituting the transformation (3.11) into the first and second equations of system (3.10), and comparing the coefficient of τ , we have U ′ 1 = X1, and V ′ 1 = Y1. (3.12) Substituting the transformation (3.11) into the third equation of system (3.10), we obtain X ′ = X ′ 0 +X ′ 1τ +O(τ2) = cX0 − U0(σU0 − σU2 0 − η) + U0V0 α+ U0 + βV0 +X ′ 1τ +O(τ2) = cX0 − U0(σU0 − σU2 0 − η) + [ (3σU2 0 − 2σU0 + η)U1 + cX1 + 2cσU0X0(1− 2U0) ] τ + U0V0 + (U0V1 + V0U1)τ +O(τ2) α+ U0 + βV0 + (U1 + βV1)τ +O(τ2) +O(τ2), i.e., [ α+ U0 + βV0 + (U1 + βV1)τ ]( U0V0 α+ U0 + βV0 +X ′ 1τ ) +O(τ2) = [ α+ U0 + βV0 + (U1 + βV1)τ ][ (3σU2 0 − 2σU0 + η)U1 + cX1 + 2cσU0X0(1− 2U0) ] τ + U0V0 + (U0V1 + V0U1)τ +O(τ2). EJDE-2024/33 TRAVELING WAVES FOR PREDATOR-PREY SYSTEMS 9 Comparing the coefficient of τ , one has X ′ 1 = [ 3σU2 0 − 2σU0 + η + αV0 + βV 2 0 (α+ U0 + βV0)2 ] U1 + αU0 + U2 0 (α+ U0 + βV 2 0 ) 2 V1 + cX1 + 2cσU0X0(1− 2U0). (3.13) Substituting the transformation (3.11) into the fourth equation of system (3.10) and comparing the coefficient of τ , we have Y ′ 1 = −γ δ αV0 + βV 2 0 (α+ U0 + βV0)2 U1 + (1 δ − γ δ αU0 + U2 0 (α+ U0 + βV0)2 ) V1 + c δ Y1. (3.14) Combining equations (3.12), (3.13) and (3.14), we obtain the following differential equation system determining U1, V1, X1 and Y1, dψ(ξ) dξ − P (ξ)ψ(ξ) = Q(ξ), (3.15) where ψ(ξ) =  U1(ξ) V1(ξ) X1(ξ) Y1(ξ)  , P (ξ) =  0 0 1 0 0 0 0 1 3σU2 0 − 2σU0 + η + αV0+βV 2 0 (α+U0+βV0)2 αU0+U2 0 (α+U0+βV 2 0 )2 c 0 −γ δ αV0+βV 2 0 (α+U0+βV0)2 1 δ − γ δ αU0+U2 0 (α+U0+βV0)2 0 c δ  , Q(ξ) = (0, 0, 2cσU0X0(1− 2U0), 0) T . Next we proof that the system (3.15) admits a solution satisfying (U1, V1, X1, Y1)(±∞) = 0. Let l = d dξ − P (ξ). (3.16) Denote L2 as the space of square integral functions, with inner production∫ +∞ −∞ (M(ξ), N(ξ))dξ, where M(ξ) = (U1(ξ), V1(ξ), X1(ξ), Y1(ξ)) T , N(ξ) = (U1(ξ), V1(ξ), X1(ξ), Y1(ξ)) T , and (·, ·) being the Euclidean inner product on R4. According to Fredholm or- thogonality theory, we have that the system (3.15) will have a solution if and only if ∫ +∞ −∞ (M(ξ), Q(ξ))dξ = 0, for all functions M(ξ) ∈ R4 in the kernel of the adjoint of the operator l. It can be readily verified that the adjoint operator l∗ is l∗ = − d dξ − PT (ξ), 10 Y. HUA, X. LIN, J. LIU, H. LU EJDE-2024/33 where PT (ξ) =  0 0 3σU2 0 − 2σU0 + η + αV0+βV 2 0 (α+U0+βV0)2 −γ δ αV0+βV 2 0 (α+U0+βV0)2 0 0 αU0+U2 0 (α+U0+βV 2 0 )2 1 δ − γ δ αU0+U2 0 (α+U0+βV0)2 1 0 c 0 0 1 0 c δ  . (3.17) To compute ker l∗, we need to find that all M(ξ) satisfying l∗M(ξ) = 0, i.e., dM(ξ) dξ = −PT (ξ)M(ξ). (3.18) Then the persistence question reduces to the solvability of equation (3.18). It is evident that the zero solution is a solution of the equation (3.18). Since the matrix PT (ξ) is nonconstant, it is difficult to find the general solution of equation (3.18). Nevertheless, we are only focusing on solutions that satisfyM(±∞) = 0, and in fact, the sole such solution is the zero solution. Recall that (U0(ξ), V0(ξ), X0(ξ), Y0(ξ)) is the solution of system (2.3) obtained in Lemma 2.1. Although we have no explicit expression for it, we know that (U0(ξ), V0(ξ), X0(ξ), Y0(ξ)) tends to P1(u1, 0, 0, 0) as ξ → −∞. Letting ξ → −∞, the matrix −PT finally becomes a constant matrix 0 0 3σu21 − 2σu1 + η 0 0 0 u1 α+u1 1 δ (1− γu1 α+u1 ) 1 0 c 0 0 1 0 c δ  (3.19) The characteristic equation of the above matrix is (λ2 − cλ− 3σu21 + 2σu1 − η)(δλ2 − cλ+ γu1 α+ u1 − 1) = 0. (3.20) By direct computation, it can be found that the equation (3.20) has three posi- tive real eigenvalues and one negative eigenvalue. Consequently, the sole solution that satisfies M(−∞) = 0 is the zero solution. This implies that the Fredholm orthogonality condition holds trivially∫ +∞ −∞ (M(ξ), Q(ξ))dξ = ∫ +∞ −∞ (0, Q(ξ))dξ = 0, and the solutions of the system (3.15) exist, which satisfies (U1, V1, X1, Y1)(±∞) = 0. Therefore, for 0 < τ ≪ 1, the system (3.10) exists one heteroclinic orbit (3.11) connecting the equilibrium P1(u1, 0, 0, 0) to P∗(u∗, v∗, 0, 0). This means that the system (1.8) with local strong delay kernel admits a traveling wave, and we obtain the following theorem. Theorem 3.1. If c ≥ 2 √ δ( γu1 α+u1 − 1), then for any sufficiently small τ > 0, the system (1.8) with local strong delay kernel admits a traveling wave (u, v)(x, t)= (U, V )(ξ) connecting from the steady-state E1(u1, 0) to E∗(u∗, v∗), where ξ = x+ct. Remark 3.2. The result for τ = 0 is presented in [7]. Theorem 3.1 indicates that the traveling wave connecting from the steady-state E1(u1, 0) to E∗(u∗, v∗) still exists for any sufficiently small τ > 0. EJDE-2024/33 TRAVELING WAVES FOR PREDATOR-PREY SYSTEMS 11 4. Traveling waves for the system with nonlocal delay This section focuses on studying the existence of traveling waves for the system (1.8) with nonlocal delay (1.6) by the geometric singular perturbation theory and the Fredholm orthogonality theory. We mainly consider the case of nonlocal strong delay kernel, i.e., taking f(x, t) = 1√ 4πt e− x2 4t t τ2 e −t/τ , the case of the nonlocal weak delay kernel can be studied similarly. 4.1. Perturbation analysis with nonlocal delay. We define ϕ(x, t) = ∫ t −∞ ∫ +∞ −∞ 1√ 4π(t− s) e− (x−y)2 4(t−s) t− s τ2 e− t−s τ u(y, s) dy ds. By a direct calculation, we have ∂ϕ ∂t − ∂2ϕ ∂x2 = 1 τ (κ− ϕ), (4.1) where κ(x, t) = ∫ t −∞ ∫ +∞ −∞ 1√ 4π(t− s) e− (x−y)2 4(t−s) 1 τ e− t−s τ u(y, s) dy ds. Similarly, it follows that ∂κ ∂t − ∂2κ ∂x2 = 1 τ (u− κ). (4.2) Combining equations (4.1) and (4.2), we obtain ϕtt = 2ϕtxx − ϕxxxx + 2 τ (ϕxx − ϕt) + 1 τ2 (u− ϕ). (4.3) Then system (1.8) can be reformulated as ut = uxx + σuϕ(1− ϕ)− ηu− uv α+ u+ βv , vt = δvxx + γuv α+ u+ βv − v, ϕtt = 2ϕtxx − ϕxxxx + 2 τ (ϕxx − ϕt) + 1 τ2 (u− ϕ). (4.4) Introducing the traveling wave coordinate (u, v, ϕ)(x, t) = (U, V, ϕ̄)(ξ), ξ = x+ ct, where the constant c > 0 is the wave speed, the system (4.4) is transformed into cU ′ = U ′′ + σUϕ̄(1− ϕ̄)− ηU − UV α+ U + βV , cV ′ = δV ′′ + γUV α+ U + βV − V, ϕ̄′′′′ − 2cϕ̄′′′ + c2ϕ̄′′ − 2 τ (ϕ̄′′ − cϕ̄′)− 1 τ2 (U − ϕ̄) = 0, (4.5) 12 Y. HUA, X. LIN, J. LIU, H. LU EJDE-2024/33 which can be rewritten as the eight-dimensional system U ′ = X, V ′ = Y, X ′ = cX − σUϕ̄(1− ϕ̄) + ηU + UV α+ U + βV , Y ′ = 1 δ [ cY − γUV α+ U + βV + V ] , ϕ̄′ = ϕ̄1, ϕ̄′1 = ϕ̄2, ϕ̄′2 = ϕ̄3, ϕ̄′3 = 2cϕ̄3 − c2ϕ̄2 + 2 τ (ϕ̄2 − cϕ̄1) + 1 τ2 (U − ϕ̄). (4.6) Set ε = √ τ and define the new variables Z = ϕ̄, Z1 = εϕ̄1, Z2 = ε2ϕ̄2, Z3 = ε3ϕ̄3, then system (4.6) becomes U ′ = X, V ′ = Y, X ′ = cX − σUZ(1− Z) + ηU + UV α+ U + βV , Y ′ = 1 δ [ cY − γUV α+ U + βV + V ] , εZ ′ = Z1, εZ ′ 1 = Z2, εZ ′ 2 = Z3, εZ ′ 3 = 2cεZ3 + (2− c2ε2)Z2 − 2cεZ1 + U − Z. (4.7) For small ε > 0, system (4.7) is a slow system in which (Z,Z1, Z2, Z3) are the fast variables and (U, V,X, Y ) are the slow variables. Let ε = 0, then system (4.7) is constrained to the critical manifold S0 = {(U, V,X, Y, Z, Z1, Z2, Z3) ∈ R8 : Z = U,Z1 = Z2 = Z3 = 0}. (4.8) For a point P ∈ S0, the matrix of the first partial derivatives with respect to the fast variables (Z,Z1, Z2, Z3) is J2 =  0 1 0 0 0 0 1 0 0 0 0 1 −1 0 2 0  . (4.9) A computation shows that the eigenvalues of the matrix J2 are −1,−1, 1, 1, hence the critical manifold S0 is normally hyperbolic. According to Lemma 2.2, we obtain that there exists a slow manifold Sε, O(ε) close and diffeomorphic to S0 for 0 < EJDE-2024/33 TRAVELING WAVES FOR PREDATOR-PREY SYSTEMS 13 ε≪ 1, which can be expressed as Sε = { (U, V,X, Y, Z, Z1, Z2, Z3) ∈ R8 : Z = U + f1(U, V,X, Y )ε+ f2(U, V,X, Y )ε2 +O(ε3), Z1 = g1(U, V,X, Y )ε+ g2(U, V,X, Y )ε2 +O(ε3), Z2 = h1(U, V,X, Y )ε+ h2(U, V,X, Y )ε2 +O(ε3), Z3 = r1(U, V,X, Y )ε+ r2(U, V,X, Y )ε2 +O(ε3) } , (4.10) where fi, gi, hi, ri (i = 1, 2) are smooth functions defined on a compact domain. Substituting equation (4.10) into system (4.7), we have εX + f ′1ε 2 + f ′2ε 3 +O(ε4) = g1ε+ g2ε 2 +O(ε3), g′1ε 2 + g′2ε 3 +O(ε4) = h1ε+ h2ε 2 +O(ε3), h′1ε 2 + h′2ε 3 +O(ε4) = r1ε+ r2ε 2 +O(ε3), r′1ε 2 + r′2ε 3 +O(ε4) = (2h1 − f1)ε+ (2cr1 + 2h2 − 2cg1 − f2)ε 2 +O(ε3). Comparing the coefficients of ε and ε2, one has f1 = 0, f2 = 2UV α+ U + βV − 2(σU − σU2 − η)U, h1 = 0, h2 = cX − (σU − σU2 − η)U + UV α+ U + βV , g1 = X, g2 = 0, r1 = 0, r2 = 0. Thus system (4.7) restricted to Sε can be reformulated as U ′ = X, V ′ = Y, X ′ = cX − σU(U + f2ε 2)(1− U − f2ε 2) + ηU + UV α+ U + βV +O(ε3), Y ′ = 1 δ [cY − γUV α+ U + βV + V ], (4.11) which is a regular perturbation of the system (2.3). It is evident that the system (4.11) is simplified to system (2.3) when ε = 0. 4.2. Analysis by the Fredholm orthogonality theory. This section we prove that system (4.11) admits a heteroclinic orbit connecting equilibrium P1(u1, 0, 0, 0) to P∗(u∗, v∗, 0, 0) for 0 < ε≪ 1. Let (U0, V0, X0, Y0)(ξ) be the heteroclinic orbit of system (2.3) obtained in Lemma 2.1, connecting the equilibrium P1(u1, 0, 0, 0) to P∗(u∗, v∗, 0, 0). To solve system (4.11) for 0 < τ ≪ 1, we set U = U0 + Ū1ε 2 +O(ε3), V = V0 + V̄1ε 2 +O(ε3), X = X0 + X̄1ε 2 +O(ε3), Y = Y0 + Ȳ1ε 2 +O(ε3), (4.12) 14 Y. HUA, X. LIN, J. LIU, H. LU EJDE-2024/33 Substituting the transformation (4.12) into system (4.11) and comparing the coef- ficient of ε2, the differential equation system determining Ū1, V̄1, X̄1, Ȳ1 is dψ̄(ξ) dξ − P̄ (ξ)ψ̄(ξ) = Q̄(ξ), (4.13) where ψ̄(ξ) =  Ū1(ξ) V̄1(ξ) X̄1(ξ) Ȳ1(ξ)  , P̄ (ξ) =  0 0 1 0 0 0 0 1 3σU2 0 − 2σU0 + η + αV0+βV 2 0 (α+U0+βV0)2 αU0+U2 0 (α+U0+βV 2 0 )2 c 0 −γ δ αV0+βV 2 0 (α+U0+βV0)2 1 δ − γ δ αU0+U2 0 (α+U0+βV0)2 0 c δ  , Q̄(ξ) = (0, 0, σ(2U2 0 − U0)f2(U0, V0, X0, Y0), 0) T . By an analysis similar to the one in Section 3.2, it follows that system (4.11) admits a heteroclinic orbit (4.12) connecting the equilibrium P1(u1, 0, 0, 0) to P∗(u∗, v∗, 0, 0) for 0 < ε≪ 1. Therefore, the system (1.8) with nonlocal strong delay kernel admits a traveling wave, and we obtain the following theorem. Theorem 4.1. If c ≥ 2 √ δ( γu1 α+u1 − 1), then for any sufficiently small τ > 0, the system (1.8) with nonlocal strong delay kernel admits a traveling wave (u, v)(x, t)= (U, V )(ξ) connecting from the steady-state E1(u1, 0) to E∗(u∗, v∗), where ξ = x+ct. 5. Asymptotic behavior This section focus on analyzing the asymptotic behaviors of traveling waves for the system (1.2) with the method of the asymptotic theory [6]. Let φ(ξ) = (u, v)(x, t) = (U, V )(ξ) be the traveling wave of system (1.2) satisfying the boundary conditions (U, V )(−∞) = E1(u1, 0), (U, V )(+∞) = E∗(u∗, v∗). (5.1) Differentiating the system (2.2) with respect to ξ and denoting φ′(ξ) = (U ′, V ′)(ξ) = (φ1, φ2)(ξ), we have cφ′ 1 = φ′′ 1 + (2σU − 3σU2 − η)φ1 − (αV + βV 2)φ1 + (αU + U2)φ2 (α+ U + βV )2 , cφ′ 2 = δφ′′ 2 + γ (αV + βV 2)φ1 + (αU + U2)φ2 (α+ U + βV )2 − φ2. (5.2) Note that the traveling wave φ(ξ) satisfies the boundary conditions (5.1), hence the limiting system of system (5.2) as ξ → −∞ is cφ′ 1− = φ′′ 1− − (σu1 − 2η)φ1− − u1 α+ u1 φ2−, cφ′ 2− = δφ′′ 2− + ( γu1 α+ u1 − 1)φ2−, (5.3) EJDE-2024/33 TRAVELING WAVES FOR PREDATOR-PREY SYSTEMS 15 which is equivalent to the system φ′ 1− = φ3−, φ′ 2− = φ4−, φ′ 3− = (σu1 − 2η)φ1− + u1 α+ u1 φ2− + cφ3−, φ′ 4− = 1 δ [ (1− γu1 α+ u1 )φ2− + cφ4− ] . (5.4) The system (5.4) can be rewritten as T ′ − = A1T−, (5.5) where T−(ξ) =  φ1−(ξ) φ2−(ξ) φ3−(ξ) φ4−(ξ)  , A1 =  0 0 1 0 0 0 0 1 σu1 − 2η u1 α+u1 c 0 0 1 δ (1− γu1 α+u1 ) 0 c δ  . (5.6) The eigenvalues of the matrix A1 are Λ1 = c+ √ c2 + 4σu1 − 8η 2 , Λ2 = c+ √ c2 +∆(u1) 2δ , Λ3 = c− √ c2 +∆(u1) 2δ , Λ4 = c− √ c2 + 4σu1 − 8η 2 , where ∆(u1) = 4δ(1− γu1 α+u1 ). It is evident that Λi > 0 (i = 1, 2, 3), and Λ4 < 0 for u1 = σ+ √ σ2−4ση 2σ , σ > 4η, and c ≥ 2 √ δ( γu1 α+u1 − 1) > 0. The general solutions of the system (5.5) can be expressed as (φ1−, φ2−, φ3−, φ4−) T (ξ) = 4∑ i=1 aiρie Λiξ, where ρi are the corresponding eigenvectors to the eigenvalues Λi (i = 1, 2, · · · , 4), and ai (i = 1, 2, · · · , 4) are arbitrary constants. Since (φ1−, φ2−, φ3−, φ4−) T (−∞) = (0, 0, 0, 0), we have a4 = 0 and (φ1−, φ2−, φ3−, φ4−) T (ξ) = 3∑ i=1 aiρie Λiξ. Therefore, we obtain the asymptotic behavior( φ1− φ2− ) = (∑3 i=1 µi(mi + o(1))eΛiξ∑3 i=1 µi(ni + o(1))eΛiξ ) , ξ → −∞, (5.7) where mi, ni (i = 1, 2, 3) are constants, and µi (i = 1, 2, 3) can not be zero simul- taneously, see [6]. On the other hand, the limiting system of system (5.2) as ξ → +∞ is cφ′ 1+ = φ′′ 1+ − (σu∗ − 2γ + 1 γ k∗ − 2η)φ1+ + βk∗ − 1 γ φ2+, cφ′ 2+ = δφ′′ 2+ + (γk∗ − k∗)φ1+ − βk∗φ2+, (5.8) 16 Y. HUA, X. LIN, J. LIU, H. LU EJDE-2024/33 where k∗ = v∗ γu∗ = (γ − 1)u∗ − α γβu∗ > 0. (5.9) Then system (5.8) is equivalent to φ′ 1+ = φ3+, φ′ 2+ = φ4+, φ′ 3+ = (σu∗ − 2γ + 1 γ k∗ − 2η)φ1+ + 1− βk∗ γ φ2+ + cφ3+, φ′ 4+ = 1 δ [ (k∗ − γk∗)φ1+ + βk∗φ2+ + cφ4+ ] , (5.10) which can be reformulated as T ′ + = A2T+, (5.11) where T+(ξ) =  φ1+(ξ) φ2+(ξ) φ3+(ξ) φ4+(ξ)  , A2 =  0 0 1 0 0 0 0 1 σu∗ − 2γ+1 γ k∗ − 2η 1−βk∗ γ c 0 1−γ δ k∗ β δ k ∗ 0 c δ  . (5.12) The general solutions of system (5.11) has the form (φ1+, φ2+, φ3+, φ4+) T (ξ) = 4∑ i=1 ãiρ̃ie Λ̃iξ, where ρ̃i are the corresponding eigenvectors to the eigenvalues Λ̃i (i = 1, 2, · · · , 4), and ãi (i = 1, 2, · · · , 4) are arbitrary constants. The characteristic equation of the matrix A2 is Λ̃4 − (c+ c δ )Λ̃3 +K2Λ̃ 2 +K1Λ̃ +K0 = 0, where K0 = [ σβu∗ − (3β + 1 γ )k∗ − 2βη + 1 ]k∗ δ , K1 = [ σu∗ + (β − 1 γ − 2)k∗ − 2η ] c δ , K2 = −σu∗ + (2 + 1 γ − β δ )k∗ + 2η + c2 δ . According to the Viete theorem, the equality (5.9) and u∗ is a positive root of the cubic equation (2.4), we have Λ̃1 + Λ̃2 + Λ̃3 + Λ̃4 = c+ c δ > 0, Λ̃1Λ̃2Λ̃3Λ̃4 = K0 = σβk∗ δu∗ ∆(u∗), where ∆(u∗) = (u∗) 2 − 2ηγ2β2 + 4γ2β − 3γβ + γ − 1 σγ2β2 u∗ + (3γβ + 1)α σγ2β2 . EJDE-2024/33 TRAVELING WAVES FOR PREDATOR-PREY SYSTEMS 17 It is evident that Λ̃1Λ̃2Λ̃3Λ̃4 and ∆(u∗) have the same symbol. For simplicity, we shall only consider the situation of ∆(u∗) < 0 here. The situation of ∆(u∗) ≥ 0 can be discussed similarly. When ∆(u∗) < 0, the distribution of eigenvalues involves four cases: Case 1: Re(Λ̃1) = Re(Λ̃2) > 0, Λ̃3 < 0, Λ̃4 > 0, or Λ̃1 ≥ Λ̃2 ≥ Λ̃4 > 0, Λ̃3 < 0. Since (φ1+, φ2+, φ3+, φ4+) T (+∞) = (0, 0, 0, 0), (5.13) we have ã1 = ã2 = ã4 = 0 and (φ1+, φ2+, φ3+, φ4+) T (ξ) = ã3ρ̃3e Λ̃3ξ. Therefore, we deduce the asymptotic behavior( φ1+ φ2+ ) = ( µ̃3(m̃3 + o(1))eΛ̃3ξ µ̃3(ñ3 + o(1))eΛ̃3ξ ) , ξ → +∞, (5.14) where m̃3, ñ3 are two constants, and µ̃3 can not be zero simultaneously. Case 2: Re(Λ̃1) = Re(Λ̃2) < 0, Λ̃3 < 0, Λ̃4 > 0, or Λ̃1 < Λ̃2 < Λ̃3 < 0, Λ̃4 > 0. From condition (5.13), we obtain ã4 = 0 and (φ1+, φ2+, φ3+, φ4+) T (ξ) = 3∑ i=1 ãiρ̃ie Λ̃iξ. Thus, we achieve the asymptotic behavior( φ1+ φ2+ ) = (∑3 i=1 µ̃i(m̃i + o(1))eΛ̃iξ∑3 i=1 µ̃i(ñi + o(1))eΛ̃iξ ) , ξ → +∞, (5.15) where m̃i, ñi (i = 1, 2, 3) are constants, and µ̃i (i = 1, 2, 3) can not be zero simul- taneously. Case 3: Λ̃1 = Λ̃2 < 0, Λ̃3 < 0, Λ̃4 > 0. From condition (5.13), it follows that ã4 = 0 and (φ1+, φ2+, φ3+, φ4+) T (ξ) = 2∑ i=1 ãiρ̃i{I + (A2 − Λ̃iI)ξ}eΛ̃iξ + ã3ρ̃3e Λ̃3ξ. Thus, we achieve the asymptotic behavior( φ1+ φ2+ ) = ( µ̃3(m̃3 + o(1))eΛ̃3ξ + ∑2 i=1 µ̃i{(m̃i + o(1)) + (m̃ii + o(1))ξ}eΛ̃iξ µ̃3(ñ3 + o(1))eΛ̃3ξ + ∑2 i=1 µ̃i{(ñi + o(1)) + (ñii + o(1))ξ}eΛ̃iξ ) , as ξ → +∞, where m̃i, ñi, m̃ii, ñii (i = 1, 2, 3) are constants, and µ̃i (i = 1, 2, 3) can not be zero simultaneously. Case 4: Λ̃1 = Λ̃2 = Λ̃3 < 0, Λ̃4 > 0. By condition (5.13), one has ã4 = 0 and (φ1+, φ2+, φ3+, φ4+) T (ξ) = 3∑ i=1 ãiρ̃i{I + (A2 − Λ̃iI)ξ}eΛ̃iξ. Consequently, we obtain the asymptotic behavior( φ1+ φ2+ ) = (∑3 i=1 µ̃i{(m̃i + o(1)) + (m̃ii + o(1))ξ}eΛ̃iξ∑3 i=1 µ̃i{(ñi + o(1)) + (ñii + o(1))ξ}eΛ̃iξ ) , ξ → +∞, (5.16) where m̃i, ñi, m̃ii, ñii (i = 1, 2, 3) are constants, and µ̃i (i = 1, 2, 3) can not be zero simultaneously. From equalities (5.7) and (5.14)-(5.16), we have the following result. 18 Y. HUA, X. LIN, J. LIU, H. LU EJDE-2024/33 Theorem 5.1. If c ≥ 2 √ δ( γu1 α+u1 − 1) and ∆(u∗) < 0, then there exist constants R, R̄, L1, L̄1, L2, L̄2 such that system (1.2) admits a traveling wave φ(ξ) = (u, v)(x, t) = (U, V )(ξ) with the following asymptotic properties: φ(ξ) = ( u1 + (R+ o(1))eΛiξ (R̄+ o(1))eΛiξ ) , ξ → −∞, (5.17) where Λi can be Λ1,Λ2,Λ3. For cases 1 and case 2, we have φ(ξ) = ( u∗ − (L1 + o(1))eΛ̃jξ v∗ − (L2 + o(1))eΛ̃jξ ) , ξ → +∞, (5.18) where for case 1, Λ̃j is Λ̃3, and for case 2, Λ̃j can be one of Λ̃1, Λ̃2, Λ̃3. For cases 3 and case 4, we have φ(ξ) = ( u∗ − ((L1 + o(1))eΛ̃jξ + (L̄1 + o(1))ξeΛ̃1ξ) v∗ − ((L2 + o(1))eΛ̃jξ + (L̄2 + o(1))ξeΛ̃1ξ) ) , ξ → +∞, (5.19) where for case 3, Λ̃j can be one of Λ̃1, Λ̃3, and for case 4, Λ̃j is Λ̃1. The asymptotic behaviors of traveling waves for the system (1.8) with local delay kernels (1.5) or nonlocal delay kernels (1.7) can be investigated similarly. Acknowledgments. This work was supported by the Natural Science Foundation of China (Grant No. 12271220). References [1] M. Alfaro, L. Girardin, F. Hamel, et al.; When the Allee threshold is an evolutionary trait: Persistence vs. extinction, J. Math. Pures Appl. (9), 155 (2021), 155-191. [2] W. Allee; Animal Aggregations: A study in General Sociology, University of Chicago Press, (1931). [3] P. Amster, G. 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Differential Equations, 2022 (82) (2022), 1-16. [34] Z. Zhu, X. Liu; Existence of traveling wave solutions in a singularly perturbed predator-prey equation with spatial diffusion, Discrete Contin. Dyn. Syst., 44 (2024), 78-115. Yang Hua School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China Email address: huayangjs0924@163.com Xiaojie Lin (corresponding author) School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China Email address: linxiaojie1973@163.com Jiang Liu School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China Email address: jiangliu@jsnu.edu.cn Haixia Lu School of Life Sciences, Jiangsu Normal University, Xuzhou 221116, China Email address: luhaixia76@126.com 1. Introduction 2. Preliminaries 3. Traveling waves for the system with local delay 3.1. Perturbation analysis with local delay 3.2. Analysis by the Fredholm orthogonality theory 4. Traveling waves for the system with nonlocal delay 4.1. Perturbation analysis with nonlocal delay 4.2. Analysis by the Fredholm orthogonality theory 5. Asymptotic behavior Acknowledgments References