Loading [MathJax]/jax/element/mml/optable/GeneralPunctuation.js
Research article Special Issues

Representations of modified type 2 degenerate poly-Bernoulli polynomials

  • Research on the degenerate versions of special polynomials provides a new area, introducing the λ-analogue of special polynomials and numbers, such as λ-Sheffer polynomials. In this paper, we propose a new variant of type 2 Bernoulli polynomials and numbers by modifying a generating function. Then we derive explicit expressions and their representations that provide connections among existing λ-Sheffer polynomials. Also, we provide the explicit representations of the proposed polynomials in terms of the degenerate Lah-Bell polynomials and the higher-order degenerate derangement polynomials to confirm the presented identities.

    Citation: Jongkyum Kwon, Patcharee Wongsason, Yunjae Kim, Dojin Kim. Representations of modified type 2 degenerate poly-Bernoulli polynomials[J]. AIMS Mathematics, 2022, 7(6): 11443-11463. doi: 10.3934/math.2022638

    Related Papers:

    [1] Xiaoying Wang, Xingfu Zou . Pattern formation of a predator-prey model with the cost of anti-predator behaviors. Mathematical Biosciences and Engineering, 2018, 15(3): 775-805. doi: 10.3934/mbe.2018035
    [2] Xin-You Meng, Yu-Qian Wu . Bifurcation analysis in a singular Beddington-DeAngelis predator-prey model with two delays and nonlinear predator harvesting. Mathematical Biosciences and Engineering, 2019, 16(4): 2668-2696. doi: 10.3934/mbe.2019133
    [3] Shiqiang Feng, Dapeng Gao . Existence of traveling wave solutions for a delayed nonlocal dispersal SIR epidemic model with the critical wave speed. Mathematical Biosciences and Engineering, 2021, 18(6): 9357-9380. doi: 10.3934/mbe.2021460
    [4] Renji Han, Binxiang Dai, Lin Wang . Delay induced spatiotemporal patterns in a diffusive intraguild predation model with Beddington-DeAngelis functional response. Mathematical Biosciences and Engineering, 2018, 15(3): 595-627. doi: 10.3934/mbe.2018027
    [5] G.A.K. van Voorn, D. Stiefs, T. Gross, B. W. Kooi, Ulrike Feudel, S.A.L.M. Kooijman . Stabilization due to predator interference: comparison of different analysis approaches. Mathematical Biosciences and Engineering, 2008, 5(3): 567-583. doi: 10.3934/mbe.2008.5.567
    [6] Ran Zhang, Shengqiang Liu . Traveling waves for SVIR epidemic model with nonlocal dispersal. Mathematical Biosciences and Engineering, 2019, 16(3): 1654-1682. doi: 10.3934/mbe.2019079
    [7] Xixia Ma, Rongsong Liu, Liming Cai . Stability of traveling wave solutions for a nonlocal Lotka-Volterra model. Mathematical Biosciences and Engineering, 2024, 21(1): 444-473. doi: 10.3934/mbe.2024020
    [8] Paulo Amorim, Bruno Telch, Luis M. Villada . A reaction-diffusion predator-prey model with pursuit, evasion, and nonlocal sensing. Mathematical Biosciences and Engineering, 2019, 16(5): 5114-5145. doi: 10.3934/mbe.2019257
    [9] Jian Fang, Na Li, Chenhe Xu . A nonlocal population model for the invasion of Canada goldenrod. Mathematical Biosciences and Engineering, 2022, 19(10): 9915-9937. doi: 10.3934/mbe.2022462
    [10] Yan Wang, Minmin Lu, Daqing Jiang . Viral dynamics of a latent HIV infection model with Beddington-DeAngelis incidence function, B-cell immune response and multiple delays. Mathematical Biosciences and Engineering, 2021, 18(1): 274-299. doi: 10.3934/mbe.2021014
  • Research on the degenerate versions of special polynomials provides a new area, introducing the λ-analogue of special polynomials and numbers, such as λ-Sheffer polynomials. In this paper, we propose a new variant of type 2 Bernoulli polynomials and numbers by modifying a generating function. Then we derive explicit expressions and their representations that provide connections among existing λ-Sheffer polynomials. Also, we provide the explicit representations of the proposed polynomials in terms of the degenerate Lah-Bell polynomials and the higher-order degenerate derangement polynomials to confirm the presented identities.



    Nowadays predator-prey models have been widely applied in biological and ecological phenomena. The most general prey-predator population model is represented by

    {˙x(t)=xG(x)yP(x,y),˙y(t)=yH(x,y),

    where x(t) and y(t) denote the density of the prey and predator at time t, respectively. G(x) is the per capita growth rate of the prey in the absence of predator, P(x,y) represents the functional response of predators and H(x,y) measures the growth rate of predators.

    A prototype of G(x) is the logistic growth pattern of G(x)=r(1xN), where r>0 denotes the prey intrinsic growth rate and N means the carrying capacity in the absence of predator [1]. One of known growth rate of predators is the Leslie-Gower type: H(x,y)=α(1kyx) [2,3], where α is the intrinsic growth rates of predator and k is the conversion factor of prey into predators.

    Lotka-Volterra response was used by Lotka [4] in studying a hypothetical chemical reaction and by Volterra [5] in modeling a predator-prey interaction. Lotka-Volterra response function is a straight line through the origin and is unbounded. The solutions of Lotka-Volterra model are not structural stable, thus a small perturbation can have a very marked effect [6]. The Holling-type Ⅱ functional responses function is P(x,y)=cxa+bx, where c is the maximum number of prey consumed per predator per unit time [7,8]. When a=1 and b=0, the functional response is of Lotka-Volterra type. In 1975, Beddington [9] and DeAngelis et al. [10] developed a predator-prey model of the mutual interference effects, in which the relationship between predators' searching efficiency and both prey and predator is presented. The Beddington-DeAngelis (B-D) functional response is defined by

    P(x,y)=sx1+ax+by,

    where s,a,b>0, s is the consumption rate, a means the saturation constant for an alternative prey and b stands for the predator interference. The predator-prey models with the B-D functional response have been well-studied in the literature, for example, see [11,12,13,14] and references therein.

    From the view of human needs, the exploitation of biological resources and harvest of population are commonly practised in the fields of fishery, wildlife, and forestry management. Many mathematical models have been proposed and developed to better describe the relationship between predator and prey populations by taking into account the harvesting, for instance, see [14,15,16,17,18]. In a very general way, harvesting for predator-prey models can be divided into three types. If the harvesting function h(t) is a constant, it is called constant-rate or constant yield harvesting. It arises when a quota is specified (for example, through permits, as in deer hunting seasons in many areas, or by agreement as sometimes occurring in whaling) [19,20]. If the function h(t) is a linear function of population size, it is called proportional or constant-effort harvesting [16,17,18]. The harvesting function h(t) can be of nonlinear form, for example, one of which is the so-called Michaelis-Menten type harvesting used in ecology and economics [21,22].

    Movements of some individuals usually cannot be restricted to a small area, and they are often free, so integral operators have been widely applied to model the long-distance dispersal problem [23]. That is, the diffusion process depends on the distance between two niches of population, such as the model:

    ut(x,t)=RJ(xy)(u(y,t)u(x,t))dy+f(u),

    where RJ(xy)(u(y,t)u(x,t))dy represents the nonlocal dispersal process [24,25]. Such model arises not only in biological phenomena, but also in many other fields, such as phase transition modelling [25,26,27,28].

    There is, however, considerable evidence that time delay should not be neglected in biological and ecological phenomena. The growth rate of population of species and the response of one species to the interactions with other species are mediated by some time delay. Other causes of response delays include differences in resource consumption with respect to age structure, migration and diffusion of populations, gestation and maturation periods, delays in behavioral response to environmental changes, and dependence of a population on a food supply that requires time to recover from grazing [15,25]. Hence, in order to make the modeling of interactions between predator and prey more realistic, time delay is often necessarily incorporated into predator-prey models [22,28,29,30,31].

    The purpose of this paper is to study the existence and nonexistence of traveling wave solution of a nonlocal dispersal delayed predator-prey model with the B-D functional response and harvesting:

    {ut=d1((Ju)(x,t)u(x,t))+ru(x,t)(1u(x,t)K)su(x,t)v(x,tτ)1+au(x,t)+bv(x,tτ)qu(x,t),vt=d2((Jv)(x,t)v(x,t))+v(x,t)(αβv(x,t)u(x,tτ)), (1.1)

    where

    (Jw)(x,t)=RJ(y)w(xy,t)dy,

    q represents the prey harvesting, τ denotes the time delay, and a,b,r,d1,d2,s,K,α and β are positive real constants. To reduce the number of parameters in system (1.1), we make the following transformations:

    ˉt=rt, ˉτ=rτ, ˉu=uK, ˉv=svr, ¯d1=d1r, ¯d2=d2r,
    ˉa=aK, ˉb=rbs, ˉα=αr, ˉβ=βsK, ˉq=qr.

    For the sake of convenience, we ignore the bars on u,v and other parameters, then system (1.1) can be re-expressed as

    {ut=d1(Juu)+u(1u)uv(x,tτ)1+au+bv(x,tτ)qu,vt=d2(Jvv)+v(αβvu(x,tτ)). (1.2)

    Biologically, we require 0<q<1. It is easy to see that system (1.2) has two spatially constant equilibria (1q,0) and (u,v), where u=(1q)κβα+(κβα)2+4βκ2κ, v=αuβ and κ=(aβ+bα)(1q).

    In biology and ecology, traveling wave solutions are often used to describe the spatial-temporal process where the predator invades the territory of prey and they eventually coexist [25]. A solution of system (1.2) is called a traveling wave with the speed c>0 if there exist positive function ϕ1 and ϕ2 defined on R such that

    u(x,t)=ϕ1(z),v(x,t)=ϕ2(z),z=x+ct.

    Here ϕ1 and ϕ2 represent the wave profiles and (ϕ1, ϕ2) satisfies the resultant system:

    {cϕ1(z)=d1(Jϕ1(z)ϕ1(z))+ϕ1(z)(1ϕ1(z))ϕ1(z)ϕ2(zcτ)1+aϕ1(z)+bϕ2(zcτ)qϕ1(z),cϕ2(z)=d2(Jϕ2(z)ϕ2(z))+ϕ2(z)(αβϕ2(z)ϕ1(zcτ)), (1.3)

    and

    Jϕ(z)=RJ(y)ϕ(zy)dy.

    Our primary interest lies in the traveling wave solution of system (1.3) connecting (1q, 0) and (u, v) with the asymptotic behavior:

    limz(ϕ1(z),ϕ2(z))=(1q,0),   limz+(ϕ1(z),ϕ2(z))=(u,v). (1.4)

    The asymptotic behavior of traveling wave solution plays an important role in dispersion models of biological populations, because it describes the propagation processes of different species and enables us to understand how some species migrate from one area into another area until the density attains a certain value.

    Recently, the existence of traveling wave solution for the nonlocal dispersal systems with the time delay has been extensively studied [28,29,30,31,32,33]. We can see that system (1.3) is non-monotone system and Schauder's fixed point theorem is a quiet powerful technique for constructing a suitable invariant set (see, for example [31,33,34,35,36]). To explore the existence of traveling wave solution of nonlocal dispersal systems with c>c, we need to construct an invariant cone in a large bounded domain with the initial functions [33,34,35], where the nonlocal dispersal kernel function J is assumed to be compactly supported. For the existence of traveling wave solution at the critical point c=c, Corduneanu's theorem and the limiting method are useful techniques [33,36].

    Throughout this paper, for the nonlocal dispersal kernel function J of system (1.3), we make the following assumptions:

    (G1) J is a smooth function in R, Lebesgue measurable with JC1(R) and

    J(x)=J(x)0,  RJ(x)dx=1.

    (G2) RJ(x)eλxdx<+, λR.

    For convenience, we assume the parameters of system (1.3) satisfying

    0<d1d2,0<q<1,b>1,a>1q,0<bαβ.

    The rest of this paper is structured as follows. We construct an appropriate pair of upper-lower solutions of system (1.3) for c>c in Section 2. We apply Schauder's fixed point theorem to investigate the existence of traveling wave solution for c>c and develop the contracting rectangles method to study the asymptotic behavior of system (1.3) in Section 3. The existence of traveling wave solution for c=c is discussed by means of Corduneanu's theorem and Lebesgue's dominated convergence theorem in Section 4. Section 5 is dedicated to the nonexistence of traveling wave for 0<c<c. A brief conclusion is given in Section 6.

    Definition 2.1. Assume that Z:={z1,z2,,zm}R contains finite points of R. We say that the functions (¯ϕ1, ¯ϕ2) and (ϕ_1, ϕ_2) are a pair of upper-lower solutions of system (1.3), if for any zRZ, ¯ϕi(z) and ϕ_i(z) (i=1,2) are bounded and continuous such that

    {F(¯ϕ1,ϕ_2)(z)=d1(J¯ϕ1(z)¯ϕ1(z))c¯ϕ1(z)+¯ϕ1(z)(1¯ϕ1(z)) ¯ϕ1(z)ϕ_2(zcτ)1+a¯ϕ1(z)+bϕ_2(zcτ)q¯ϕ1(z)0,F(ϕ_1,¯ϕ2)(z)=d1(Jϕ_1(z)ϕ_1(z))cϕ_1(z)+ϕ_1(z)(1ϕ_1(z)) ϕ_1(z)¯ϕ2(zcτ)1+aϕ_1(z)+b¯ϕ2(zcτ)qϕ_1(z)0,F(¯ϕ1,¯ϕ2)(z)=d2(J¯ϕ2(z)¯ϕ2(z))c¯ϕ2(z)+¯ϕ2(z)(αβ¯ϕ2(z)¯ϕ1(zcτ))0,F(ϕ_1,ϕ_2)(z)=d2(Jϕ_2(z)ϕ_2(z))cϕ_2(z)+ϕ_2(z)(αβϕ_2(z)ϕ_1(zcτ))0. (2.1)

    Define

    fσ(d,c,λ)=d(RJ(y)eλydy1)cλ+σ,

    where σ0. By a direct calculation, for c>0 and λ>0 we have

    (F1) f0(d1,c,0)=0 and fα(d2,c,0)>0;

    (F2) fσc=λ<0, fσλ|λ=0=c<0 and fσd=RJ(y)eλydy1>0;

    (F3) 2fσλ2>0.

    From (F1)–(F3), it follows that there exist c>0 and λ>0 such that [35]

    fα(d2,c,λ)=0andfα(d2,c,λ)λ|(c,λ)=0.

    Lemma 2.1. There exist c>c and positive constants 0<λ2<λ<λ3<λ1 such that

    f0(d1,c,λ){=0λ=0, λ=λ1>0λ(λ1,+)<0λ(0,λ1),fα(d2,c,λ){=0λ=λ2, λ=λ3>0λ(0,λ2)(λ3,+)<0λ(λ2,λ3).

    Proof. We only need to show λ1>λ3. It is easy to see fα(d2,c,λ3)=f0(d1,c,λ1)=0 and f0(d1,c,λ1)<fα(d1,c,λ1). Due to d2d1 and (F2), we have fα(d1,c,λ1)fα(d2,c,λ1). It indicates fα(d2,c,λ3)<fα(d2,c,λ1), i.e., λ1>λ3.

    Now, we will construct an appropriate pair of upper-lower solutions for system (1.3). We fix c>c. For any given constant m>1, it is easy to check that the function

    g(z)=eλ2zmeθz

    has a unique zero point at z0=lnmθλ2 where θ(λ2,min{2λ2,λ3}), and a unique maximum point at zM=lnmλ2(θλ2)<z0. Clearly, g is continuous on R and positive on (,z0). For any given yR we let

    Θ(z)=zJ(yx)g(x)dxg(z)2

    with z[zM,z0]. Since Θ(z) is nondecreasing for z[zM,z0] and Θ(z0)>0, we can find a sufficiently small δ(0,α(b1)β) and z2(zM,z0) such that

    δ=g(z2),Θ(z2)>0.

    Let p and m satisfy the following conditions:

    (A1) p>1f0(d1,c,λ2).

    (A2) m>β(b1)fα(d2,c,θ).

    Then, we introduce ¯ϕ1(z),¯ϕ2(z),ϕ_1(z),ϕ_2(z) as follows:

    ¯ϕ1(z)=1q  zR, (2.2)
    ϕ_1(z)={(1q)(11b)zz1,(1q)(11b(eλ1z+peλ2z))zz1, (2.3)
    ¯ϕ2(z)={1qb z0,1qbeλ2z z0, (2.4)
    ϕ_2(z)={1qbδzz2,1qb(eλ2zmeθz)zz2, (2.5)

    where z1<0 is defined by eλ1z1+peλ2z1=1.

    Lemma 2.2. Assume c>c. Then (¯ϕ1, ¯ϕ2) and (ϕ_1, ϕ_2) defined by (2.2)–(2.5) are a pair of upper-lower solutions of system (1.3).

    Proof. Firstly, we show that

    F(¯ϕ1,ϕ_2)(z)0

    holds for zR. For any zR, we have ¯ϕ1(z)=1qE and

    F(¯ϕ1,ϕ_2)(z)=(1q)q(1q)ϕ_2(zcτ)1+a(1q)+bϕ_2(zcτ)q(1q)=(1q)ϕ_2(zcτ)1+a(1q)+bϕ_2(zcτ)0.

    For zz1, we would like to show that

    F(ϕ_1,¯ϕ2)(z)0.

    When z>z1, we have ϕ_1(z)=(1q)(11b),¯ϕ2(z)1qb and

    F(ϕ_1,¯ϕ2)(z)(1q)b1b[1(1q)b1b1qb+a(1q)(b1)+b(1q)q]0.

    In view of f0(d1,c,λ2)<f0(d2,c,λ2)<fα(d2,c,λ2)=0 and (A1), for z<z1<0 we have ϕ_1=(1q)[11b(eλ1z+peλ2z)], ¯ϕ2=1qbeλ2z and

    F(ϕ_1,¯ϕ2)(z)d1(RJ(y)(1q)[11b(eλ1(zy)+peλ2(zy))]dy(1q)[11b(eλ1z+peλ2z)])+(1q)cb(λ1eλ1z+pλ2eλ2z)+(1q)2[11b(eλ1z+peλ2z)](1q)2[11b(eλ1z+peλ2z)]2ϕ_1¯ϕ2(zcτ)1+aϕ_1+b¯ϕ2=1qbeλ1z[d1(RJ(y)eλ1ydy1)cλ1]1qbpeλ2z[d1(RJ(y)eλ2ydy1)cλ2]+(1q)2b(eλ1z+peλ2z)[11b(eλ1z+peλ2z)]ϕ_1¯ϕ2(zcτ)1+aϕ_1+b¯ϕ2(zcτ)>1qbpeλ2z[d1(RJ(y)eλ2ydy1)cλ2]¯ϕ2(zcτ)=1qbpeλ2z[d1(RJ(y)eλ2ydy1)cλ2]1qbeλ2(zcτ)=1qbeλ2z[(p)(d1(RJ(y)eλ2ydy1)cλ2)eλ2cτ]>1qbeλ2z[(p)f0(d1,c,λ2)1]>0. (2.6)

    Now, we show

    F(¯ϕ1,¯ϕ2)(z)0

    for z0. In the case of z>0, we have ¯ϕ1=1q and ¯ϕ2=1qb. Then

    F(¯ϕ1,¯ϕ2)(z)1qb[αβ1qb1q]=1qb[αβb]0.

    For z<0, we obtain ¯ϕ2=1qbeλ2z and

    F(¯ϕ1,¯ϕ2)(z)d2(RJ(y)1qbeλ2(zy)dy1qbeλ2z)1qbcλ2eλ2z+1qbeλ2z[αβbeλ2z]=1qbeλ2z[d2(RJ(y)eλ2ydy1)cλ2+α]β(1q)b2e2λ2z=β(1q)b2e2λ2z0.

    Finally, to show

    F(ϕ_1,ϕ_2)(z)0

    for zz2, we use the inequality ϕ_1(1q)(11b) and ϕ_2=1qbδ if z>z2. Then

    F(ϕ_1,ϕ_2)(z)d2(1q)b[+zz2J(y)(eλ2(zy)meθ(zy))dy+zz2J(y)δdyδ]+1qbδ[αβ(1q)(11b)1qbδ]d2(1q)b(z2J(zy)(eλ2zmeθz)dyδ2)+1qbδ[αβδb1]=d2(1q)bΘ(z2)+1qbδ[αβδb1]1qbδ[αβδb1]0,

    due to 0<δ<α(b1)β.

    On the other hand, if z<z2, we have ϕ_2=1qb(eλ2zmeθz) and thus

    F(ϕ_1,ϕ_2)(z)d2(1q)b[RJ(y)(eλ2(zy)meθ(zy))dy(eλ2zmeθz)]c(1q)b(λ2eλ2zmθeθz)+1qb(eλ2zmeθz)[αβ(1q)(11b)1qb(eλ2zmeθz)]=1qbeλ2z[d2(RJ(y)eλ2ydy1)cλ2+α]m1qbeθz[d2(RJ(y)eθzdy1)cθ+α]β(1q)b(b1)(eλ2zmeθz)2>m1qbeθz[d2(RJ(y)eθzdy1)cθ+α]β(1q)b(b1)e2λ2z=1qbeθz[(m)(d2(RJ(y)eθzdy1)cθ+α)βb1e(2λ2θ)z]>1qbeθz[(m)(d2(RJ(y)eθzdy1)cθ+α)βb1]>1qbeθz[(m)fα(d2,c,θ)βb1]>0.

    The last inequality holds due to θ(λ2,min{2λ2,λ3}) and condition (A2).

    In this section, we start with discussing the existence of traveling wave solution for system (1.3) with condition (1.4) by using the upper-lower solutions of system (1.3), which is defined in the preceding section, to construct an invariant set.

    Let C be a set of bounded and uniformly continuous functions from R to R2 and

    Γ={(ϕ1,ϕ2)C:ϕ_i(z)ϕi¯ϕi(z), zR, i=1,2},

    where ¯ϕi(z) and ϕ_i(z) (i = 1, 2) are defined by (2.2)–(2.5). Thus for any (ϕ1,ϕ2)Γ, we have (1q)b1bϕ1(z)1q and  0ϕ2(z)1qb.

    For Φ=(ϕ1,ϕ2)Γ, we define

    {H1(ϕ1,ϕ2)(z):=d1Jϕ1(z)+F1(ϕ1(z),ϕ2(zcτ)),H2(ϕ1,ϕ2)(z):=d2Jϕ2(z)+F2(ϕ1(zcτ),ϕ2(z)),

    where

    {F1(y1,y2)=(γd1)y1+y1(1y1y21+ay1+by2q),F2(y1,y2)=(γd2)y2+y2(αβy2y1),

    for some constant γ. For any fixed γ>max{d1+(1q)(1+1b),d2+2βb1α}, it follows that F1 is nondecreasing in y1 and is decreasing in y2 for y1[(1q)b1b, 1q] and y2[0, 1qb]. Also, F2 is nondecreasing with respect to y1 and y2 for y1[(1q)b1b, 1q] and y2[0, 1qb].

    Define an operator P=(P1,P2):ΓC by

    {P1(ϕ1,ϕ2)(z)=1czeγ(zy)cH1(ϕ1,ϕ2)(y)dy,P2(ϕ1,ϕ2)(z)=1czeγ(zy)cH2(ϕ1,ϕ2)(y)dy.

    Apparently, a fixed point of P is a solution of system (1.3). Let ρ(0,γc) and |||| denote the Euclidean norm in R2. We define

    Bρ(R,R2)={ΦC:supzR||Φ(z)||eρ|z|<}

    and

    |Φ|ρ:=supzR||Φ(z)||eρ|z|.

    It is easy to see that (Bρ(R,R2)),||ρ) is a Banach space. Clearly, Γ is nonempty, bounded, convex and closed in Bρ(R,R2).

    Lemma 3.1. P:ΓΓ.

    Proof. For any Φ(z)=(ϕ1,ϕ2)(z)Γ, owing to the monotonicity of F1 and F2 we have

    {H1(ϕ1,ϕ2)(z)d1Jϕ_1(z)+F_1(z)=:H_1(z), zR,H1(ϕ1,ϕ2)(z)d1J¯ϕ1(z)+¯F1(z)=:¯H1(z), zR,

    and

    {H2(ϕ1,ϕ2)(z)d2Jϕ_2(z)+F_2(z)=:H_2(z), zR,H2(ϕ1,ϕ2)(z)d2J¯ϕ2(z)+¯F2(z)=:¯H2(z), zR,

    in which ¯F1, F_1, ¯F2 and F_2 are defined by

    {¯F1(z)=(γd1)¯ϕ1(z)+¯ϕ1(z)(1q¯ϕ1(z)ϕ_2(zcτ)1+a¯ϕ1(z)+bϕ_2(zcτ)),F_1(z)=(γd1)ϕ_1(z)+ϕ_1(z)(1qϕ_1(z)¯ϕ2(zcτ)1+aϕ_1(z)+b¯ϕ2(zcτ)),

    and

    {¯F2(z)=(γd2)¯ϕ2(z)+¯ϕ2(z)(αβ¯ϕ2(z)¯ϕ1(zcτ)),F_2(z)=(γd2)ϕ_2(z)+ϕ_2(z)(αβϕ_2(z)ϕ_1(zcτ)).

    Let

    P_1(z)=1czeγ(zy)cH_1(y)dy,  ¯P1(z)=1czeγ(zy)c¯H1(y)dy, zR,
    P_2(z)=1czeγ(zy)cH_2(y)dy,  ¯P2(z)=1czeγ(zy)c¯H2(y)dy, zR.

    Obviously, P_i(z)Pi(z)¯Pi(z) (i=1,2). It suffices to prove that

    ϕ_i(z)P_i(z),¯Pi(z)¯ϕi(z), zR, i=1,2.

    We denote z0= and zm+1=. For any zRZ, there exists a k{0,1,2,...,m} such that z(zk,zk+1), and

    ¯P1(z)=1czeγ(zy)c¯H1(y)dy=(ki=11czizi1+1czzk)eγ(zy)c¯H1(y)dy(ki=11czizi1+1czzk)eγ(zy)c[c¯ϕ1(y)dy+γ¯ϕ1(y)]=¯ϕ1(z).

    Due to the continuity of both ¯P1(z) and ¯ϕ1(z), we get

    ¯P1(z)¯ϕ1(z), zR.

    Similarly, we have

    ϕ_1(z)P_1(z), zR,

    and

    ϕ_2(z)P2(ϕ)(z)¯ϕ2(z), zR.

    Consequently, we obtain P(Γ)Γ.

    Lemma 3.2. P: ΓΓ is continuous with respect to ||ρ.

    Proof. For any Φ=(ϕ1,ϕ2) and Ψ=(ψ1,ψ2)Γ, we have

    |H1(ϕ1,ϕ2)(z)H1(ψ1,ψ2)(z)|d1RJ(zy)|ϕ1(y)ψ1(y)|dy+(γd1+1q)|ϕ1(z)ψ1(z)|+|ϕ1(z)+ψ1(z)||ϕ1(z)ψ1(z)|+|ϕ1(z)ϕ2(zcτ)1+aϕ1(z)+bϕ2(zcτ)ψ1(z)ψ2(zcτ)1+aψ1(z)+bψ2(zcτ)|

    and

    |ϕ1(z)ϕ2(zcτ)1+aϕ1(z)+bϕ2(zcτ)ψ1(z)ψ2(zcτ)1+aψ1(z)+bψ2(zcτ)|<1qb(2q)(1+a(1q)b1b)2|ϕ1(z)ψ1(z)|+(1q)(1+a(1q))(1+a(1q)b1b)2|ϕ2(zcτ)ψ2(zcτ)|<1qb(2q)|ϕ1(z)ψ1(z)|+(1q)(1+a(1q))|ϕ2(zcτ)ψ2(zcτ)|<1b(1q)(2q)|ϕ1(z)ψ1(z)|+a(1q)(2q)|ϕ2(zcτ)ψ2(zcτ)|.

    A straightforward calculation yields

    |P1(ϕ1,ϕ2)(z)P1(ψ1,ψ2)(z)|eρ|z|d1eρ|z|czeγ(zs)c(RJ(sy)|ϕ1(y)ψ1(y)|dy)ds+(γd1+1q)eρ|z|czeγ(zy)c|ϕ1(y)ψ1(y)|dy+2(1q)eρ|z|czeγ(zy)c|ϕ1(y)ψ1(y)|dy+(1q)(2q)eρ|z|cbzeγ(zy)c|ϕ1(y)ψ1(y)|dy+a(1q)(2q)eρ|z|czeγ(zy)c|ϕ2(ycτ)ψ2(ycτ)|dy=d1eρ|z|czeγ(zs)c(RJ(sy)|ϕ1(y)ψ1(y)|dy)ds+[γd1+(1q)(3+2qb)]eρ|z|czeγ(zy)c|ϕ1(y)ψ1(y)|dy+a(1q)(2q)eρ|z|czeγ(zy)c|ϕ2(ycτ)ψ2(ycτ)|dy.

    We further have

    eρ|z|czeγ(zs)c(RJ(sy)|ϕ1(y)ψ1(y)|dy)ds=eρ|z|czeγ(zs)c(RJ(sy)eρ|y||ϕ1(y)ψ1(y)|eρ|y|dy)ds|ΦΨ|ρcze(γcρ)(zs)(RJ(y)eρ|y|dy)ds2RJ(y)eρydyγcρ|ΦΨ|ρ

    and

    eρ|z|czeγ(zy)c|ϕ1(y)ψ1(y)|dy=eρ|z|czeγ(zy)ceρ|y||ϕ1(y)ψ1(y)|eρ|y|dy|ΦΨ|ρcze(γcρ)(zy)dy1γcρ|ΦΨ|ρ.

    Processing in an analogous manner, we can derive

    eρ|z|czeγ(zy)c|ϕ2(ycτ)ψ2(ycτ)|dyeρcτγcρ|ΦΨ|α.

    We now choose

    L1=2d1RJ(y)eρydy+γd1+(1q)[3+(1b+aeρcτ)(2q)]γcρ

    such that

    |P1(ϕ1,ϕ2)(z)P1(ψ1,ψ2)(z)|eρ|z|L1|ΦΨ|α. (3.1)

    On the other hand, we have

    |P2(ϕ1,ϕ2)(z)P2(ψ1,ψ2)(z)|eρ|z|d2eρ|z|czeγ(zs)c(RJ(ys)|ϕ2(y)ψ2(y)|dy)ds+(γd2+α+2bβ(b1)2)eρ|z|czeγ(zy)c|ϕ2(y)ψ2(y)|dy+βeα|z|(b1)2czeγ(zy)c|ϕ1(ycτ)ψ1(ycτ)|dyL2|ΦΨ|ρ, (3.2)

    where

    L2=2d2RJ(y)eρydy+γd2+α+β(2b+eρcτ)(b1)2γcρ.

    In view of (3.1)–(3.2), there exists some constant L>0 such that

    |P(ϕ)P(Ψ)|ρL|ΦΨ|ρ.

    Hence, P is a continuous operator from Γ to Γ.

    For any given NR, let RN:=(,N] and consider the domain of the functions of the space Bρ on RN:

    Bρ(RN,R2)={ΦC|RN: supzRN||Φ(z)||e|ρ|z<}.

    Then (Bρ(RN,R2),||Nρ) is a Banach space equipped with the norm ||Nρ defined by

    |Φ|Nα:=supzRN||Φ(z)||e|ρ|z.

    Let us recall Corduneanu's Theorem [37,§2.12].

    Lemma 3.3. Let FBα(RN,R2) be a set satisfying the following conditions:

    (1) F is bounded in Bα(RN,R2));

    (2) the functions belonging to F are equicontinuous on any compact interval of RN;

    (3) the functions in F are equiconvergent, i.e., for any given ε>0, there is a corresponding Z(ε)<0 such that for z\leq Z(\varepsilon) and \Phi\in F .

    Then F is compact in B_{\alpha}\left(\mathbb{R}^-_{N}, \mathbb{R}^2\right) .

    Lemma 3.4. P(\Gamma) is compact in B_{\rho}.

    Proof. For any \Phi = (\phi_{1}, \phi_{2})\in \Gamma and n\in \mathbb{N}, we define

    \begin{equation} P^{n}(\Phi)(z) = \begin{cases} P(\Phi)(n) &z \gt n, \\ P(\Phi)(z) &z\in(-\infty, n]. \end{cases} \end{equation} (3.3)

    Clearly, P^{n}(\Gamma) is compact if P(\Gamma)(z)|_{\mathbb{R}^-_{n}} is compact. We will show that the functions belonging to P(\Gamma)(z)|_{\mathbb{R}^-_{n}} satisfy all three conditions (1)–(3) in Lemma 3.3. Since P(\Gamma)\subset\Gamma , it is easy to see that P(\Gamma)(z)|_{\mathbb{R}^-_{n}} is bounded. Indeed, for any z_1, z_2\in (-\infty, n] we deduce

    \begin{align*} &\left|P_{1}\left(\phi_{1}, \phi_{2}\right)\left(z_1\right)e^{-\rho |z_1|}-P_{1}\left(\phi_{1}, \phi_{2}\right) \left(z_2\right)e^{-\rho |z_2|}\right|\\ &\quad = \frac{1}{c}\left|e^{-\rho |z_1|}\int_{-\infty}^{z_1}e^{-\frac{\gamma\left(z_1-y\right)}{c}}H_{1} \left(\phi_{1}, \phi_{2}\right)\left(y\right)\mathrm{d}y-e^{-\alpha |z_2|}\int_{-\infty}^{z_2}e^{-\frac{\gamma\left(z_2-y\right)}{c}}H_{1} \left(\phi_{1}, \phi_{2}\right)\left(y\right)\mathrm{d}y\right|\\ &\quad = \frac{1}{c}\left|e^{-\left(\rho|z_1|+\frac{\gamma}{c}z_1 \right)}\int_{-\infty}^{z_1}e^{\frac{\gamma}{c}y}H_{1} \left(\phi_{1}, \phi_{2}\right)\left(y\right)\mathrm{d}y- e^{-\left(\rho|z_2|+\frac{\gamma}{c}z_2 \right)}\int_{-\infty}^{z_2}e^{\frac{\gamma}{c}y}H_{1} \left(\phi_{1}, \phi_{2}\right)\left(y\right)\mathrm{d}y\right|\\ &\quad\leq \frac{1}{c}e^{-\frac{\gamma}{c} z_1} \left|\int_{z_1}^{z_2}e^{\frac{\gamma}{c}y}H_{1} \left(\phi_{1}, \phi_{2}\right) \left(y\right)\mathrm{d}y\right|\\ &\qquad +\frac{1}{c} \left(e^{-\rho |z_2|}\left|e^{-\frac{\gamma}{c}z_1}-e^{-\frac{\gamma}{c}z_2 }\right| +e^{-\frac{\gamma}{c} z_1}\left|e^{-\rho |z_1|}-e^{-\rho |z_2|}\right|\right) \cdot \left|\int_{-\infty}^{z_2}e^{\frac{\gamma}{c}y}H_{1} \left(\phi_{1}, \phi_{2}\right)\left(y\right)\mathrm{d}y\right|\\ &\quad\leq (1-q)e^{\frac{\gamma}{c}|z_2-z_1|} \left[\left(1+\frac{\gamma}{c}\right)|z_2-z_1|+1\right]. \end{align*}

    Similarly, we have

    \begin{align*} \left|P_{2}\left(\phi_{1}, \phi_{2}\right)\left(z_1\right)e^{-\rho |z_1|}-P_{2}\left(\phi_{1}, \phi_{2}\right) \left(z_2\right)e^{-\rho |z_2|}\right| \leq \frac{1-q}{b\gamma}\left(\gamma+\alpha -\frac{\beta}{b}\right)e^{\frac{\gamma}{c}|z_2-z_1|} \left[\left(1+\frac{\gamma}{c}\right)|z_2-z_1|+1\right]. \end{align*}

    This implies that \left.P(\Gamma)(z)\right|_{\mathbb{R}^-_n} is equicontinuous on any compact interval of \mathbb{R}^-_{n} .

    For any \Phi(z) = (\phi_1(z), \phi_2(z))\in \Gamma , we find

    \begin{align*} (1-q)\left(1-\frac{1}{b}\left(e^{\lambda_{1}z}+pe^{\lambda_{2}z}\right)\right)\leq \phi_1(z)\leq 1-q, \ \ \frac{1-q}{b}\left(e^{\lambda_{2}z}-me^{\theta z}\right)\leq \phi_1(z)\leq\frac{1-q}{b}e^{\lambda_{2}z} \end{align*}

    for z < \min\{z_1, \ z_2\} . That is,

    \lim\limits_{z\rightarrow-\infty}\phi_1(z) = 1-q, \quad \lim\limits_{z\rightarrow-\infty}\phi_2(z) = 0.

    Then

    |\phi_1(z)-(1-q)|e^{-\rho|z|} \lt \frac{1-q}{b}\left(e^{(\lambda_{1}+\rho) z}+pe^{(\lambda_{2}+\rho) z}\right), \ \ |\phi_2(z)-0|e^{-\rho|z|} \lt \frac{1-q}{b}e^{(\lambda_{2}+\rho) z}

    for z < \min\{z_1, \ z_2\} . That is, condition (3) is satisfied. According to Lemma 3.3, P^n(\Gamma)(z) is compact in the sense of the norm |\cdot|_{\rho} . Note that

    |P^{n}(\Phi)(z)-P(\Phi)(z)|e^{-\rho |z|}\leq 2(1-q)\sqrt{1+\left(\frac{\gamma+\alpha -\frac{\beta}{b}}{b\gamma}\right)^2}e^{-\rho n}\rightarrow0, \ \, as \ n\rightarrow \infty.

    Hence, P^{n}(\Phi)(z) converge to P(\Phi)(z) with respect to the norm |\cdot|_{\rho} , and P(\Gamma) is compact.

    From Lemmas 3.1–3.4 and Schauder's fixed point theorem, we can see that P has a fixed point \Phi\in\Gamma such that P(\Phi) = \Phi , which is a solution of system (1.3). Hence, we obtain the following theorem immediately.

    Theorem 3.5. Assume that conditions (G1)–(G2) hold. Then for any fixed c > c^* , system (1.3) has a positive solution (\phi_1(z), \phi_2(z))\in \Gamma . That is, \underline{\phi}_{i}(z)\leq{\phi_{i}(z)}\leq\overline{\phi}_{i}(z) (i = 1, 2) , where \overline{\phi}_{i} and \underline{\phi}_{i} (i = 1, 2) are defined by (2.2)–(2.5).

    We now discuss the asymptotic behavior of traveling wave solution described in Theorem 3.5. For z\rightarrow-\infty , it is easy to see that

    \lim\limits_{z\rightarrow -\infty} \phi_1(z) = 1-q, \quad \lim\limits_{z\rightarrow -\infty} \phi_2(z) = 0.

    By applying the contracting rectangles method, we analyze the asymptotic behavior of traveling wave solution as z\rightarrow\infty . We define

    \begin{eqnarray} \left\{ \begin{aligned} E_{1}(\xi, \eta)&: = \xi\left(1-q E-\xi-\frac{\eta}{1+a\xi+b\eta}\right), \\ E_{2}(\xi, \eta)&: = \eta(\alpha-\beta\frac{\eta}{\xi}), \end{aligned} \right. \end{eqnarray} (3.4)

    and

    \begin{equation} \begin{split} &u_{1}(\theta): = u^*\theta, \qquad\qquad\qquad\qquad\qquad u_{2}(\theta): = \left(1+\frac{\beta a}{\alpha b}(1-\epsilon)(1-\theta)\right)u^*, \\ &v_{1}(\theta): = \begin{cases} v^*\frac{\theta}{2(1-\epsilon)} &\theta \lt 2\epsilon\\ v^*\frac{\theta-\epsilon}{1-\epsilon} &\theta\geq2\epsilon \end{cases}, \qquad\quad\ v_{2}(\theta): = v^*+\frac{a}{b}u^*(1-\theta), \end{split} \end{equation} (3.5)

    for \theta\in[0, 1] , where (u^*, v^*) is the equilibrium point of system (1.3) and 0 < \epsilon < \min \left\{\frac{1}{4}, \frac{bv^*}{1+au^*}, 1-\frac{1}{a}\right\} .

    Theorem 3.6. The following three statements are true.

    (C1) u_1(\theta) and v_1(\theta) are continuous and strictly increasing while u_{2}(\theta) and v_2(\theta) are continuous and strictly decreasing for \theta\in[0, 1] .

    (C2) For \theta\in[0, 1] , we have

    \begin{equation*} \left\{ \begin{aligned} u_{1}(0)\leq u_1(\theta)\leq u_{1}(1)& = u^{\ast} = u_{2}(1)\leq u_{2}(\theta)\leq u_{2}(0), \\ v_{1}(0)\; \leq v_{1}(\theta)\leq v_{1}(1)& = v^{\ast} = v_{2}(1)\leq v_{2}(\theta)\leq \; v_{2}(0). \end{aligned} \right. \end{equation*}

    (C3) If \xi_{1} = u_{1}(\theta_{0}) , \eta_{1} = v_{1}(\theta_{0}) for any \theta_{0}\in(0, 1) and

    u_1(\theta_0))\leq \xi \leq u_{2}(\theta_{0}), \ \ v_1(\theta_0)\leq \eta \leq v_{2}(\theta_{0}),

    then E_{1}(\xi_{1}, \eta) > 0 and E_{2}(\xi, \eta_{1}) > 0 .

    If \xi_{2} = u_2(\theta_0) , \eta_{2} = v_2(\theta_0) for any \theta_{0}\in(0, 1) and

    u_1(\theta_0))\leq \xi \leq u_2(\theta_0), \ \ v_1(\theta_0)\leq \eta \leq v_2(\theta_0),

    then E_{1}(\xi_{2}, \eta) < 0 and E_{2}(\xi, \eta_{2}) < 0.

    Proof. It is easy to see that (C1)–(C2) are true.

    To prove (C3), we claim that for any \theta_0\in(0, 1) , there holds E_{1}(\xi_{1}, \eta) > 0 with \xi_{1} = u_1(\theta_0) = u^*\theta_0 and v_1(\theta_0)\leq \eta\leq v_2(\theta_0) . As E_{1}(\xi, \eta) is decreasing in \eta , we only need to show that E_{1}(u^*\theta_0, v_2(\theta_0)) > 0 .

    Let

    \tilde{v}(\theta) = -\frac{a}{b}u^*\theta+\frac{a-b}{b^{2}}+ \frac{\frac{a}{b}(1-q)-\frac{a-b}{b^{2}}}{1-b(1-q)+bu^*\theta}.

    Then E_{1}\left(u^*\theta, \tilde{v}(\theta)\right)\equiv0 for \theta\in[0, 1] . In view of a > \frac{1}{q} , it follows that

    \begin{equation*} \begin{aligned} v_2(\theta_0)& = v^*+\frac{a}{b}u^*(1-\theta_0)\\ & = -\frac{a}{b}u^*\theta_0+\frac{a-b}{b^{2}}+\frac{\frac{a}{b}(1-q)-\frac{a-b}{b^{2}}}{1-b(1-q)+bu^*}\\ & \lt -\frac{a}{b}u^*\theta_0+\frac{a-b}{b^{2}}+\frac{\frac{a}{b}(1-q)-\frac{a-b}{b^{2}}}{1-b(1-q)+bu^*\theta_0}\\ & = \tilde{v}(\theta_0). \end{aligned} \end{equation*}

    Thus, E_{1}(u^*\theta_0, v_2(\theta_0)) > E_{1}(u^*\theta_0, \tilde{v}(\theta_0)) = 0 .

    To show E_{2}(\xi, \eta_{1}) > 0 for any \theta_0\in(0, 1) , \eta_{1} = v_1(\theta_0) and u_1(\theta_0))\leq \xi\leq u_2(\theta_0) , we know that E_2(\xi, \eta) is nondecreasing in \xi . So it is equivalent to prove E_2(u_1(\theta_0), v_1(\theta_0)) > 0 . When 2\epsilon\leq \theta_0 < 1 , in view of v_1(\theta_0) = v^*\frac{\theta_0-\epsilon}{1-\epsilon} we have

    \begin{align*} E_{2}(u_1(\theta_0), v_1(\theta_0)) = v_1(\theta_0)\left[\alpha-\beta\frac{v^*\frac{\theta_0-\epsilon}{1-\epsilon}}{u^*\theta_0}\right] = \alpha v_1(\theta_0)\frac{\epsilon(1-\theta_0)}{\theta_0(1-\epsilon)} \gt 0. \end{align*}

    For 0 < \theta_0 < 2\epsilon , using v_1(\theta_0) = v^*\frac{\theta_0}{2(1-\epsilon)} we have

    \begin{align*} E_{2}(u_1(\theta_0), v_1(\theta_0)) = v_1(\theta_0)\left[\alpha -\beta \frac{v^*\frac{\theta_0}{2(1-\epsilon)}}{u^*\theta_0}\right] = \alpha v_1(\theta_0)\frac{1-2\epsilon}{2(1-\epsilon)} \gt 0. \end{align*}

    For any \theta_0\in(0, 1) , to show E_{1}(\xi_{2}, \eta) < 0 , where \xi_{2} = u_2(\theta_0) and v_1(\theta_0)\leq \eta\leq v_2(\theta_0) , it suffices to prove that E_{1}(u_2(\theta_0), v_1(\theta_0)) < 0 . Let \varphi(\theta): = E_{1}(u_{2}(\theta), v_1(\theta))/u_{2}(\theta) . Then \varphi(1) = 0 . We proceed by considering two cases.

    \underline{\mathrm{Case\ 1}} : for 2\epsilon\leq \theta < 1 , from (3.5) we have v_1(\theta) = v^*\frac{\theta-\epsilon}{1-\epsilon} , u_{2}(\theta) = \left(1+\frac{\beta a}{\alpha b}(1-\epsilon)(1-\theta)\right)u^* for \theta\in[2\epsilon, 1) , and then

    \begin{align*} \frac{\mathrm{d}\varphi}{\mathrm{d}\theta}& = \frac{\mathrm{d}}{\mathrm{d} \theta}\left(1-q-u_{2}(\theta)-\frac{v_1(\theta)}{1+au_{2}(\theta)+bv_1(\theta)}\right)\\ & = \frac{\rho_{1}(\theta)}{(1+au_{2}(\theta)+bv_{1}(\theta))^{2}}, \end{align*}

    where

    \begin{equation*} \begin{aligned} \rho_{1}(\theta)& = -\frac{\mathrm{d}u_{2}}{\mathrm{d}\theta}(1+au_{2}(\theta)+bv_{1}(\theta))^{2} -\frac{\mathrm{d}v_{1}}{\mathrm{d}\theta}(1+au_{2}(\theta))+av_{1}(\theta)\frac{\mathrm{d}u_{2}}{\mathrm{d}\theta}\\ & = \frac{\beta a}{\alpha b}\left(1-\epsilon\right)u^*(1+au_{2}(\theta)+bv_{1}(\theta))^{2} -\frac{v^*}{1-\epsilon}(1+au^*)-\frac{\beta a^2}{\alpha b}u^*v^*(1-\epsilon). \end{aligned} \end{equation*}

    Since \alpha b\leq\beta and 0 < \epsilon < 1-\frac{1}{a} , we get

    \frac{\mathrm{d}}{\mathrm{d} \theta}(1+au_{2}(\theta)+bv_{1}(\theta)) = \frac{bv^*}{1-\epsilon}-\frac{\beta a^2}{\alpha b}u^*(1-\epsilon) = \frac{bv^*}{1-\epsilon}\left[1-\left(\frac{\beta a}{\alpha b}\right)^2(1-\epsilon)^2\right] \lt 0.

    It is easy to see that \inf\limits_{\theta\in[2\epsilon, 1)}\rho_{1}(\theta) = \rho_{1}(1) . In view of 0 < \epsilon < \min \left\{\frac{1}{4}, \frac{bv^*}{1+au^*}, 1-\frac{1}{a}\right\} , we have

    \begin{align*} \rho_{1}(1) & = \frac{\beta a}{\alpha b}(1-\epsilon)u^* \left[(1+au^*+bv^*)^{2}-av^*\right]-\frac{v^*}{1-\epsilon}(1+au^*)\\ & \gt u^*\left[(1+au^*+bv^*)^{2}-av^*-\frac{1+au^*}{(1-\epsilon)}\right]\\ & \gt u^*\left[1+2au^*+2bv^*-av^*-(1+2\epsilon)(1+au^*)\right]\\ & \gt 2u^*\left[bv^*-\epsilon(1+au^*)\right]\\ & \gt 0. \end{align*}

    This implies that for \theta\in[2\epsilon, 1) , \rho_{1}(\theta) > 0 holds and \varphi(\theta) is nondecreasing. That is, \varphi(\theta) < 0 . Moreover, E_{1}(u_2(\theta), v_1(\theta)) = \varphi(\theta)u_2(\theta) < 0 for \theta\in[2\epsilon, 1) .

    \underline{\mathrm{Case\ 2}} : for 0 < \theta < 2\epsilon , from (3.5) we have v_1(\theta) = v^*\frac{\theta}{2(1-\epsilon)} and u_{2}(\theta) = \left(1+\frac{\beta a}{\alpha b}(1-\epsilon)(1-\theta)\right)u^* for \theta\in(0, 2\epsilon) . Then we get

    \begin{align*} \frac{\mathrm{d}\varphi}{\mathrm{d}\theta} = \frac{\rho_2(\theta)}{(1+au_{2}(\theta)+bv_{1}(\theta))^{2}}, \end{align*}

    where

    \rho_{2}(\theta) = \frac{\beta a}{\alpha b}\left(1-\epsilon\right)u^*(1+au_{2}(\theta)+bv_{1}(\theta))^{2} -\frac{v^*}{2(1-\epsilon)}(1+au^*)-\frac{\beta a^2}{2\alpha b}u^*v^*,

    and \inf\limits_{\theta\in(0, 2\epsilon)}\rho_{2}(\theta) = \rho_{2}(2\epsilon) . In view of 0 < \epsilon < \min \left\{\frac{1}{4}, \frac{bv^*}{1+au^*}, 1-\frac{1}{a}\right\} , there holds

    \begin{align*} \rho_{2}(2\epsilon) & \gt u^*\left[1+au^*\left(1+\frac{\beta a}{\alpha b}(1-\epsilon)(1-2\epsilon)\right)+\frac{\epsilon bv^*}{2(1-\epsilon)}\right]^{2}-\frac{v^*}{2(1-\epsilon)}(1+au^*)-\frac{\beta a^2}{2\alpha b}u^*v^*\\ & \gt u^*\left[1+2au^*+2u^*\frac{\beta a^2}{\alpha b}(1-3\epsilon)-\frac{1+2\epsilon}{2}(1+au^*) -\frac{\beta a^2}{2\alpha b}u^*\right]\\ & = u^*\left[\left(1-\frac{1+2\epsilon}{2}\right)+\left(2-\frac{1+2\epsilon}{2}\right)au^* +\left(2(1-3\epsilon)-\frac{1}{2}\right)\frac{\beta a^2}{\alpha b}u^* \right]\\ & \gt 0. \end{align*}

    Since \varphi(2\epsilon) < 0 , for \theta\in(0, 2\epsilon) we have \rho_{2}(\theta) > 0 and \varphi(\theta) < 0 . This leads to E_{1}(u_2(\theta), v_1(\theta)) = \varphi(\theta)u_2(\theta) < 0 for \theta\in(0, 2\epsilon) . Hence, E_{1}(u_2(\theta), v_1(\theta)) < 0 for \theta\in(0, 1) .

    To prove E_{2}(\xi, \eta_{2}) < 0 for \theta_0\in(0, 1) , \eta_{2} = v_2(\theta_0) and u_1(\theta_0))\leq \xi\leq u_2(\theta_0) , from (3.5) we deduce

    \begin{align*} E_{2}(u_2(\theta_0), v_2(\theta_0)) & = v_2(\theta_0)\left[\alpha-\beta\frac{v^*+\frac{a}{b}u^*(1-\theta_0)}{u^*+\frac{\beta a}{\alpha b}(1-\epsilon)(1-\theta_0)u^*}\right]\\ & \lt v_2(\theta_0)\left[\alpha-\beta\frac{v^*+\frac{a}{b}u^*(1-\theta_0)}{u^*+\frac{\beta a}{\alpha b}u^*(1-\theta_0)}\right]\\ & = v_2(\theta_0)\left[\alpha-\beta\frac{v^*+\frac{a}{b}u^*(1-\theta_0)} {\frac{\beta}{\alpha}(v^*+\frac{a}{b}u^*(1-\theta_0))}\right]\\ & = 0. \end{align*}

    Hence, E_2(u_2(\theta), v_2(\theta)) < 0 for any \theta\in(0, 1) .

    Theorem 3.7. Assume that conditions (G1)–(G2) hold and \Phi = (\phi_{1}, \phi_{2})\in \Gamma is a solution of system (1.3). Then we have

    \begin{equation} \lim\limits_{z\rightarrow\infty}(\phi_{1}(z), \phi_{2}(z)) = (u^{\ast}, v^{\ast}). \end{equation} (3.6)

    Proof. From (3.5), we observe

    \begin{align*} &u_{1}(0) = 0, \qquad u_2(0) = u^*+\frac{\beta a}{\alpha b}(1-\epsilon)u^* \gt 2u^* \gt 1-q, \\ &v_1(0) = 0, \qquad v_2(0) = v^*+\frac{a}{b}u^* \gt \frac{v^*+u^*}{b} \gt \frac{1-q}{b}. \end{align*}

    In view of (\phi_{1}, \phi_{2})\in \Gamma for z > > 0 , it follows that

    (1-q)\left(1-\frac{1}{b}\right)\leq \phi_{1}(z)\leq 1-q, \quad \frac{1-q}{b}\delta\leq \phi_2(z)\leq \frac{1-q}{b}.

    So we have

    \begin{equation} \begin{aligned} u_1(\theta_0))\leq\liminf\limits_{z\rightarrow\infty}\phi_{1}(z)\leq \limsup\limits_{z\rightarrow\infty}\phi_{1}(z)\leq u_2(\theta_0), \\ v_1(\theta_0)\leq\liminf\limits_{\xi\rightarrow\infty}\phi_{2}(z)\leq \limsup\limits_{z\rightarrow\infty}\phi_{2}(z)\leq v_2(\theta_0), \end{aligned} \end{equation} (3.7)

    for some \theta_0\in(0, 1) .

    Denote

    \theta^*: = \sup\{\theta\in\left[\theta_{0}, 1\right)| \ (3.7)\ \mathrm{ hold}\}.

    Then, \theta^* = 1 . Otherwise, we have \theta^* < 1 in (3.7). Namely, at least one of the following equalities is true:

    u_{1}\left(\theta^*\right) = \liminf\limits_{z\rightarrow\infty}\phi_{1} \left(z\right), \ \, u_{2}\left(\theta^*\right) = \limsup\limits_{z\rightarrow\infty}\phi_{1} \left(z\right),
    v_{1}\left(\theta^*\right) = \liminf\limits_{z\rightarrow\infty}\phi_{2} \left(z\right), \ \, v_{2}\left(\theta^*\right) = \limsup\limits_{z\rightarrow\infty}\phi_{2} \left(z\right).

    Without loss of generality, we assume that

    u_{1}(\theta^*) = \liminf\limits_{z\rightarrow\infty}\phi_{1}(z).

    It follows from Lebesgue's dominated convergence theorem that

    \begin{align*} \liminf\limits_{z\rightarrow\infty}\phi_{1}(z) = &\liminf\limits_{z\rightarrow\infty} \frac{1}{\gamma}\left[\gamma\phi_{1}(z)+ E_1(\phi_{1}(z), \phi_{2}(z-c\tau) )\right]\\ \geq& \liminf\limits_{z\rightarrow\infty}\phi_{1}(z)+\frac{1}{\gamma} E_{1}(\liminf\limits_{z\rightarrow\infty}\phi_{1}(z), \limsup\limits_{z\rightarrow\infty}\phi_{2}(z)). \end{align*}

    That is,

    E_{1}(\liminf\limits_{z\rightarrow\infty}\phi_{1}(z), \ \limsup\limits_{z\rightarrow\infty}\phi_2(z))\leq0.

    This implies that E_{1}(u_{1}(\theta^*), \ \eta)\leq0 with v_{1}\left(\theta^*\right)\leq \eta\leq v_{2}\left(\theta^*\right) , which yields a contradiction to (C3) of Theorem 3.6. The other three cases can be proceeded in an analogous manner.

    Let z\in\mathbb{R}^-_N with N\in \mathbb{R} . We define

    C_l\left(\mathbb{R}^-_{N}, \mathbb{R}^2\right) = \left\{(\phi_1, \phi_2)\in \mathcal{C}|_{\mathbb{R}^-_{N}}:\ \lim\limits_{z\rightarrow-\infty}\phi_{1}(z) = \phi_{1}(-\infty), \quad \lim\limits_{z\rightarrow-\infty}\phi_{2}(z) = \phi_{2}(-\infty)\right\}.

    It is not difficult to see that C_l\left(\mathbb{R}^-_{N}, \mathbb{R}^2\right) is isomorphic to C\left(\left[\frac{N}{N-1}, 1\right], \mathbb{R}^2\right) . Indeed, if x(s)\in C\left(\left[\frac{N}{N-1}, 1\right], \mathbb{R}^2\right) , then y(t) = x(s) for t = \frac{s}{s-1} , s\in \left[\frac{N}{N-1}, 1\right) , and y(t)\in C_l\left(\mathbb{R}^-_{N}, \mathbb{R}^2\right) . That is, C_l\left(\mathbb{R}^-_{N}, \mathbb{R}^2\right) is a Banach space equipped with the superemum norm.

    Theorem 4.1. When c = c^* , system (1.3) has a positive traveling wave solution satisfying (1.4).

    Proof. Let \{c_n\} be a decreasing sequence with c_n < c^*+1 and \lim\limits_{n\rightarrow\infty} c_n = c^* . Then for each c_n , system (1.3) has a positive traveling wave solution \left(\phi_{1n}\left(z\right), \, \phi_{2n}\left(z\right)\right) satisfying (1.4) and

    (1-q)\frac{b-1}{b}\leq\phi_{1n}(z)\leq1-q, \ \ 0\leq\phi_{2n}(z)\leq\frac{1-q}{b}.

    Since a traveling wave solution is invariant in the sense of phase shift, we can assume that

    \phi_{1n}(0) = (1-q)\iota_1, \ \phi_{1n}(z) \gt (1-q)\iota_1\ \text{for}\ z \lt 0 \ \text{and}\ \phi_{2n}(0) = \iota_2, \ \phi_{2n}(z) \lt \iota_2\ \text{for}\ z \lt 0,

    with \frac{b-1}{b} < \iota_1 < 1 and 0 < \iota_2 < \frac{1-q}{b} . From (1.4), we know that the above expressions are admissible.

    For n\in\mathbb{N} , it is evident that (\phi_{1n}(z), \, \phi_{2n}(z)) are equipcontinuous, bounded and equipconvergent in C_l\left(\mathbb{R}^-_{N}, \mathbb{R}^2\right) . According to Lemma 3.3, \{(\phi_{1n}(z), \, \phi_{2n}(z))\} has a subsequence, still denoted by \{(\phi_{1n}(z), \, \phi_{2n}(z))\} , such that

    \phi_{1n}(z)\rightarrow\phi_{1}(z), \ \, \phi_{2n}(z)\rightarrow\phi_{2}(z), \ \mathrm{as} \ n\rightarrow\infty

    and

    \lim\limits_{z\rightarrow-\infty}\phi_{1}(z) = 1-q, \quad \lim\limits_{z\rightarrow-\infty}\phi_{2}(z) = 0.

    Here, (\phi_{1}(z), \, \phi_{2}(z))\in C_l\left(\mathbb{R}^-_{N}, \mathbb{R}^2\right) is continuous and the above limits converge uniformly on \mathbb{R}^-_N . It follows from Lebesgue's dominated convergence theorem that

    \lim\limits_{n\rightarrow\infty} J*\phi_{in}(z) = \phi_{i}(z) , \quad i = 1, 2

    on z\in \mathbb{R}^-_N . Thus, (\phi_{1}(z), \, \phi_{2}(z)) is a solution to system (1.3) which satisfies

    \phi_{1}(0) = (1-q)\iota_1, \ \phi_{1}(z) \gt (1-q)\iota_1 \ \text{for}\ z \lt 0\ \text{and}\ \phi_{2}(0) = \iota_2, \ \phi_{2}(z) \lt \iota_2 \ \text{for}\ z \lt 0,

    and

    (1-q)\frac{b-1}{b}\leq\phi_{1}(z)\leq1-q, \ \, \ 0\leq\phi_{2}(z)\leq\frac{1-q}{b}.

    From \phi_{2}(0) = \iota_2 > 0 , \liminf\limits_{z\rightarrow-\infty}\phi_{2}(z) > 0 holds. By virtue of Theorem 3.7, we obtain

    \lim\limits_{z\rightarrow+\infty}\phi_1(z) = u^*, \ \ \lim\limits_{z\rightarrow+\infty}\phi_2(z) = v^*.

    Consider the Cauchy problem:

    \begin{equation} \left\{ \begin{aligned} &\frac{\partial u\left(x, \, t\right)}{\partial t} = d\left(J \ast u\left(x, \, t\right)-u\left(x, \, t\right)\right)+u\left(x, \, t\right)\left(1-ru\left(x, \, t\right)\right), \\ &u\left(x, \, 0\right) = u_0\left(x\right), \ \, x\in\mathbb{R}, \end{aligned} \right. \end{equation} (5.1)

    where J satisfies condition (G1), r > 0 is constant and the initial value u_0\left(x\right) is uniformly continuous and bounded for x\in\mathbb{R} .

    Lemma 5.1. [32] Assume that 0\leq u_0\left(x\right)\leq\frac{1}{r} . Then system (5.1) admits a solution for x\in\mathbb{R} and t > 0 . If \omega\left(x, \, 0\right) is uniformly continuous and bounded, and \omega\left(x, \, 0\right) satisfies

    \begin{equation*} \left\{ \begin{aligned} &\frac{\partial\omega\left(x, \, t\right)}{\partial t}\geq\left(\leq\right)d\left(J \ast\omega\left(x, \, t\right)-\omega\left(x, \, t\right)\right) +\omega\left(x, \, t\right)\left(1-r\omega\left(x, \, t\right)\right), \\ &\omega\left(x, \, 0\right)\geq\left(\leq\right)u_0\left(x\right), \ \, x\in\mathbb{R}, \\ \end{aligned} \right. \end{equation*}

    then we have

    \omega\left(x, \, t\right)\geq\left(\leq\right)u\left(x, \, t\right), \ \, x\in\mathbb{R}, \ t \gt 0.

    Lemma 5.2. [32] Assume that u_0\left(x\right) > 0 . Then for any 0 < c < c^* there holds

    \liminf\limits_{t\rightarrow{\infty}}\inf\limits_{|x| \lt ct} u\left(x, \, t, \ u_0\left(x\right)\right) = \limsup\limits_{t\rightarrow{\infty}}\sup\limits_{|x| \lt ct} u\left(x, \, t, \ u_0\left(x\right)\right) = \frac{1}{r}.

    Theorem 5.3. For any speed 0 < c < c^* , there is no nontrivial positive solution \left(\phi_{1}\left(z\right), \, \phi_{2}\left(z\right)\right) of system (1.3) satisfying condition (1.4).

    Proof. Suppose on the contrary that there exists some 0 < c_{1} < c^*, such that system (1.3) has a positive solution (\phi_{1}(z), \phi_{2}(z)) satisfying condition (1.4). Then \phi_{1}(z) is bounded on \mathbb{R} and we can find a positive constant K such that \psi(x, t) = \phi_2(x+ct) satisfies

    \begin{equation*} \left\{ \begin{aligned} &\frac{\partial \psi\left(x, t\right)}{\partial t}\geq d_2\left(J \ast \psi\left(x, t\right)-\psi\left(x, t\right)\right)+\alpha\psi\left(x, t\right)\left(1-K\psi \left(x, t\right)\right), \\ &\psi\left(x, 0\right) = \phi_{2}\left(x\right) \gt 0. \end{aligned} \right. \end{equation*}

    Let x(t) = -\frac{c_1+c^*}{2}t . From Lemmas 5.1 and 5.2 it follows that

    \liminf\limits_{t\rightarrow{\infty}}\inf\limits_{2|x| = (c_1+c^*)t}\psi(x, t)\geq\frac{1}{K}.

    Meanwhile, in view of x(t)+c_1t = \frac{c_1-c^*}{2}t , we see z = x(t)+c_1t\rightarrow -\infty as t\rightarrow +\infty , and

    \limsup\limits_{t\rightarrow{\infty}}\psi(x(t), t) = \lim\limits_{z\rightarrow{-\infty}}\phi_2(z) = 0.

    This yields a contradiction.

    In this paper, we have studied the existence and nonexistence of traveling wave solution of a nonlocal delayed predator-prey model with the B-D functional response and harvesting. As we see, model (1.3) is nonmonotone or not quasimonotone. We employed Schauder's fixed point theorem and the upper-lower solutions method to discuss the existence of traveling wave solution for the speed c > c^* . Then, we investigated the asymptotic behavior of traveling wave solution by construction of the upper-lower solutions at -\infty and by developing the contacting rectangles technique at +\infty . For the special case of c = c^* , one usually can not establish the existence of traveling wave solution directly by constructing a pair of upper-lower solutions. One of available methods is the limiting argument together with the Arzela-Ascoli Theorem [33,36,39]. In this study we have presented not only the existence of traveling wave solution but also the asymptotic behavior of traveling wave solution at -\infty by Corduneanu's theorem. The nonexistence of traveling wave solution of system (1.3) with condition (1.4) was investigated by applying the comparison principle of nonlocal dispersal equations.

    It is remarkable that for the parameters of system (1.3), we only need b > 1 and 0 < b\alpha\leq \beta to prove Theorem 3.5. These conditions were used to construct a pair of suitable upper-lower solutions of system (1.3). For a > 1 and 0 < a\alpha\leq \beta , we could also construct the appropriate upper-lower solutions of system (1.3) in a similar way. To obtain the asymptotic behavior of traveling wave solution as z\rightarrow\infty , we additionally needed a > \frac{1}{q} .

    When q = 0 in model (1.3), it means that there does not have any prey harvesting. By assuming b > 1 , 0 < b\alpha\leq \beta and a > \frac{b\alpha}{\beta} , we can derive the same results as Theorems 3.5 and 3.7 in an analogous manner.

    We are grateful to the anonymous referees for their valuable comments. This work is supported by National Science Foundation of China under 11601029. All authors declare no conflicts of interest in this paper.

    The authors declare there is no conflicts of interest.



    [1] S. Araci, Degenerate poly type 2-Bernoulli polynomials, Math. Sci. Appl. E-Notes, 9 (2021), 1–8. https://doi.org/10.36753/mathenot.839111 doi: 10.36753/mathenot.839111
    [2] A. Bayad, D. S. Kim, Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math., 20 (2010), 23–28.
    [3] A. Bayad, M. Hajli, On the multidimensional zeta functions associated with theta functions, and the multidimensional Appell polynomials, Math. Method. Appl. Sci., 43 (2020), 2679–2694 https://doi.org/10.1002/mma.6075 doi: 10.1002/mma.6075
    [4] A. Bayad, T. Kim, Results on values of Barnes polynomials, Rocky Mountain J. Math., 43 (2013), 1857–1869. https://doi.org/10.1216/RMJ-2013-43-6-1857 doi: 10.1216/RMJ-2013-43-6-1857
    [5] L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math., 7 (1956), 28–33. https://doi.org/10.1007/BF01900520 doi: 10.1007/BF01900520
    [6] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Util. Math., 15 (1979), 51–88.
    [7] U. Duran, M. Acikgoz, On generalized degenerate Gould-Hopper based fully degenerate Bell polynomials, JMCS, 21 (2020), 243–257. http://doi.org/10.22436/jmcs.021.03.07 doi: 10.22436/jmcs.021.03.07
    [8] M. Hajli, The spectral properties of a continuous family of zeta functions, Int. J. Number Theory, 16 (2020), 693–717. https://doi.org/10.1142/S1793042120500359 doi: 10.1142/S1793042120500359
    [9] H. Haroon, W. A. Khan, Degenerate Bernoulli numbers and polynomials associated with degenerate Hermite polynomials, Commun. Korean Math. Soc., 33 (2018), 651–669. https://doi.org/10.4134/CKMS.c170217 doi: 10.4134/CKMS.c170217
    [10] M. Hajli, On a formula for the regularized determinant of zeta functions with application to some Dirichlet series, Q. J. Math., 71 (2020), 843–865. https://doi.org/10.1093/qmathj/haaa006 doi: 10.1093/qmathj/haaa006
    [11] G. W. Jang, T. Kim, A note on type 2 degenerate Euler and Bernoulli polynomials, Adv. Stud. Contemp. Math., 29 (2019), 147–159.
    [12] L. C. Jang, D. S. Kim, T. Kim, H. Lee, p-adic integral on Z_p associated with degenerate Bernoulli polynomials of the second kind, Adv. Differ. Equ., 2020 (2020), 278. https://doi.org/10.1186/s13662-020-02746-2 doi: 10.1186/s13662-020-02746-2
    [13] M. Kaneko, Poly-Bernoulli numbers, J. Theor. Nombr. Bordx., 9 (1997), 221–228.
    [14] W. A. Khan, R. Ali, K. A. H. Alzobyd, N. Ahmed, A new family of degenerate poly-Genocchi polynomials with its certain properties, J. Funct. Spaces, 2021 (2021), 6660517. https://doi.org/10.1155/2021/6660517 doi: 10.1155/2021/6660517
    [15] D. S. Kim, T. Kim, Lah-Bell numbers and polynomials, Proc. Jangjeon Math. Soc., 23 (2020), 577–586.
    [16] D. S. Kim, T. Kim, A note on a new type of degenerate Bernoulli numbers, Russ. J. Math. Phys., 27 (2020), 227–235. https://doi.org/10.1134/S1061920820020090 doi: 10.1134/S1061920820020090
    [17] D. S. Kim, T. Kim, Degenerate Sheffer sequences and \lambda-Sheffer sequences, J. Math. Anal. Appl., 493 (2021), 124521. https://doi.org/10.1016/j.jmaa.2020.124521 doi: 10.1016/j.jmaa.2020.124521
    [18] T. Kim, D. S. Kim, Degenerate Laplace transform and degenerate gamma function, Russ. J. Math. Phys., 24 (2017), 241–248. https://doi.org/10.1134/S1061920817020091 doi: 10.1134/S1061920817020091
    [19] T. Kim, D. S. Kim, A note on type 2 Changhee and Daehee polynomials, RACSAM, 113 (2019), 2763–2771. https://doi.org/10.1007/s13398-019-00656-x doi: 10.1007/s13398-019-00656-x
    [20] T. Kim, D. S. Kim, Degenerate polyexponential functions and degenerate Bell polynomials, J. Math. Anal. Appl., 487 (2020), 124017. https://doi.org/10.1016/j.jmaa.2020.124017 doi: 10.1016/j.jmaa.2020.124017
    [21] T. Kim, D. S. Kim, J. Kwon, H. Lee, Degenerate polyexponential functions and type 2 degenerate poly-Bernoulli numbers and polynomials, Adv. Differ. Equ., 2020 (2020), 168. https://doi.org/10.1186/s13662-020-02636-7 doi: 10.1186/s13662-020-02636-7
    [22] T. Kim, D. S. Kim, H. Lee, L. Jang, A note on degenerate derangement polynomials and numbers, AIMS Mathematics, 6 (2021), 6469–6481. https://doi.org/10.3934/math.2021380 doi: 10.3934/math.2021380
    [23] T. Kim, A note on degenerate Stirling polynomials of the second kind, Proc. Jangjeon Math. Soc., 20 (2017), 319–331.
    [24] T. K. Kim, D. S. Kim, H. I. Kwon, A note on degenerate Stirling numbers and their applications, Proc. Jangjeon Math. Soc., 21 (2018), 195–203.
    [25] F. Qi, J. L. Wang, B. N. Guo, Simplifying differential eqpuations concerning degenerate Bernoulli and Euler numbers, T. A. Razmadze Math. In., 172 (2018), 90–94. https://doi.org/10.1016/j.trmi.2017.08.001 doi: 10.1016/j.trmi.2017.08.001
    [26] S. Roman, The umbral calculus (Pure and Applied Mathematics), London: Academic Press, 1984.
  • This article has been cited by:

    1. Shao-Yue Mi, Bang-Sheng Han, Yinghui Yang, Spatial dynamics of a nonlocal predator–prey model with double mutation, 2022, 15, 1793-5245, 10.1142/S1793524522500358
    2. Lili Jia, Changyou Wang, Xiaojuan Zhao, Wei Wei, Dynamic Behavior of a Fractional-Type Fuzzy Difference System, 2022, 14, 2073-8994, 1337, 10.3390/sym14071337
    3. Boumediene Guenad, Rassim Darazirar, Salih Djilali, Ibrahim Alraddadi, Traveling waves in a delayed reaction–diffusion SIR epidemic model with a generalized incidence function, 2024, 0924-090X, 10.1007/s11071-024-10413-4
    4. Zhihong Zhao, Huan Cui, Yuwei Shen, Traveling waves of a modified Holling-Tanner predator–prey model with degenerate diffusive, 2024, 75, 0044-2275, 10.1007/s00033-024-02339-z
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1860) PDF downloads(67) Cited by(3)

Figures and Tables

Figures(3)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog