Research article

Traveling wave solutions to a cubic predator-prey diffusion model with stage structure for the prey

  • Received: 20 April 2022 Revised: 14 June 2022 Accepted: 20 June 2022 Published: 04 July 2022
  • MSC : 34C37, 35C07

  • In this paper, we investigate the traveling wave solutions to a cubic predator-prey diffusion model with stage structure for the prey. Firstly, using the upper and lower solutions method we prove the existence and non-existence of weak traveling wave solutions. Furthermore, we prove that the weak traveling wave solutions are actually traveling wave solutions under additional conditions by using Lyapunov function method and LaSalle's invariance principle.

    Citation: Yujuan Jiao, Jinmiao Yang, Hang Zhang. Traveling wave solutions to a cubic predator-prey diffusion model with stage structure for the prey[J]. AIMS Mathematics, 2022, 7(9): 16261-16277. doi: 10.3934/math.2022888

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  • In this paper, we investigate the traveling wave solutions to a cubic predator-prey diffusion model with stage structure for the prey. Firstly, using the upper and lower solutions method we prove the existence and non-existence of weak traveling wave solutions. Furthermore, we prove that the weak traveling wave solutions are actually traveling wave solutions under additional conditions by using Lyapunov function method and LaSalle's invariance principle.



    In [2], Cao and Fu considered the following cubic predator-prey diffusion model with stage structure for the prey:

    {x1t=d1Δx1+η1x2r1x1η2x1+b1x21b2x31b3x1x3,x2t=d2Δx2+η2x1r2x2,x3t=d3Δx3cx3+(αx1βx3)x3, (1.1)

    where x1 and x2 denote the densities of the immature and mature prey species, respectively, and x3 is the density of the predator species. The predators live only on the immature prey species. The constants d1, d2, d3, η1, η2, r1, r2, b1, b2, b3, c, α, β are positive. d1, d2 and d3 denote diffusion coefficients. η1 and r1 represent the birth rate and the mortality rate of the immature prey species, respectively. η2 is the conversion rate of the immature prey species to the mature prey species. b1x21b2x31 is the density restriction term of the immature prey species. b3x1 is the predation rate of the predator to the immature prey population. r2 and c are the net mortality rate of the mature prey population and the predator population, respectively. αx1 is the conversion rate of the predator, and βx3 is the density restriction term of the predator population. For more details on the backgrounds this system, see [2].

    Rescaling the system (1.1) such that

    αr2x1u1, αη2x2u2, βr2x3v, r2dtdτ, τt

    yields

    {u1t=d1Δu1+a0u2a1u1+a2u21a3u31eu1v, xRn,t>0,u2t=d2Δu2+u1u2,xRn,t>0,vt=d3Δv+(b+u1v)v,xRn,t>0, (1.2)

    where a0=η1η2/(r22),a1=(r1+η2)/r2,a2=b1/α,a3=b2/r2,e=b3/β,b=c/r2 are positive constants.

    If a0>a1, a2>e+2a3b, then the system (1.2) has a semi-trivial equilibrium (K,K,0) and the unique positive constant equilibrium (u1,u2,v), where

    K=a2+a22+4a3(a0a1)2a3,u1=u2=(a2e)+(a2e)2+4a3(eb+a0a1)2a3,v=u1b.

    Cao and Fu have obtained the following main conclusions: (1) The asymptotical stability of equilibrium points of the system (1.2) without diffusion; (2) the global existence of solutions and the stability of equilibrium points of system (1.2); (3) the existence of nonnegative classical global solutions and the global asymptotic stability of a unique positive equilibrium point of system (1.2) with cross-diffusion.

    A traveling wave solution of the system (1.2) is a special solution (u1(x,t),u2(x,t),v(x,t)) taking the form

    ui(x,t)=ui(xν+ct)(i=1,2),v(x,t)=v(xν+ct),

    where νRn is a unit vector denoting the direction of wave propagation, xν is the usual inner product in Rn, c>0 is the wave speed, and (u1(ξ),u2(ξ),v(ξ)) with ξ=xν+ct satisfies the following ODE system:

    {cu1=d1u1+a0u2a1u1+a2u21a3u31eu1v,ξR,cu2=d2u2+u1u2,ξR,cv=d3v+(b+u1v)v,ξR, (1.3)

    and

    {0<ui(ξ)K (i=1,2),  0<v(ξ)V0, ξR,(u1,u1,u2,u2,v,v)()=E0:=(K,0,K,0,0,0),(u1,u1,u2,u2,v,v)()=E:=(u1,0,u2,0,v,0), (1.4)

    where V0 positive constant. For convenience, we shall use the variable x to replace ξ and use i to denote the integers 1,2 in this paper.

    In the past several decades, the existence and non-existence of traveling wave solutions for predator-prey systems have been widely studied by many researchers. Dunbar [4,5,6] established the existence of traveling wave solutions for a reaction-diffusion system by using Lyapunov function, shooting techniques, invariant manifold theory, etc. Hsu, Yang and Yang [9] obtained the existence of traveling wave solutions for a class of diffusive predator-prey type systems by using the Wazewski theorem, LaSalle's invariance principle and Hopf bifurcation theory. Huang, Guang and Ruan [10] considered the existence of traveling front solutions and small amplitude traveling wave train solutions for a reaction-diffusion system. Ai, Du and Peng [1] studied traveling wave solutions of the generalized Holling-Tanner predator-prey model by the squeezing method and Lyapunov function method. Wang and Fu [15] established the existence of traveling wave solutions to a diffusive generalized Holling-Tanner predator-prey model by constructing the Lyapunov function. For more results, we can see [3,7,8,11,12,13,14] and references.

    Based on the idea of Ai, Du and Peng [1], in this paper, we are concerned with the existence and non-existence of traveling wave solutions of the system (1.2). We obtain the existence and non-existence of weak traveling wave solutions by using the upper and lower solutions method and the Schauder fixed point theorem. Moreover, we prove that the weak traveling wave solutions are actually traveling wave solutions under additional conditions by the using the Lyapunov function method and LaSalle's invariance principle. Although the idea was used before for other predator-prey systems, the adaptation to our problem harder, and we need more detailed and complicated estimates.

    This paper is organized as follows. In Section 2, employing the method of upper and lower solutions together with the Schauder fixed point theorem, we prove the existence and non-existence of weak traveling wave solutions for (1.3) with the main Theorem 2.1. In Section 3, we prove that the weak traveling wave solution obtained in Theorem 2.1 is also a traveling wave solution under certain conditions by using the Lyapunov function and LaSalle's invariance principle (Theorem 3.1).

    In this section, we will apply the method of upper and lower solutions together with the Schauder fixed point theorem to study the existence of weak traveling wave solutions for (1.3).

    Let

    F1(u1,u2,v)=a0u2a1u1+a2u21a3u31eu1v,
    F2(u1,u2,v)=u1u2,
    G(u1,u2,v)=(b+u1v)v.

    We give the following theorem.

    Theorem 2.1. Assume that a0>a1, a2>max{e+2a3b,2a3a1} and r=Kb>0. In addition, there exists a constant V0>0 such that 1/V01/r<min{d3/d2,1}, and let

    a22+4a0a34a1a34a3r<min{d3d1,1}.

    If there exists a small constant δ>0,

    b+u1vr2[(Ku1)+(Ku2)+v] (2.1)

    for any (u1,u2,v)[Kδ,K]2×[0,δ] holds.\\

    Then, for arbitrary cc:=4d3r, the system (1.3) has a solution (u1,u2,v) satisfying

    {0<ui(x)K,0<v(x)V0,x0,0ui(x)K,0v(x)V0,x>0,(u1,u1,u2,u2,v,v)()=E0.

    Moreover, for any 0<c<4d3r, the system (1.3) does not have a solution (u1(x),u2(x),v(x)) connecting (K,K,0) as x and satisfying v(x)>0 for sufficiently negative x.

    Now we give the definition of upper and lower solutions.

    Definition 2.1. The continuous functions (u_1,u_2,v_) and (ˉu1,ˉu2,ˉv) on R are called a pair of lower and upper solutions of the system (1.3) if they satisfy

    (i)

    0u_i(x)ˉui(x)Ui0,0v_(x)ˉv(x)V0, xR

    for some positive constants Ui0 and V0.

    (ii) There exists a set D consisting of at most finitely many real numbers such that

    (a) ˉui,u_i,ˉv,v_ are in C2(RD),

    (b) The right and left limits of u_i,ˉui,v_,ˉv all exist at each xD and satisfy

    ˉui(x)ˉui(x+),u_i(x)u_i(x+),ˉv(x)ˉv(x+),v_(x)v_(x+).

    (iii) At ±, the first and second derivatives of ˉui,ˉv,u_i,v_ have at most exponential growth.

    (iv) For every pair of continuous functions (u1,u2,v) with u_iuiˉui and v_vˉv,

    {d1ˉu1(x)cˉu1(x)+a0u2a1ˉu1+a2ˉu21a3ˉu31eˉu1v0,d2ˉu2(x)cˉu2(x)+u1ˉu20,d3ˉv(x)cˉv(x)+(b+u1ˉv)ˉv0,d1u_1(x)cu_1(x)+a0u2a1u_1+a2u_21a3u_31eu_1v0,d2u_2(x)cu_2(x)+u1u_20,d3v_(x)cv_(x)+(b+u1v_)v_0.xRD.

    Lemma 2.2. Assume that (u_1,u_2,v_) and (ˉu1,ˉu2,ˉv) are a pair of lower and upper solutions of (1.3). Then, there is a solution (u1,u2,v) of the system (1.3) satisfying

    u_i(x)ui(x)ˉui(x),v_(x)v(x)ˉv(x),xR,

    and ui, ui, v and v are bounded on R.

    Proof. Since Fi and G satisfy the Lipschitz condition on [0,U10]×[0,U20]×[0,V0], there is Λ=max{(1+a1+V0(e+1)+2a2U10+3a3U210),b+2V0+U10(e+1),a0+1}, so that for any (u1i,u2i,vi)[0,U10]×[0,U20]×[0,V0], we have

    (2.2)

    Define

    ˆFi(u1,u2,v):=Fi(u1,u2,v)+Λui,
    ˆG(u1,u2,v)=G(u1,u2,v)+Λv.

    According to (2.2), we derive that ˆF1(u1,u2,v) is nondecreasing in u1[0,U10] for each fixed (u2,v)[0,U20]×[0,V0], ˆF2(u1,u2,v) is nondecreasing in u2[0,U20] for each fixed (u1,v)[0,U10]×[0,V0], ˆG(u1,u2,v) nondecreasing in v[0,V0] for each fixed (u1,u2)[0,U10]×[0,U20], and the system (1.3) can be written as

    {d1u1cu1Λu1+ˆF1(u1,u2,v)=0,  xRn,d2u2cu2Λu2+ˆF2(u1,u2,v)=0,  xRn,d3vcvΛv+ˆG(u1,u2,v)=0,     xRn.

    Now, let

    X={(u1,u2,v)[C(R)]3:u_i(x)ui(x)ˉui(x),v_(x)v(x)ˉv(x),xR},

    and define the map T=(T1,T2,T3):X[C(R)]3 by

    Ti(u1,u2,v)(x)=1di(λ+iλi)(xeλi(xy)+xeλ+i(xy))ˆFi(u1,u2,v)(y)dy,
    T3(u1,u2,v)(x)=1d3(λ+3λ3)(xeλ3(xy)+xeλ+3(xy))ˆG(u1,u2,v)(y)dy,

    where

    λ±i=12di(c±c2+4diΛ),λ±3=12d3(c±c2+4d3Λ).

    By rather standard arguments similarly to those in the reference [1] that (U1,U2,V)=T(u1,u2,v) for each (u1,u2,v)X is the unique bounded solution of the linear equation

    {d1U1cU1ΛU1+ˆF1(u1,u2,v)=0,d2U2cU2ΛU2+ˆF2(u1,u2,v)=0,d3VcVΛV+ˆG(u1,u2,v)=0,

    and any fixed point of T in X gives a solution of the system (1.3). Therefore, it suffices to show by the Schauder fixed point theorem that T has a fixed point in X. To do so, we define the Banach space

    Cρ(R,R3)={(u1,u2,v)[C(R)]3:(u1,u2,v)ρ<}

    with the exponentially weighted norm

    (u1,u2,v)ρ=supxR|(u1(x),u2(x),v(x))|eρ|x|:=supxR[|u1(x)|+|u2(x)|+|v(x)|]eρ|x|,

    where 0<ρ<min{|λ1|,|λ2|,|λ3|}, and it follows that X is a bounded, closed and convex subset of Cρ(R,R3).

    It is easy to check that T is completely continuous on X. By applying the Schauder fixed point theorem, we conclude that T has a fixed point (u1,u2,v) in X, which gives a solution of the system (1.3).

    Note that for xR,

    ui(x)=1di(λ+iλi)(λixeλi(xy)+λ+ixeλ+i(xy))ˆFi(u1,u2,v)(y)dy.
    v(x)=1d3(λ+3λ3)(λ3xeλ3(xy)+λ+3xeλ+3(xy))ˆG(u1,u2,v)(y)dy.

    It follows that |ui(x)|M0/[di(λ+iλi)], and |v(x)|M0/[d3(λ+3λ3)] for xR, where M0=max{|a0u2a1u1+a2u21a3u31eu1v+Λu1|, |u1u2+Λu2|, |(b+u1v)v+Λv|:0u1U10,0u2U20,0vV0}. This shows that ui and v are bounded on R, and then using the equations in the system (1.3) yields the boundedness of ui and v as well. This completes the proof of Lemma 2.2.

    In the following two subsections, we will construct the upper and lower solutions of the system (1.3) under c>c:=4d3r, c=c, respectively.

    In this subsection, we always assume that c>c, λ=(cc24d3r)/(2d3), and Λ is the constant in (2.2). We will construct the upper and lower solutions with super-critical wave speed for the system (1.3).

    Now, we introduce the non-negative, continuous and bounded functions ˉui(x), u_i(x), ˉv(x) and v_(x) on R by

    ˉu1(x)=¯u2(x)=K,u_1(x)=u_2(x)={Kβeγx, xa1,0,x>a1,ˉv(x)={eλx,xa2,V0,x>a2,v_(x)={eλx(1Aeηx),xa0,0,x>a0,

    where

    a0=1ηlnA,a1=1γlnβK,a2=1λlnV0.

    Next, in Lemmas 2.3 and 2.4, we give that (ˉu1,ˉu2,ˉv) and (u_1,u_2,v_) constructed above are a pair of upper and lower solutions of the system (1.3) with super-critical wave speed.

    Lemma 2.3. Suppose all the assumptions of Theorem 2.1 are satisfied. For all xR, choose

    max{γ1,γ2}<γ<min{λ,γ+1,γ+2},
    β>max{(ΛKλ/γ1β1)γ/λ, (ΛKλ/γ1β2)γ/λ, K(1V0)γ/λ},

    where

    γ±i=12di(c±c24di(Mi1+Mi2)),  βi=cγdiγ2(Mi1+Mi2),  i=1,2,

    where

    M11=a224a3a10,M12=a0,M21=1,M22=0, (2.3)

    and then we can obtain that the following inequalities

    d1ˉu1(x)cˉu1(x)+a0u2(x)a1ˉu1(x)+a2ˉu21(x)a3ˉu31(x)eˉu1(x)v(x)0,
    d1u_1(x)cu_1(x)+a0u2(x)a1u_1(x)+a2u_21(x)a3u_31(x)eu_1(x)v(x)0,
    d2ˉu2(x)cˉu2(x)+u1(x)ˉu2(x)0,
    d2u_2(x)cu_2(x)+u1(x)u_2(x)0

    hold.

    Proof. According to (2.3) and the assumptions of Theorem 2.1, we have Mi1+Mi2<min{d3/di,1}r, so

    a0u2a1u1+a2u21a3u31M11(u1K)+M12(u2K)M11(Ku1)M12(Ku2). (2.4)

    In addition, we also obtain

    12di(cc24di(Mi1+Mi2))<λ=12d3(cc24d3r),

    and then γ is well defined. If d3/di<1, then we have Mi1+Mi2<(d3/di)r, so this inequality is clearly true. If d3/di1, then Mi1+Mi2<r(d3/di)r, which implies that an equivalent inequality

    Mi1+Mi2c+c24di(Mi1+Mi2)<rc+c24d3r

    holds. Since the choice of γ, we have βi=cγdiγ2(Mi1+Mi2)>0, so β is well defined.

    According to the definitions of a1, u_i are continuous at a1, and by the assumptions on γ, we have

    u_i(x)<ˉui(x),  u_i(a1)=γK<0=u_i(a1+), xR.

    Since ˉu1K, it follows that

     d1ˉu1(x)cˉu1(x)+a0u2(x)a1ˉu1(x)+a2ˉu21(x)a3ˉu31(x)eˉu1(x)v(x)= a0u2(x)a1K+a2K2a3K3eKva0Ka1K+a2K2a3K3eKv= 0,  xR.

    Choose β> K(1/V0)γ/λ such that a1<a2, and then we have for any x<a1 that

    u_1(x)=Kβeγx,  ˉv(x)=eλx.

    According to (2.2) and (2.4), we have

     d1u_1(x)cu_1(x)+a0u2(x)a1u_1(x)+a2u_21(x)a3u_31(x)eu_1(x)v(x) d1u_1(x)cu_1(x)+a0u2(x)a1u_1(x)+a2u_21(x)a3u_31(x)Λv(x) d1u_1(x)cu_1(x)[M11(Ku_1(x))+M12(Ku2(x))]Λˉv(x) d1u_1(x)cu_1(x)[M11(Ku_1(x))+M12(Ku_2(x))]Λˉv(x)= d1βγ2eγx+cβγeγx(M11+M12)βeγxΛeλx= βeγx[γ(cd1γ)(M11+M12)1βΛe(λγ)x] βeγx[γ(cd1γ)(M11+M12)1βΛe(λγ)a1]= βeγx[γ(cd1γ)(M11+M12)ΛKλ/γ1βλ/γ] 0,x<a1,

    where the last inequality is guaranteed by the assumptions on γ and β.

    For x>a1, since u_1(x)=0, we also have

     d1u_1(x)cu_1(x)+a0u2(x)a1u_1(x)+a2u_21(x)a3u_31(x)eu_1(x)v(x)= a0u2(x)0.

    Similarly, we have

    d2ˉu2(x)cˉu2(x)+u1(x)ˉu2(x)0, xR,
    d2u_2(x)cu_2(x)+u1(x)u_2(x)0, xR.

    The proof is completed.

    Lemma 2.4. Let the assumptions of Theorem 2.1 hold and γ satisfy Lemma 2.3, and choose

    β>max{ K(1V0)γ/λ, K},
    0<η<γ,d3(λ+η)2+c(λ+η)r>0,
    A>max{(βK)η/γ, (1δ)η/λ, (βδ)η/γ, 2(1+2β)d3(λ+η)2+c(λ+η)r}.

    Then, for all xR, the inequalities

    d3ˉv(x)cˉv(x)+(b+u1(x)ˉv(x))ˉv(x)0,
    d3v_(x)cv_(x)+(b+u1(x)v_(x))v_(x)0

    hold.

    Proof. We first point out that by the assumptions on γ, β, η, A and the definitions of u_i, ˉv, v_, a0<a1<min{0,a2}, and

    v_(x)<ˉv(x),     xR,
    v_(a0)=ηeλa0<0=v_(a0+),
    ˉv(a2)=λV0>0=ˉv(a2+).

    For x<a2, we have ˉv(x)=eλx, and then

     d3ˉv(x)cˉv(x)+(b+u1(x)ˉv(x))ˉv(x) d3ˉv(x)cˉv(x)+(b+Kˉv(x))ˉv(x)d3ˉvcˉv+(b+K)ˉv(x)= d3ˉv(x)cˉv(x)+rˉv(x)= (d3λ2cλ+r)eλx=0.

    For x>a2, since ˉv(x)=V0, we can obtain

     d3ˉv(x)cˉv(x)+(b+u1(x)ˉv(x))ˉv(x)= (b+u1(x)V0)V0 (b+Kr)r= 0.

    For x<a0, since a0<0<a1<min{0,a2}, 0<η<γ and by the choice of A, we have

    v_(x)=eλxAe(λ+η)x, ˉv(x)=eλx, Ku_i(x)=βeγx,
    v_(x)ˉv(x)eλa0<δ, Kui(x)Ku_i(x)βeγa0<δ.

    By (2.1), one can obtain that

    d3v_(x)cv_(x)+(b+u1(x)v_(x))v_(x) d3v_(x)cv_(x)+rv_(x)2[(Ku1(x))+(Ku2(x))+v_(x)]v_(x) d3v_(x)cv_(x)+rv_(x)2[(Ku1(x))+(Ku2(x))+ˉv(x)]ˉv(x) d3v_(x)cv_(x)+rv_(x)2(βeγx+βeγx+eλx)ˉv(x)= d3v_(x)cv_(x)+rv_(x)2(2β+e(λγ)x)eγxeλx d3v_(x)cv_(x)+rv_(x)2(2β+1)eγxeλx= e(λ+η)x{A[d3(λ+η)2+c(λ+η)r]2(2β+1)e(γη)x} 0.

    For x>a0, we have v_(x)=0, and thus

    d3v_(x)cv_(x)+(b+u1(x)v_(x))v_(x)=0.

    Combining the above, we have proved the assertions of Lemma 2.4.

    In this subsection, we always assume that c:=4d3r, λ=c/(2d3), M=λeV0 and Λ is the constant in (2.2). We will construct the upper and lower solutions with super-critical wave speed for the system (1.3).

    There exists a large R0>0 so that for all RR0, we define

    ˉu1=ˉu2K,u_1(x)=u_2(x)={Kβeγx,xa1,0,  x>a1,
    ˉv(x)={M|x|eλx,xa2,v0,  x>a2,v_(x)={(M|x|R|x|)eλx,xa0,0,  x>a0,

    where

    a2=1λ,a1=1γln1β,a0=R2M2.

    Next, in Lemmas 2.5 and 2.6, we will prove (ˉu1,ˉu2,ˉv) and (u_1,u_2,v_) constructed above are a pair of upper and lower solutions of the system (1.3) with critical wave speed.

    Lemma 2.5. Let the assumptions of Theorem 2.1 hold, and Mi1 and Mi2 are fixed points and satisfy Lemma 2.3. For all xR, choose auxiliary constants γ and β such that

    max{γ1,γ2}<γ<min{λ,γ+1,γ+2},β>max{eγλγ,MΛβ1(λγ)e,MΛβ2(λγ)e}.

    Then, for all xR, the inequalities

    d1ˉu1(x)cˉu1(x)+a0u2(x)a1ˉu1(x)+a2ˉu21(x)a3ˉu31(x)eˉu1(x)v(x)0,
    d1u_1(x)cu_1(x)+a0u2(x)a1u_1(x)+a2u_21(x)a3u_31(x)eu_1(x)v(x)0,
    d2ˉu2(x)cˉu2(x)+u1(x)ˉu2(x)0,
    d2u_2(x)cu_2(x)+u1(x)u_2(x)0

    hold.

    Proof. Similar to Lemma (2.3), it is easy to get

    d1ˉu1(x)cˉu1(x)+a0u2(x)a1ˉu1(x)+a2ˉu21(x)a3ˉu31(x)eˉu1(x)v(x)0,
    d2ˉu2(x)cˉu2(x)+u1(x)ˉu2(x)0.

    For x<a1, we can obtain a1<a2 from β>eγ/λγ, and then

    u_1(x)=Kβeγx, ˉv(x)=M|x|eλx,
    u_1(x)=βγ2eγx, u_1(x)=βγeγx.

    By (2.2) and (2.4), we have

     d1u_1(x)cu_1(x)+a0u2(x)a1u_1(x)+a2u_21(x)a3u_31(x)eu_1(x)v(x) d1u_1(x)cu_1(x)+a0u2(x)a1u_1(x)+a2u_21(x)a3u_31(x)Λv(x) d1u_1(x)cu_1(x)[M11(Ku_1(x))+M12(Ku_2(x))]Λˉv(x)= d1βγ2eγx+cβγeγx(M11+M12)βeγxΛM|x|eλx= βeγx[γ(cd1γ)(M11+M12)1βΛM|x|e(λγ)].

    Since |x|e(λγ)x is monotone increasing over (,1/(λγ)), and a1<1/(λγ), it follows that |x|e(λγ)x1/((λγ)e) for x<a1, and then by the choice of β, we have

     d1u_1(x)cu_1(x)++a0u2(x)a1u_1(x)+a2u_21(x)a3u_31(x)eu_1(x)v(x) βeγx[γ(cd1γ)(M11+M12)MΛβ(λγ)e] 0,x<a1.

    Moreover,

    u_1(a1)=βγeγa10=u_1(a1+).

    Similarly, we have

    d2u_2(x)cu_2(x)+u1(x)u_2(x)0,

    and

    u_2(a1)=βγeγa10=u_2(a1+).

    This ends the proof.

    Lemma 2.6. Let the hypothesis of Theorem 2.1 and Lemma 2.5 be satisfied. For all xR, the inequalities

    d3ˉv(x)cˉv(x)+(b+u1(x)ˉv(x))ˉv(x)0,
    d3v_(x)cv_(x)+(b+u1(x)v_(x))v_(x)0

    hold.

    Proof. For x<a0, we have

    (v_+Mxeλx)=(R2xλRx)eλx=R(12xxλ)eλx,
    (v_+Mxeλx)=R(14xx+1xλxλ2)eλx,

    and so

    d3v_cv_+rv_= R(d34xx+d3xλd3xλ2c2x+cxλ)eλxrRxeλx= d3R4xxeλx.

    Since a0<0, and R is large enough to ensure that a0<a1<a2 and a0<1, for x<a0, we have

    v_(x)=(M|x|R|x|)eλx, ˉv(x)=M|x|eλx, Ku_i(x)=βeγx,

    and

    v_(x)ˉv(x)<δ, Kui(x)Ku_i(x)βeγa0<δ.

    Hence, for such R and x,

     (b+u1(x)v_(x))v_(x) rv_(x)2[(Ku1(x))+(Ku2(x))+v_(x)]v_(x) rv_(x)2[(Ku_1(x))+(Ku_2(x))+ˉv(x)]ˉv(x)= rv_(x)2(βeγx+βeγx+M|x|eλx)ˉv(x)= rv_(x)2M(2β+M|x|e(λγ)x)eγx|x|eλx rv_(x)2M(2β+M|x|)|x|eγxeλx.

    So,

     d3v_cv_+(b+u1(x)v_(x))v_(x) [d3R4xx2M(2β+M|x|)|x|eγx]eλx= 14|x||x|[d3R2M(2β+M|x|)4x2|x|eγx]eλx 14|x||x|[d3R2M(2β+M)x4eγx]eλx 0.

    Using

    v_(x)=[M+R2x+λ(MxRx)]eλx,

    we have

    v_(a0)=[M+R2|a0|]eλa0=M2eλa0<0=v_(a0+).

    For x<a2=1/λ, we have

    ˉv(x)=Mxeλx, ˉv(x)=M(1+λx)eλx,

    so that

    ˉv(a2)=M/(λe)=v0, ˉv(a2)=0=ˉv(a2+),

    and then we have

    d3ˉvcˉv+(b+u1(x)v_(x))v_(x)d3ˉvcˉv+(b+Kv_(x))v_(x)d3ˉvcˉv+(b+K)v_(x)=d3ˉvcˉv+rv(x)=0.

    We thus conclude the proof of Lemma 2.6.

    Proof of Theorem 2.1. We first show that for any cc:=4d3r, the system (1.3) has a solution (u1,u2,u3) satisfying Theorem 2.1. Applying all the Lemmas above with U10=U20=K yields the existence of a solution (u1,u2,v) to the system (1.3), satisfying u_iuiˉui and v_vˉv. The definitions of u_i, ˉui, v_ and ˉv imply that (u1,u2,v)(x)(K,K,0) as x, that, after a translation in x, 0<ui(x)K and 0<v(x)V0 for x0, that 0ui(x)K and 0v(x)V0 for x>0. Using the expressions

    ui(x)=ecxdiui(0)+0xec(xy)diFi(u1(y),u2(y),v(y))dy,v(x)=ecxd3u(0)+1d0xec(xy)d3G(u1(y),u2(y),v(y))dy,

    and L'Hospital's rule, we get (u1(x),u2(x),v(x))0 as x. Therefore, (u1,u2,v) is a weak traveling wave solution to the system (1.3).

    Next, we prove that for any 0<c<4d3r, there is no weak traveling wave solution of the system (1.3). Under the assumptions, we can write in a neighborhood of (K,K,0) the v equation in (1.3) as

    d3vcv+rv+(g(u1,u2,v)r)v=0,

    where g(K,K,0)=b+K=r, and g(u1,u2,v)r0 as (u1,u2,v)(K,K,0). The characteristic equation d3λ2cλ+r=0 has a pair of complex roots λ=(c±i4d3rc2)/(2d3). Assume by contradiction there is a solution (u1,u2,v) of (1.3) satisfying (u1(x),u2(x),v(x))(K,K,0) as x and v(x)>0 for sufficiently negative x. Then, using the variation of constants formula, one can show that, for sufficiently negative x0 and x,

    v(x)=eα(xx0){v(x0)cosβ(xx0)+1β(v(x0)αv(x0))sinβ(xx0)}(1+R(x,x0)),

    where limx0supx<x0|R(x,x0)|=0. (See [1] for details.) This asymptotic expression shows that v(x) changes signs infinitely many times as x, a contradiction.

    The proof is completed.

    In this section, we prove that under certain conditions, the weak traveling wave solution obtained in Theorem 2.1 is also a traveling wave solution by using the Lyapunov function and LaSalle's invariance principle.

    Theorem 3.1. Assume that all conditions in Theorem 2.1 are satisfied, and a3(eb+a0a1)>a2e holds. Then, the system (1.3) has a traveling wave solution (u1,u2,v) satisfying (1.4) for every c4d3r.

    Proof. Let (u1,u2,v) be a weak traveling wave solution of the system (1.3). By Theorem 2.1, there are δ>0 and x0>0 such that ui(x)>δ for xR and v(x)>δ for x>x0. This implies that the orbit (u1,u1,u2,u2,v,v)(x) lies in the set Ωδ=:([δ,K]×R)2×[δ,V0]×R for x>x0. To show that (u1,u1,u2,u2,v,v)(x)(u1,0,u1,0,v,0) as x, it is necessary to define a Lyapunov function L on ((0,K]×R)2×(0,V0]×R by

    L(u1,u1,u2,u2,v,v)=cH(u1,u2,v)d1Hu1u1d2Hu2u2d3Hvv,

    where

    H(u1,u2,v)=α1(u1u1u1lnu1u1)+α2(u2u2u2lnu2u2)+α3(vvvlnvv),

    and α1, α2, α3 are non-negative constants. Then, along the orbits of the system (1.3) with x>x0, we have

    ddxL=(Hu1F1(u1,u2,v)+Hu2F2(u1,u2,v)+HvG(u1,u2,v))d12Hu21(u1)2d22Hu22(u2)2d32Hv2(v)2, (3.1)

    where

    F1(u1,u2,v)=a0u2a1u1+a2u21a3u31eu1v,F2(u1,u2,v)=u1u2, G(u1,u2,v)=(b+u1v)v.

    Then, we have

                 Hu1F1(u1,u2,v)+Hu2F2(u1,u2,v)+HvG(u1,u2,v)   =α1u1u1u1×{a0u1[u1(u2u2)u2(u1u1)]u1(u1u1)(a2+a3u1+a3u1)eu1(vv)}+α2u2u2u21u2[u2(u1u1)u1(u2u2)]+α3v[(u1u1)(vv)]vvv=α1(a2+a3u1+a3u1)(u1u1)2α3(vv)2+(α3kα1)(u1u1)(vv)+a0α1(u1u1)×[u2u1u1(u1u1)+1u1(u2u2)]+α2(u2u2)(u1u1)u2(u2u2)u1u2u2.

    Let α2=a0α1, α3=eα1. Thus,

    Hu1F1(u1,u2,v)+Hu2F2(u1,u2,v)+HvG(u1,u2,v)=α1(a2+a3u1+a3u1)(u1u1)2α3(vv)2a0α11u1[u2u1(u1u1)u1u2(u2u2)]2.

    We observe that a3(eb+a0a1)>a2e is a sufficient condition of a2+a3u1+a3u1>0. So, when a3(eb+a0a1)>a2e holds, we have

    Hu1F1(u1,u2,v)+Hu2F2(u1,u2,v)+HvG(u1,u2,v)<0.

    Since

    di2Hu2i(ui)2=diuiu2i(ui)2=diui(uiui)2,d32Hv2(v)2=d3vv2(v)2=d3v(vv)2, (3.2)

    we can obtain that dL(u1,u1,u2,u2,v,v)/dx0 for (u1,u1,u2,u2,v,v)Ωδ.

    Let ρi=ui/ui,  ρ3=v/v. Next, we prove that |ρi|ρ+i, |ρ3|ρ+3.

    Since 0<uiK and 0<vV0, there exists a positive constant ¯M>0 such that max{a1a2u1+a3u21+ev,bu1+v,1}¯M for all xR. We have

          ρ1=u1u1(u1)2u21=u1u1ρ21=cd1ρ1a0u2d1u1+1d1(a1a2u1+a3u21+ev)ρ21ρ21+cd1ρ1+¯Md1.

    Let ρ+1 be a positive constant solution of ρ=ρ2+(c/d1)+¯M/d1. According to the comparison theorem, we have ρ1(x)<ρ+1 for all xR. Similarly, if ρ1<ρ+1 occurs at some x0, then, letting ρ(x) be the solution of ρ=ρ2+(c/d1)ρ+¯M/d1 with ρ(x0)=ρ1(x0), it follows from the comparison theorem that ρ1(x)ρ(x) for xx0.

    Note that

    ρ2(x0)+cd1ρ(x0)¯Md1<(ρ+1)2+cd1(ρ+1)+¯Md1<0,

    implies ρ(x) as xx1 for some finite value x1>x0. It follows that ρ1(x) as xx2 for some x2(x0,x1], contradicting the fact that ρ1(x) is defined for all xR.

    Similarly,

          ρ2=u2u2(u2)2u22=cd2ρ2u1d2u2+1d2ρ22ρ22+cd2ρ2+¯Md2,

    and

          ρ3=vv(v)2v2=cd3ρ31d3(b+u1v)ρ23ρ23+cd3ρ3+¯Md3,

    and there exist constants ρ+2>0,ρ+3>0 such that |ρ2|ρ+2, |ρ3|ρ+3 for (u1,u2,v)(0,K]2×(0,V0]{(u1,u2,v)}.

    Since equality holds only at E, we derive that (u1,u1,u2,u2,v,v)(x)E as x by LaSalle's invariance principle. This proves Theorem 3.1.

    The main results of this work can be summarized as follows: This paper is concerned with traveling wave solutions to a cubic predator-prey diffusion model with stage structure for the prey given by system (1.2). First, employing the method of upper and lower solutions together with the Schauder fixed point theorem, we give a sharp existence result on weak traveling wave solutions for system (1.2), with minimal speed explicitly determined. Such a weak traveling wave (u1(ξ),u2(ξ),v(ξ)) connects the semi-trivial equilibrium (K,K,0) at ξ= but needs not connect the coexistence equilibrium (u1,u2,v) at ξ= (see Theorem 2.1). Then, we use the Lyapunov function method and LaSalle's invariance principle to prove that, under additional conditions, the weak traveling wave solutions of (1.2) established in Theorem 2.1 are actually traveling wave solutions; namely, they converge to the coexistence equilibrium (u1,u2,v) as ξ (see Theorem 3.1). To the best of the authors' knowledge, the results in Theorems 2.1 and 3.1 are new.

    The authors first thank the referees for their valuable comments and suggestions. In addition, Yujuan Jiao would like to express her gratitude to Dr. Shangbing Ai for his helpful discussions on an earlier draft of this work while she visited the Department of Mathematical Sciences of the University of Alabama in Huntsville in 2019. Yujuan Jiao was supported by the Fundamental Research Funds for the Central Universities (31920190057, 31920220041), and the Innovation Team of Intelligent Computing and Application of Northwest Minzu University.

    All authors declare that they have no competing interests.



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