Nonlinear Analysis: Real World Applications 11 (2010) 3666–3679

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Nonlinear Analysis: Real World Applications

journal homepage: www.elsevier.com/locate/nonrwa

Traveling waves in delayed lattice dynamical systems withcompetition interactionsI

Guo Lin ∗, Wan-Tong LiSchool of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China

a r t i c l e i n f o

Article history:Received 28 April 2009Accepted 27 January 2010

Keywords:Cross-iterationUpper and lower solutionsCompetition interactionTraveling waveMonotone dynamical systems

a b s t r a c t

This paper deals with the existence of traveling wave solutions of a class of delayedsystem of lattice differential equations, which formulates the invasion process when twocompetitive species are invaders. Employing the comparison principle of competitivesystems, a new cross-iteration scheme is given to establish the existence of traveling wavesolutions. More precisely, by the cross-iteration, the existence of traveling wave solutionsis reduced to the existence of an admissible pair of upper and lower solutions. To illustrateour main results, we prove the existence of traveling wave solutions in two delayed two-species competition systems with spatial discretization. Our results imply that the delayappeared in the interspecific competition terms do not affect the existence of travelingwave solutions.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Lattice dynamical systems are infinite systems of ordinary differential equations (so-called lattice ODE) or of differenceequations (so-called coupled map lattices), indexed by points in a lattice, such as the D-dimensional integer lattice ZDwhich incorporate some aspect of the spatial structure of the lattice. Such systems arise, on the one hand, from practicalbackgrounds, such as modeling the population growth over patchy environments [1–3] and the phase transitions [4,5]. Onthe other hand, they also arise as the spatial discretization of partial differential equations; we refer to [6–10].In the past decades, more and more evidence indicates that the traveling wave solutions play an important role in the

study of lattice dynamical systems. More precisely, the traveling wave solutions can determine the long term behavior ofthe corresponding initial value problems of lattice dynamical systems, which partly arise from the stability of travelingwave solutions; e.g., we can refer to [11–15]. At the same time, the traveling wave solutions in lattice differential equationsmay describe many important phenomena in physical systems, population dynamics and other fields; see [4,16,17] and thereferences cited therein.For the single lattice differential equation, a typical example is

dun(t)dt= D (un+1(t)− 2un(t)+ un−1(t))+ f (un(t)), n ∈ Z, t > 0, (1.1)

which was initially used in Bell and Cosner [1] to model myelinated axons in nerve systems. They studied the long timebehavior of solutions to (1.1) for some nonlinear function f . The traveling wave solutions were analytically discussed in[18] and numerically computed in [19]. Keener [2] analyzed propagation and its failure for (1.1). In particular, when the

I Supported by NNSF of China (10871085, 10926090), NSF of Gansu Province of China (096RJZA051) and The Fundamental Research Fund for Physicsand Mathematic of Lanzhou University (LZULL200902).∗ Corresponding author.E-mail address: ling@lzu.edu.cn (G. Lin).

1468-1218/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2010.01.013

G. Lin, W.-T. Li / Nonlinear Analysis: Real World Applications 11 (2010) 3666–3679 3667

nonlinear term in (1.1) is of bistable type, the study on travelingwave fronts of such lattice differential equations is extensiveand intensive, and has resulted in many interesting and significant results, some of which have revealed some essentialdifferences between adiscretemodel and its continuous version—parabolic equations (see [20]). For details, see, for example,[1,2,4,5,8,21–25], and the references therein.When the nonlinear term is ofmonostable type, Zinner et al. [26] addressed theexistence and minimal speed of traveling wave front for discrete Fisher equation. Recently, Chen and Guo [12,27] discusseda more general case of equation

dun(t)dt= D (g(un+1(t))− 2g(un(t))+ g(un−1(t)))+ f (un(t)), n ∈ Z, t > 0, (1.2)

where g(u) is increasing and f (u) is monostable, and the authors established the existence, uniqueness and stability as wellas minimal wave speed for (1.2).It is well known that, inmodeling population growth and transition of signals in the nerve systems, temporal delays seem

to be inevitable, accounting for thematuration time of the species under consideration and the time needed for the signals totravel along axons and to cross synapses. Based on such a consideration, Wu and Zou [28] first studied the existence of trav-eling wave solutions of delayed lattice differential equations. Liang and Zhao [29] further considered the spatial–temporalpattern of these models by monotone dynamical systems. For more results, see [3,7,13,14,30–35].For systems of lattice differential equations, a simple example is

dun(t)dt= d1 (un+1(t)− 2un(t)+ un−1(t))+ un(t)(r1 − b1vn(t)),

dvn(t)dt= d2 (vn+1(t)− 2vn(t)+ vn−1(t))+ vn(t)(−r2 + b2un(t)),

which was proposed by Renshaw [36] to model morphogenesis growth, and is also called the Turing model. In 1995,Anderson and Sleeman [37] investigated a spatially discretized Fitzhugh–Nagumo system with monotone reaction terms,and studied the existence and propagation failure of traveling wave fronts; also see Nekorkin et al. [38]. Recently, Huangand Lu [39] and Huang et al. [40] considered the following delayed lattice systems

dun(t)dt=

m∑j=1

aj[g(un+j(t))− 2g(un(t))+ g(un−j(t))] + f1(unt , vnt),

dvn(t)dt=

m∑j=1

bj[g(vn+j(t))− 2g(vn(t))+ g(vn−j(t))] + f2(unt , vnt),

where n ∈ Z, t > 0, un and vn are continuous functions, g is a continuous function, f1 and f2 are continuous functions definedon functional space C([−τ , 0],R2) and valued in R, unt ∈ C([−τ , 0],R) defined by un(t + s) is a continuous function fors ∈ [−τ , 0], so for vnt . By using the idea of Huang et al. [31] and Wu and Zou [28] for delayed lattice differential equations,Huang and Lu [39], Huang et al. [40] and Lin et al. [41] established the existence of traveling wave solutions connectingtrivial equilibrium (0, 0)with nontrivial one (k1, k2), if the reaction terms satisfy the so-called (exponential) quasimonotonecondition or the partial (exponential) quasimonotone condition. For related results on reaction–diffusion equations withdelays, we refer to [42–53] and references cited therein.However, it is quite common that the reaction terms in somemodelsmay not satisfy themonotone conditionsmentioned

above, such as the following two systems of lattice ODEsdun(t)dt= d1[un+1(t)− 2un(t)+ un−1(t)] + r1un(t) [1− a1un(t)− b1vn(t − τ1)] ,

dvn(t)dt= d2[vn+1(t)− 2vn(t)+ vn−1(t)] + r2vn(t) [1− b2un(t − τ2)− a2vn(t)] ,

(1.3)

and dun(t)dt= d1[un+1(t)− 2un(t)+ un−1(t)] + r1un(t) [1− a1un(t − τ1)− b1vn(t − τ2)] ,

dvn(t)dt= d2[vn+1(t)− 2vn(t)+ vn−1(t)] + r2vn(t) [1− b2un(t − τ3)− a2vn(t − τ4)] ,

(1.4)

which can be regarded as spatially discrete versions of the diffusion competition systems with delays considered in [54].It is evident that (1.3) and (1.4) do not satisfy the monotone conditions in [28,29,31,39–41] when we are interested in thetraveling wave solutions connecting (0, 0) with positive equilibrium (if it exists). Furthermore, (1.3) and (1.4) can satisfythe monotone condition in [29,31,41] by change of variables, but the new interested equilibria are not ordered such thatthe known results fail. Due to the ecology sense of trivial and coexistence equilibria of (1.3) and (1.4), such a traveling wavesolution is very important in modeling the population invasions when two species are competitive invaders. Therefore, itis worthwhile to further explore this topic for lattice differential systems including (1.3) and (1.4), and this constitutes thepurpose of this paper. For the related topics in reaction–diffusion systems, we refer to [46,54–57].

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In order to focus on the mathematical ideas and for the sake of simplicity, we consider the following delayed system oftwo lattice ODEs

dun(t)dt= g1(un+1(t))+ g1(un−1(t))− 2g1(un(t))+ f1(unt , vnt),

dvn(t)dt= g2(vn+1(t))+ g2(vn−1(t))− 2g2(vn(t))+ f2(unt , vnt),

(1.5)

where n ∈ Z, t ∈ R, g1, g2 : R → R, let τ > 0 be the maximal delay involved by (1.5), then fi : C([−τ , 0],R2) → R.Furthermore, we also give the following assumptions.

(A1) There exist k1 > 0, k2 > 0 such that fi(̂0, 0̂) = fi(̂k1, k̂2) = 0, i = 1, 2, herein ·̂ denotes the constant valuedfunction in C([−τ , 0],R), and there existMi > ki such that [0,M1] × [0,M2] is a positive invariant region of thefollowing ODEs

du(t)dt= f1(u(t), v(t)),

dv(t)dt= f2(u(t), v(t)).

(A2) There exists a constant L = L(M1,M2) > 0 such that

|f1(φ1t , ψ1t)− f1(φ2t , ψ2t)| ≤ L ‖Φ − Ψ ‖ ,|f2(φ1t , ψ1t)− f2(φ2t , ψ2t)| ≤ L ‖Φ − Ψ ‖

for any Φ = (φ1, ψ1) and Ψ = (φ2, ψ2) ∈ C([−τ , 0],R2) with 0 ≤ φit , ψit ≤ Mi, where ‖ · ‖ denotes thesupremum norm in C([−τ , 0],R2).

(A3) gi : [0,Mi] → R is Lipschitz continuous and nondecreasing with i = 1, 2. For the sake of convenience, we alwaysassume that the Lipschitz constant is Q > 0.

Moreover, we propose the following monotone conditions including (1.3) and (1.4) as two special cases, which will becalled competitive quasimonotone condition (CQM)

(CQM) There exist two positive numbers β1 and β2 such that{f1(φ1t , ψ2t)− f1(φ2t , ψ1t)+ β1(φ1(t)− φ2(t)) ≥ 2[g1(φ1(t))− g1(φ2(t))],f2(φ1t , ψ1t)− f2(φ2t , ψ2t)+ β2(ψ1(t)− ψ2(t)) ≥ 2[g2(ψ1(t))− g2(ψ2(t))]

for any t ∈ R, where (φ1t , ψ1t) and (φ2t , ψ2t) satisfy the following conditions:

(i) (φ1t , ψ1t), (φ2t , ψ2t) ∈ C([−τ , 0],R2);(ii) 0 ≤ φ2t ≤ φ1t ≤ M1, 0 ≤ ψ2t ≤ ψ1t ≤ M2,

or exponential competitive quasimonotone condition (ECQM)

(ECQM) There exist two positive numbers β1 and β2 such that{f1(φ1t , ψ2t)− f1(φ2t , ψ1t)+ β1(φ1(t)− φ2(t)) ≥ 2[g1(φ1(t))− g1(φ2(t))],f2(φ1t , ψ1t)− f2(φ2t , ψ2t)+ β2(ψ1(t)− ψ2(t)) ≥ 2[g2(ψ1(t))− g2(ψ2(t))]

for all t ∈ R, where (φ1t , ψ1t) and (φ2t , ψ2t) satisfy the following conditions:

(i) (φ1t , ψ1t), (φ2t , ψ2t) ∈ C([−τ , 0],R2);(ii) 0 ≤ φ2t ≤ φ1t ≤ M1, 0 ≤ ψ2t ≤ ψ1t ≤ M2;(iii) eβ1t(φ1(t)− φ2(t)) and eβ2t(ψ1(t)− ψ2(t)) are nondecreasing for t ∈ R.

Since the nonlinear functions f1 and f2 in (1.5) have differentmonotonicitywith respect to the first and second argumentsin the first and second equations, respectively, following Leung [55], Li et al. [54], Lin et al. [46], Pao [56] and Ye and Li [57],we will introduce definitions of the upper and lower solutions and a new cross-iteration scheme, which are different fromthat defined in [28,29,31,39–41]. Subsequently, we will construct a subset in the Banach space C(R,R2) equipped with thesupremum norm, and reduce the existence of traveling wave solutions to the existence of an admissible pair of upper andlower solutions. As applications, we establish the existence of traveling wave solutions of (1.3) and (1.4).This paper is organized as follows. Section 2 is devoted to some necessary notations, definitions and discussions. In

Section 3, we establish a cross-iteration scheme to obtain the existence of a traveling wave solution if the system (1.5)satisfies (CQM). In Section 4, we use the exponential partial ordering of the profile set to consider the existence of travelingwave solutions if (1.5) satisfies (ECQM). Finally, we apply our main results to lattice differential systems (1.3) and (1.4) andprove the existence of traveling wave solutions.

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2. Preliminaries

In this section, we first list the usual notations for the standard partial ordering in R2. That is, for u = (u1, u2) and v =(v1, v2), we denote u ≤ v if ui ≤ vi, i = 1, 2, and u < v if u ≤ v but u 6= v. We also denote [u, v] = {w ∈ R2, u ≤ w ≤ v}.Let C(R,R2) be a set, which consists of the bounded and uniform continuous functions defined on R. Define

‖x‖C(R,R2) = max1≤i≤2

supt∈R{|xi (t)|} , x = (x1, x2) ∈ C(R,R2),

then C(R,R2) is a Banach space with supremum norm ‖ · ‖. Denote

C[0,M](R,R2) = {u : u = (u1, u2) ∈ C(R,R2), 0 ≤ ui ≤ Mi, i = 1, 2}.

A traveling wave solution of (1.5) is a special translation invariant solution of the form un(t) = φ(t − nc), vn(t) =ψ(t − nc), where φ,ψ ∈ C1(R,R) are the so-called profiles of the wave that propagate through the one-dimensionalspatial lattice at a constant velocity 1/c > 0. Thus, φ,ψ satisfy the following mixed functional differential equations (if westill denote traveling wave coordinate t − nc as t)

dφ(t)dt= g1(φ(t + c))+ g1(φ(t − c))− 2g1(φ(t))+ f1(φt , ψt),

dψ(t)dt= g2(ψ(t + c))+ g2(ψ(t − c))− 2g2(ψ(t))+ f2(φt , ψt).

(2.1)

Becausewe are interested in the invasion process of two competitive invaders,we also require thatφ,ψ satisfy the followingasymptotic boundary conditions

limt→−∞

(φ(t), ψ(t)) = 0, limt→+∞

(φ(t), ψ(t)) = K . (2.2)

Remark 2.1. In this paper, we shall consider the invasion waves, so c > 0 is required. Otherwise, c < 0 is also admissible,see the results of bistable equations. Moreover, for the sake of convenience, we denote the traveling wave coordinate ast − cn, but the true wave speed is 1/c. In fact, c in above definition denotes the period that the wave profile moves perdistance, so 1/c is the wave speed that describes the phase shift of wave profile per time.

Define the operator H = (H1,H2) : C(R,R2)→ C(R,R2) by{H1(φ, ψ)(t) = f1(φt , ψt)+ β1φ(t)+ g1(φ(t + c))+ g1(φ(t − c))− 2g1(φ(t)),H2(φ, ψ)(t) = f2(φt , ψt)+ β2ψ(t)+ g2(ψ(t + c))+ g2(ψ(t − c))− 2g2(ψ(t)).

(2.3)

Then (2.1) can be rewritten as follows{φ′(t)+ β1φ(t)− H1(φ, ψ)(t) = 0, t ∈ R,ψ ′(t)+ β2ψ(t)− H2(φ, ψ)(t) = 0, t ∈ R. (2.4)

Furthermore, according to (2.4), define F = (F1, F2) : C[0,M](R,R2)→ C(R,R2) byF1(φ, ψ)(t) = e−β1t

∫ t

−∞

eβ1sH1(φ, ψ)(s)ds, t ∈ R,

F2(φ, ψ)(t) = e−β2t∫ t

−∞

eβ2sH2(φ, ψ)(s)ds, t ∈ R.(2.5)

It is evident that F is well defined. Moreover, if (2.5) has a fixed point (φ, ψ), then it is a solution of (2.1) and (2.2), and so itis also a traveling wave solution of (1.5) connecting (0, 0) with (k1, k2). Therefore, it is sufficient to prove the existence offixed points of F satisfying (2.2), which is our main goal in what follows.

3. The Case (CQM)

In this section, we shall study the existence of traveling wave solutions of (2.1) when the delayed reaction terms f1 and f2satisfy the condition (CQM). For the purpose, we give the following definition of a pair of upper and lower solutions of (2.1).

Definition 3.1. Assume that ρ = (φ, ψ), ρ = (φ, ψ) ∈ C[0,M](R,R2) and (CQM) or (ECQM) hold. If ρ ′, ρ ′ exist onR \T andρ ′, ρ ′, ρ, ρ satisfy{

φ′(t) ≥ g1(φ(t + c))+ g1(φ(t − c))− 2g1(φ(t))+ f1(φt , ψ t), t ∈ R \ T,

ψ′(t) ≥ g2(ψ(t + c))+ g2(ψ(t − c))− 2g2(ψ(t))+ f2(φt , ψ t), t ∈ R \ T,{

φ′(t) ≤ g1(φ(t + c))+ g1(φ(t − c))− 2g1(φ(t))+ f1(φt , ψ t), t ∈ R \ T,ψ ′(t) ≤ g2(ψ(t + c))+ g2(ψ(t − c))− 2g2(ψ(t))+ f2(φt , ψ t), t ∈ R \ T,

3670 G. Lin, W.-T. Li / Nonlinear Analysis: Real World Applications 11 (2010) 3666–3679

respectively, where T = {T1, T2, . . . , Tk}with T1 < T2 < · · · < Tk. Then ρ is called an upper solution and ρ is called a lowersolution of (2.1).

To discuss the fixed point of (2.5) by Schauder’s fixed point theorem, we also assume that (2.1) has a pair of upper andlower solutions ρ = (φ, ψ) and ρ = (φ, ψ) ∈ C(R,R2) such that

(P1) 0 ≤ ρ ≤ ρ ≤ M;(P2) limt→−∞ ρ(t) = limt→−∞ ρ(t) = 0, limt→∞ ρ(t) = limt→∞ ρ(t) = K .

The operators H and F defined in Section 2 enjoy the following comparison principle.

Lemma 3.2. Assume that (CQM) holds. Then

H1(φ1, ψ2)(t) ≤ H1(φ2, ψ1)(t), H2(φ2, ψ1)(t) ≤ H2(φ1, ψ2)(t),F1(φ1, ψ2)(t) ≤ F1(φ2, ψ1)(t), F2(φ2, ψ1)(t) ≤ F2(φ1, ψ2)(t),

where (φ1, ψ1), (φ2, ψ2) ∈ C[0,M](R,R2) satisfy (φ1, ψ1) ≤ (φ2, ψ2).

The proof of the lemma is trivial, so we omit it here.

Lemma 3.3. Assume that (A2) and (A3) hold. Then

F = (F1, F2) : C[0,M](R,R2)→ C(R,R2)

is continuous with respect to the norm ‖·‖ .

Proof. We first prove that F1 maps C[0,M](R,R2) into C(R,R). For t ∈ R, δ > 0 and any (φ, ψ) ∈ C[0,M](R,R2), F(φ, ψ) istotally bounded by Lemma 3.2, and

|F1(φ, ψ)(t + δ)− F1(φ, ψ)(t)|

=

∣∣∣∣e−β1(t+δ) ∫ t+δ

−∞

eβ1sH1(φ, ψ)(s)ds− e−β1t∫ t

−∞

eβ1sH1(φ, ψ)(s)ds∣∣∣∣

≤

∫ t

−∞

∣∣[e−β1(t+δ)+β1s − e−β1t+β1s]H1(φ, ψ)(s)∣∣ ds+ ∫ t+δ

t

∣∣e−β1(t+δ)+β1sH1(φ, ψ)(s)∣∣ ds= (1− e−β1δ)

∫ t

−∞

∣∣e−β1t+β1sH1(φ, ψ)(s)∣∣ ds+ ∫ t+δ

t

∣∣e−β1(t+δ)+β1sH1(φ, ψ)(s)∣∣ ds.Note that

∫ t−∞

∣∣e−β1t+β1sH1(φ, ψ)(s)∣∣ ds and ∣∣e−β1(t+δ)+β1sH1(φ, ψ)(s)∣∣ are uniformly bounded for (φ, ψ) ∈ C[0,M](R,R2)and t ∈ R, we obtain

|F1(φ, ψ)(t + δ)− F1(φ, ψ)(t)| → 0, δ→ 0

uniformly for t ∈ R and (φ, ψ) ∈ C[0,M](R,R2), that is, F1(φ, ψ)(t) is equicontinuous. Similarly, F2(φ, ψ)(t) is equicontin-uous for t ∈ R and (φ, ψ) ∈ C[0,M](R,R2).We now prove the mapping F is continuous. Assume thatΦ = (φ, ψ),Ψ = (φ1, ψ1) ∈ C[0,M](R,R2). Then

|F1(φ1, ψ1)(t)− F1(φ, ψ)(t)|

=

∣∣∣∣e−β1t ∫ t

−∞

eβ1s[H1(φ1, ψ1)− H1(φ, ψ)](s)ds∣∣∣∣

≤ e−β1t∫ t

−∞

eβ1s |[f1(φ1s, ψ1s)− f1(φs, ψs)]| ds+ β1e−β1t∫ t

−∞

eβ1s |φ1 − φ| ds

+

∫ t

−∞

e−β1(t−s) [|g1(φ1(s+ c))− g1(φ(s+ c))| + |g1(φ1(s− c))− g1(φ(s− c))| + 2 |g1(φ1(s))− g1(φ(s))|] ds

≤ (L+ β1) ‖Φ − Ψ ‖∫ t

−∞

e−β1(t−s)ds+ 4Q ‖Φ − Ψ ‖∫ t

−∞

e−β1(t−s)ds

≤L+ β1 + 4Q

β1‖Φ − Ψ ‖ .

For any fixed ε > 0, let 2δ = β1ε/[L+ β1 + 4Q ]. If ‖Φ − Ψ ‖ < δ, then

|F1(φ1, ψ1)(t)− F1(φ, ψ)(t)| < ε/2. (3.1)

G. Lin, W.-T. Li / Nonlinear Analysis: Real World Applications 11 (2010) 3666–3679 3671

Note that the above inequality holds for t ∈ R, so (3.1) implies that

supt∈R|F1(φ1, ψ1)(t)− F1(φ, ψ)(t)| < ε.

Similarly, we can prove the mapping F2 is continuous. The proof is complete. �

Define the profile set Γ as follows

Γ (ρ, ρ) = {(φ, ψ) ∈ C(R,R2) : ρ ≤ (φ, ψ) ≤ ρ},

in which we shall prove the existence of traveling wave solutions. Due to (P1) and (P2), the following result is clear.

Lemma 3.4. Γ (ρ, ρ) is bounded and closed with respect to the supremum norm, and it is a nonempty and convex subset ofC(R,R2).

Lemma 3.5. Assume that (CQM) holds. Then F(Γ (ρ, ρ)

)⊂ Γ (ρ, ρ).

Proof. By Lemma 3.2, it suffices to prove that{φ ≤ F1(φ, ψ) ≤ F1(φ, ψ) ≤ φ,ψ ≤ F2(φ, ψ) ≤ F2(φ, ψ) ≤ ψ.

(3.2)

According to the definition of upper and lower solutions, we obtain

φ′(t)+ β1φ(t)− H1(φ, ψ) ≥ 0, t ∈ R \ T.

Let T0 = −∞ and Tk+1 = +∞. Then∫ t

−∞

e−β1(t−s)H1(φ, ψ)(s)ds ≤

(i−1∑j=1

∫ Tj

Tj−1+

∫ t

Ti−1

)e−β1(t−s)

[φ′(s)+ β1φ(s)

]ds

= φ(t), Ti−1 < t < Ti,

where i = 1, 2, . . . , k + 1, and the continuity of F1(φ, ψ)(t) and φ(t) implies F1(φ, ψ)(t) ≤ φ(t) for all t ∈ R. Similarly,we can prove (3.2) for F1(φ, ψ), F2(φ, ψ) and F2(φ, ψ). The proof is complete. �

Lemma 3.6. Assume that (A2), (A3) and (CQM) hold. Then F(Γ (ρ, ρ)

)⊂ Γ (ρ, ρ) is compact with respect to the supremum

norm.

Proof. For any ε > 0, (P2) implies that there exists T = T (ε) > 0 (which only depends on ρ and ρ) such that

supt>T|x(t)− K | < ε/2 and sup

t<−T|x(t)| < ε (3.3)

for any x ∈ Γ (ρ, ρ).Moreover, the proof of Lemma 3.3 indicates that F(Γ (ρ, ρ)) is equicontinuous. By the equicontinuityand Ascoli–Arzela lemma, there exist N = N(ε) and

A ={x1, x2, . . . , xN ∈ F

(Γ (ρ, ρ)

)}such that A is a finite ε-net of F

(Γ (ρ, ρ)

)for t ∈ [−T , T ]. (3.3) further implies that A is a finite ε-net of F

(Γ (ρ, ρ)

)for all

t ∈ R.That is, F

(Γ (ρ, ρ)

)is precompact with respect to the supremum norm. The proof is complete. �

Theorem 3.7. Assume that (CQM) and (A1)–(A3) hold. If (2.1) has a pair of upper and lower solutions satisfying (P1) and (P2),then (2.1) admits a solution satisfying (2.2), which is a traveling wave solution of (1.5).

Proof. By virtue of Lemmas 3.3–3.6 and Schauder’s fixed point theorem, F admits a fixed point (φ∗, ψ∗) ∈ Γ(ρ, ρ

), which

is a solution of (2.1).In order to prove that such a solution is a traveling wave solution, we need to verify the asymptotic boundary condition

(2.2). In fact, it is clear by the assumptions (P1) and (P2). The proof is complete. �

Remark 3.8. Reviewing our proof process, what we need is the upper and lower solutions of F . However, it is very difficultto verify the definition of upper and lower solutions for such an operator (see [49] for the discussion of a scalar equation),so we introduce the upper and lower solutions of differential equations, which is easy to check in practice.

3672 G. Lin, W.-T. Li / Nonlinear Analysis: Real World Applications 11 (2010) 3666–3679

4. The Case (ECQM)

In this section, we study the existence of traveling wave solutions of (1.5) if the delayed terms satisfy (ECQM), and weassume that (2.1) has a pair of upper and lower solutions ρ = (φ, ψ) and ρ = (φ, ψ) ∈ C(R,R2) satisfying (P1), (P2) and

(P3) eβ1t [φ(t)− φ(t)] and eβ2t [ψ(t)− ψ(t)] are nondecreasing for t ∈ R.

By the condition (ECQM), the following comparison principle is clear.

Lemma 4.1. Assume that (ECQM) holds. Then

H1(φ1, ψ2)(t) ≤ H1(φ2, ψ1)(t), H2(φ2, ψ1)(t) ≤ H2(φ1, ψ2)(t), t ∈ R,F1(φ1, ψ2)(t) ≤ F1(φ2, ψ1)(t), F2(φ2, ψ1)(t) ≤ F2(φ1, ψ2)(t), t ∈ R,

where (φ1, ψ1), (φ2, ψ2) satisfy (i) (φ1, ψ1), (φ2, ψ2) ∈ C[0,M](R,R2) with (φ1, ψ1) ≤ (φ2, ψ2) and (ii) eβ1s[φ1(s) − φ2(s)]and eβ2s[ψ1(s)− ψ2(s)] are nondecreasing for s ∈ R.

Lemma 4.2. Assume that (A2) and (A3) hold. Then

F = (F1, F2) : C[0,M](R,R2)→ C(R,R2)

is continuous with respect to the norm ‖·‖ .

The proof of Lemma 4.2 is similar to that of Lemma 3.3 and is omitted here.Now, we define the following profile set

Γ ∗(ρ, ρ) =

(φ, ψ) ∈ C(R,R2) :(i) ρ ≤ (φ, ψ) ≤ ρ,(ii) eβ1s[φ(s)− φ(s)], eβ1s[φ(s)− φ(s)],eβ2s[ψ(s)− ψ(s)] and eβ2s[ψ(s)− ψ(s)]are nondecreasing for s ∈ R

.Obviously, Γ ∗(ρ, ρ) is convex, nonempty, closed and bounded with respect to the supremum norm if (P1)–(P3) are true.We shall consider the existence traveling wave solutions in Γ ∗.

Lemma 4.3. Assume that (ECQM) holds. Then F(Γ ∗(ρ, ρ)

)⊂ Γ ∗(ρ, ρ).

Proof. Similar to the proof of Lemma 3.4, it is evident that (i) of Γ ∗ holds. We now prove (ii) of Γ ∗. In fact, for any(φ, ψ) ∈ Γ ∗,

dds

{eβ1s

[F1(φ, ψ)(s)− φ(s)

]}=dds

{∫ s

−∞

eβ1θ[H1(φ, ψ)(θ)− β1eβ1sφ(s)

]dθ}

≥ eβ1s[H1(φ, ψ)(s)− H1(φ, ψ)(s)

]≥ 0, s ∈ R \ T.

By continuity of F1(φ, ψ)(s) and φ(s) in s ∈ R, we see that eβ1s[F1(φ, ψ)(s)− φ(s)

]is nondecreasing in s ∈ R.

Similarly, we can prove that

eβ2s[F2(φ, ψ)(s)− ψ(s)

], eβ1s

[φ(s)− F1(φ, ψ)(s)

], eβ2s

[ψ(s)− F2(φ, ψ)(s)

]are nondecreasing in s ∈ R. The proof is complete. �

Repeating the proof of Lemma 3.5, we can obtain the following result.

Lemma 4.4. Assume that (A2), (A3) and (ECQM) hold. Then F(Γ ∗(ρ, ρ)

)⊂ Γ ∗(ρ, ρ) is compact with respect to the supre-

mum norm.

Now we state our main result in this section, of which the proof is similar to that of Theorem 3.7 and is omitted here.

Theorem 4.5. Assume that (A1)–(A3) and (ECQM) hold. If (2.1) has a pair of upper and lower solutions satisfying (P1)–(P3),then (2.1) has a solution satisfying (2.2), which is a traveling wave solution of (1.5).

G. Lin, W.-T. Li / Nonlinear Analysis: Real World Applications 11 (2010) 3666–3679 3673

5. Applications

In this section, we shall employ our conclusions in Sections 3 and 4 to establish the existence of traveling wave solutionsfor systems (1.3) and (1.4).

Example 5.1. We consider the existence of traveling wave solutions of (1.3), that is{u′n(t) = d1[un+1(t)+ un−1(t)− 2un(t)] + r1un(t) [1− a1un(t)− b1vn(t − τ1)] ,v′n(t) = d2[vn+1(t)+ vn−1(t)− 2vn(t)] + r2vn(t) [1− a2vn(t)− b2un(t − τ2)] ,

(5.1)

where di, ri, ai, bi and τi are positive constants for i = 1, 2. Assume that c > 0. Let un(t) = φ(t − cn) = φ(s) and vn(t) =ψ(t − cn) = ψ(s), s ∈ R, and denote the coordinate s as t , then the corresponding wave system is{

φ′(t) = d1[φ(t + c)+ φ(t − c)− 2φ(t)] + r1φ(t) [1− a1φ(t)− b1ψ(t − τ1)] ,ψ ′(t) = d2[ψ(t + c)+ ψ(t − c)− 2ψ(t)] + r2ψ(t) [1− a2ψ(t)− b2φ(t − τ2)] .

(5.2)

If a1 > b2 and a2 > b1, then (5.2) has a positive equilibrium K = (k1, k2), where

k1 =a2 − b1

a1a2 − b1b2> 0, k2 =

a1 − b2a1a2 − b1b2

> 0. (5.3)

In the remainder of this section, we always letMi = 1ai, then (A1)–(A3) are clear. Due to our main purpose in this paper, we

are interested in the solution of (5.2) satisfying the following asymptotic boundary conditions

limt→−∞

(φ(t), ψ(t)) = 0, limt→+∞

(φ(t), ψ(t)) = K . (5.4)

For any (φ, ψ) ∈ C(R,R2) and t ∈ R, denote f = (f1, f2) by{f1(φt , ψt) = r1φ(t) [1− a1φ(t)− b1ψ(t − τ1)] ,f2(φt , ψt) = r2ψ(t) [1− a2ψ(t)− b2φ(t − τ2)] .

We now verify that f satisfies (CQM).

Lemma 5.2. The functional f satisfies (CQM).

Proof. Assume that 0 ≤ φ1(t) ≤ φ2(t) ≤ M1, 0 ≤ ψ1(t) ≤ ψ2(t) ≤ M2 for t ∈ R. Then

f1(φ2t , ψ1t)− f1(φ1t , ψ2t)= r1 {φ2(t) [1− a1φ2(t)− b1ψ1(t − τ1)]− φ1(t) [1− a1φ1(t)− b1ψ2(t − τ1)]}≥ r1 {φ2(t) [1− a1φ2(t)]− φ1(t) [1− a1φ1(t)]}≥ 2r1(φ2(t)− φ1(t)).

Let β1 = 2r1 + 2d1, then f1 satisfies (CQM).Similarly, we can prove that f2 satisfies (CQM). The proof is complete. �

In order to apply Theorem 3.7, we need to construct a pair of upper and lower solutions of (5.2). Define

∆1(γ , c) , d1(eγ c + e−γ c − 2)− γ + r1 (5.5)

and

∆2(γ , c) , d2(eγ c + e−γ c − 2)− γ + r2. (5.6)

Then the following result is clear.

Lemma 5.3. There exists a constant c∗ > 0 such that for any c ∈ (0, c∗), (5.5) and (5.6) have two distinct positive roots,respectively. More precisely,

(i) there exist 0 < γ1(c) < γ2(c) such that

∆1(γ , c)

{= 0, γ = γ1(c), γ2(c),> 0, 0 < γ < γ1(c) or γ > γ2(c),< 0, γ1(c) < γ < γ2(c),

(ii) there exist 0 < γ3(c) < γ4(c) such that

∆2(γ , c)

{= 0, γ = γ3(c), γ4(c),> 0, 0 < γ < γ3(c) or γ > γ4(c)< 0, γ3(c) < γ < γ4(c).

3674 G. Lin, W.-T. Li / Nonlinear Analysis: Real World Applications 11 (2010) 3666–3679

Let

η ∈

(1,min

{γ2(c)γ1(c)

,γ4(c)γ3(c)

,γ1(c)+ γ3(c)

γ1(c),γ1(c)+ γ3(c)

γ3(c)

}). (5.7)

By constants in Lemma 5.3 and (5.7), we now define the following functions

l1(t) = eγ1t − qeηγ1t and l2(t) = eγ3t − qeηγ3t for q > 1. (5.8)

It is clear that l1(t) and l2(t) have global maximumsm1 > 0 andm2 > 0, respectively.Define

N∗1 : mes{t : l1(t) ≥

m1N∗1, t ∈ R

}= c∗,

N∗2 : mes{t : l2(t) ≥

m2N∗2, t ∈ R

}= c∗,

N∗ = max{N∗1 ,N∗

2 },

wheremes denotes the Lebesgue measure on R. From (5.8), N∗ is well defined.Denote

t1 = max{t : l1(t) =

m1N∗

}and t3 = max

{t : l2(t) =

m2N∗

}.

Let q > 0 be large enough such that there exist ε0 > 0, ε1 > 0 and ε2 > 0 satisfying{a1ε1 − b1ε4 > ε0, a2ε4 − b2ε1 > ε0,a1ε3 − b1ε2 > ε0, a2ε2 − b2ε3 > ε0

(5.9)

for any

ε3 ∈(k1 −

m1N∗, k1 −

m12N∗

), ε4 ∈

(k2 −

m2N∗, k2 −

m22N∗

).

In fact, the above choice is admissible since a1 > b2 and a2 > b1.Let γ ∈ (0, γ ∗) such that

k1 − (k1 −m1/N∗)eγ∗t1 >

m12N∗

, k2 − (k2 −m2/N∗)eγ∗t3 >

m22N∗

.

Now, we define

φ(t) ={min{eγ1t , 1/a1} for t ≤ t2,k1 + ε1e−γ t for t > t2,

φ(t) ={eγ1t − qeηγ1t for t ≤ t1,k1 − (k1 −m1/N∗)e−γ t for t > t1,

ψ(t) ={min{eγ3t , 1/a2} for t ≤ t4,k2 + ε2e−γ t for t > t4,

ψ(t) ={eγ3t − qeηγ3t for t ≤ t3,k2 − (k2 −m2/N∗)e−γ t for t > t3.

Obviously, these functions satisfy

φ(t) ≥ φ(t), ψ(t) ≥ ψ(t) for t ∈ R,

and limt→−∞

(φ(t), ψ(t)

)= limt→−∞

(φ(t), ψ(t)

)= (0, 0),

limt→∞

(φ(t), ψ(t)

)= limt→∞

(φ(t), ψ(t)

)= (k1, k2).

Moreover, it is evident that min{t2, t4} − τ1 − τ2 > max{t1, t3} holds if q > 1 is large enough.We now prove that (φ, ψ) and (φ, ψ) are a pair of upper and lower solutions of (5.2).

Lemma 5.4. Assume that (5.9) and 0 < c < c∗ hold. If q > 1 is sufficiently large and γ > 0 is sufficiently small, then (φ, ψ)and (φ, ψ) are a pair of upper and lower solutions of (5.2).

Proof. For φ(t), we need to prove

φ′(t) ≥ d1

[φ(t + c)+ φ(t − c)− 2φ(t)

]+ r1φ(t)

[1− a1φ(t)− b1ψ(t − τ1)

](5.10)

for t ∈ R such that φ(t) is differential.

G. Lin, W.-T. Li / Nonlinear Analysis: Real World Applications 11 (2010) 3666–3679 3675

If t < t2 and φ(t) = 1/a1, then the result is clear. Otherwise, φ′(t) = γ1eγ1t and

d1[φ(t + c)+ φ(t − c)− 2φ(t)

]+ r1φ(t)

[1− a1φ(t)− b1ψ(t − τ1)

]≤ d1

[φ(t + c)+ φ(t − c)− 2φ(t)

]+ r1φ(t)

≤ eγ1t[d1(eγ1c + e−γ1c − 2

)+ r1

]= 0.

If t > t2, then φ′(t) = −ε1γ e−γ t and

d1[φ(t + c)+ φ(t − c)− 2φ(t)

]+ r1φ(t)

[1− a1φ(t)− b1ψ(t − τ1)

]≤ e−γ t

{d1[ε1eγ c + ε1e−γ c − 2ε1

]− r1(k1 + ε1e−γ t)

[a1ε1 − b1(k2 −m2/N∗)eγ τ1

]}.

Thus, it suffices to show that

−ε1γ ≥ d1[ε1eγ c + ε1e−γ c − 2ε1

]− r1(k1 + ε1e−γ t)

[a1ε1 − b1(k2 −m2/N∗)eγ τ1

].

Let

I1(γ ) = d1[ε1eγ c + ε1e−γ c − 2ε1

]− r1(k1 + ε1e−γ t)

[a1ε1 − b1(k2 −m2/N∗)eγ τ1

]+ ε1γ .

Note that

I1(0) = r1(k1 + ε1)(−a1ε1 + b1(k2 −m2/N∗)

)< 0,

then the continuity of I1(γ ) implies that there exists γ ∗1 > 0 such that

I1(γ ) ≤ 0 for 0 < γ ≤ γ ∗1 ,

which implies (5.10) holds for t > t2. This completes the proof of (5.10).For φ(t), we shall verify that

φ′(t) ≤ d1[φ(t + c)+ φ(t − c)− 2φ(t)

]+ r1φ(t)

[1− a1φ(t)− b1ψ(t − τ1)

](5.11)

for t ∈ R \ t1.If t < t1, then φ′(t) =

(eγ1t − qeηγ1t

)′= γ1eγ1t − ηγ1qeηγ1t , and

d1[φ(t + c)− 2φ(t)+ φ(t − c)

]+ r1φ(t)

[1− a1φ(t)− b1ψ(t − τ1)

]≥ eγ1t

{d1[eγ1c + e−γ1c − 2

]+ r1

}− qeηγ1t

{d1[eηγ1c + e−ηγ1c − 2

]+ r1

}− r1

(eγ1t − qeηγ1t

) [a1(eγ1t − qeηγ1t

)+ b1eγ3(t−τ1)

].

Namely, we shall prove that

γ1eγ1t − ηγ1qeηγ1t ≤ eγ1t{d1[eγ1c + e−γ1c − 2

]+ r1

}− qeηγ1t

{d1[eηγ1c + e−ηγ1c − 2

]+ r1

}− r1

(eγ1t − qeηγ1t

) [a1(eγ1t − qeηγ1t

)+ b1eγ3(t−τ1)

]. (5.12)

By Lemma 5.3 and (5.7), (5.12) is equivalent to

−∆1(ηλ1, c)qeηγ1t ≤ −r1(eγ1t − qeηγ1t

) [a1(eγ1t − qeηγ1t

)+ b1eγ3(t−τ1)

].

Let q ≥ 1− r1(a1+b1)∆1(ηλ1,c)

, then (5.12) holds.If t > t1, then φ(t) = k1 − (k1 −m1/N∗)e−γ t and ψ(t) ≤ k2 + ε2e−γ t such that

d1[φ(t + c)− 2φ(t)+ φ(t − c)

]+ r1φ(t)

[1− a1φ(t)− b1ψ(t − τ1)

]≥ d1(k1 −m1/N∗)e−γ t

[2− eγ c − e−γ c

]+ r1e−γ t

[k1 − (k1 −m1/N∗)e−γ t

] (a1(k1 −m1/N∗)− b1ε2eγ τ

)≥ e−γ t

{d1(k1 −m1/N∗)

[2− eγ c − e−γ c

]+m12N∗

(a1(k1 −m1/N∗)− b1ε2eγ τ

)}.

Let

I2(γ ) = d1(k1 −m1/N∗)[2− eγ c − e−γ c

]+m12N∗

(a1(k1 −m1/N∗)− b1ε2eγ τ

).

Then it is independent of t , and there exists γ ∗2 ∈ (0, γ∗

1 ) such that

I2(γ ) > 0 for all γ ∈ (0, γ ∗1 ), (5.13)

which indicates that (5.12) is true.

3676 G. Lin, W.-T. Li / Nonlinear Analysis: Real World Applications 11 (2010) 3666–3679

In a similar way, we can prove the following inequalities

ψ′(t) ≥ d2

[ψ(t + c)+ ψ(t − c)− 2ψ(t)

]+ r2ψ(t)

[1− a2ψ(t)− b2φ(t − τ2)

],

ψ ′(t) ≤ d2[ψ(t + c)+ ψ(t − c)− 2ψ(t)

]+ r2ψ(t)

[1− a2ψ(t)− b2φ(t − τ2)

]except for several points. The proof is complete. �

Remark 5.5. To obtain (5.13), the positive lower boundedness ofψ(t) is very important, which is applied to get the estimateI2(γ ). Moreover, we may choose the parameters in the following order: q,N∗, γ . It is evident the choice is admissible.

By Theorem 3.7 and Lemmas 5.2 and 5.4, we have the following existence theorem.

Theorem 5.6. Assume that 0 < c < c∗ holds. Then (5.1) admits a traveling wave solution (φ, ψ), which connects 0 with K .Furthermore,

limt→−∞

(φ(t)e−γ1(c)t , ψ(t)e−γ3(c)t) = (1, 1),

where c∗, γ1(c), γ3(c) are defined by Lemma 5.3.

Example 5.7. We consider the existence of traveling wave solutions of (1.4), that is{u′n(t) = d1[un+1(t)+ un−1(t)− 2un(t)] + r1un(t) [1− a1un(t − τ1)− b1vn(t − τ2)] ,v′n(t) = d2[vn+1(t)+ vn−1(t)− 2vn(t)] + r2vn(t) [1− b2un(t − τ3)− a2vn(t − τ4)] ,

(5.14)

where all the constants are positive. If we denote (φ, ψ) as the wave profiles, then the corresponding wave system of (5.14)is {

φ′(t) = d1[φ(t + c)+ φ(t − c)− 2φ(t)] + r1φ(t) [1− a1φ(t − τ1)− b1ψ(t − τ2)] ,ψ ′(t) = d2[ψ(t + c)+ ψ(t − c)− 2ψ(t)] + r2ψ(t) [1− b2φ(t − τ3)− a2ψ(t − τ4)] .

(5.15)

Similar to that in Example 5.1, we also consider the asymptotic boundary conditions (5.4) in this example.For any (φ, ψ) ∈ C(R,R2), define f = (f1, f2) by{

f1(φt , ψt) = r1φ(t) [1− a1φ(t − τ1)− b1ψ(t − τ2)] ,f2(φt , ψt) = r2ψ(t) [1− a2ψ(t − τ4)− b2φ(t − τ3)] .

Lemma 5.8. If τ1 > 0 and τ4 > 0 are sufficiently small, then f satisfies (ECQM).

Proof. Assume that 0 ≤ φ1(t) ≤ φ2(t) ≤ M1, 0 ≤ ψ1(t) ≤ ψ2(t) ≤ M2 and eβ1t(φ2(t) − φ1(t)) and eβ2t(ψ2(t) − ψ1(t))are nondecreasing with respect to t ∈ R. Then

f1(φ2t , ψ1t)− f1(φ1t , ψ2t)− 2d1(φ2(t)− φ1(t))≥ f1(φ2t , ψ1t)− f1(φ1t , ψ1t)− 2d1(φ2(t)− φ1(t))= [r1 − r1b1ψ1(t − τ2)− r1a1φ1(t − τ1)− 2d1](φ2(t)− φ1(t))− r1a1φ2(t)eβ1τ1e−β1τ1(φ2(t − τ1)− φ1(t − τ1))≥ [−r1b1M2 − 2d1 − r1a1φ2(t)eβ1τ1 ](φ2(t)− φ1(t))≥ −

(r1b1M2 + 2d1 + r1a1M1eβ1τ1

)(φ2(t)− φ1(t)).

Since τ1 is sufficiently small, we can choose β1 > 0 such that

r1b1M2 + 2d1 + r1a1M1eβ1τ1 < β1,

which implies

f1(φ2t , ψ1t)− f1(φ1t , ψ2t) ≥ (2d1 − β1)(φ2(t)− φ1(t)).

Similarly, we can prove that f2 satisfies (ECQM) if τ4 is sufficiently small. The proof is complete. �

In order to use our Theorem 4.5, we need to construct proper upper and lower solutions. Similar to Example 5.1, wedefine the continuous functions same to those in Example 5.1.

Lemma 5.9. Under the assumptions of Lemma 5.4, if τ1 > 0 and τ4 > 0 are sufficiently small, then (φ, ψ) and (φ, ψ) are apair of upper and lower solutions of (5.15).

G. Lin, W.-T. Li / Nonlinear Analysis: Real World Applications 11 (2010) 3666–3679 3677

Proof. For φ(t), we need to prove

φ′(t) ≥ d1

[φ(t + c)+ φ(t − c)− 2φ(t)

]+ r1φ(t)

[1− a1φ(t − τ1)− b1ψ(t − τ2)

](5.16)

for t ∈ R \ t2.If t < t2 and t > t2 + τ1, then the proof is similar to that in (5.10), so we omit it here.We now prove (5.16) for t2 < t < t + τ1. By the proof of Lemma 5.4, we can choose small γ such that

φ′(t) > d1

[φ(t + c)+ φ(t − c)− 2φ(t)

]+ r1φ(t)

[1− a1φ(t − τ1)− b1ψ(t − τ2)

]for t = t2 + τ1. Furthermore, since φ, φ

′and ψ are uniformly continuous and independent of τ1, then

φ′(t) ≥ d1

[φ(t + c)+ φ(t − c)− 2φ(t)

]+ r1φ(t)

[1− a1φ(t − τ1)− b1ψ(t − τ2)

]is clear if τ1 is small enough. This completes the proof of (5.16).In a similar way, we can prove

ψ′(t) ≥ d2

[ψ(t + c)+ ψ(t − c)− 2ψ(t)

]+ r2ψ(t)

[1− a2ψ(t − τ4)− b2φ(t − τ3)

],

φ′(t) ≤ d1[φ(t + c)+ φ(t − c)− 2φ(t)

]+ r1φ(t)

[1− a1φ(t − τ1)− b1ψ(t − τ2)

],

ψ ′(t) ≤ d2[ψ(t + c)+ ψ(t − c)− 2ψ(t)

]+ r2ψ(t)

[1− a2ψ(t − τ4)− b2φ(t − τ3)

],

for t ∈ R \ {t1, t3, t4} if τ1, τ4 are small enough. The proof is complete. �

By Theorem 4.5 and Lemmas 5.8 and 5.9, we have the following existence theorem.

Theorem 5.10. Assume that 0 < c < c∗ holds and τ1 > 0 and τ4 > 0 are sufficiently small. Then (5.14) has a traveling wavesolution (φ, ψ) which connects 0 with K . Furthermore,

limt→−∞

(φ(t)e−γ1(c)t , ψ(t)e−γ3(c)t) = (1, 1),

where c∗, γ1(c), γ3(c) are defined by Lemma 5.3.

Remark 5.11. For systems (5.1) and (5.14), the existence of traveling wave solutions which connect the trivial equilibrium(0, 0) and the positive equilibrium (k1, k2) indicates that there is a transition zone moving from the steady state with nospecies to the steady state with the coexistence of both species, which also implies that both invasions of competitionspecies are successful. Theorems 5.6 and 5.10 further show that the delay which appeared in the interspecific competitionterms does not affect the existence of traveling waves. Namely, the delay which appeared in the interspecific competitionterms has less effect on the population evolution. We shall formulate the invasion process by spreading speed and provethe sharpness of c∗ defined by Lemma 5.3 in our future papers.

Remark 5.12. In Remark 2.1, we said that 1/c is the wave speed, which can also be understood by the correspondingreaction–diffusion equations since one origin of lattice dynamical systems is the spatial discretization of reaction–diffusionsystems. Consider the famous Fisher equation

∂z(x, t)∂t

= d1z(x, t)+ rz(x, t) [1− z(x, t)] , x ∈ R, t > 0,

herein all the parameters are positive. Let the wave coordinate be x + ct , it is well known the equation admits nontrivialtraveling wave solutions connecting 0 with 1 if the wave speed c ≥ 2

√dr . If we denote the wave coordinate by t + cx

analogous to that in this paper, we may prove the existence of traveling wave solutions connecting 0 with 1 if the wavespeed 1/c ≥ 2

√dr , and the discussion is similar to that in this section. From the viewpoint of the admissible set of wave

parameters, we also see that 1/c is the wave speed when the wave coordinate is defined by cx+ t .

Acknowledgement

The authors are very grateful to an anonymous referee for his/her helpful comments and suggestions.

3678 G. Lin, W.-T. Li / Nonlinear Analysis: Real World Applications 11 (2010) 3666–3679

References

[1] J. Bell, C. Conser, Threshold behaviour and propagation for nonlinear differential-difference systemsmotivated bymodeling myelinated axons, Quart.Appl. Math. 42 (1984) 1–14.

[2] J.P. Keener, Propagation and its failure to coupled systems of discrete excitable cells, SIAM J. Appl. Math. 47 (1987) 556–572.[3] P. Weng, H. Huang, J. Wu, Asymptotic speed of propagation of wave front in a lattice delay differential equation with global interaction, IMA J. Appl.Math. 68 (2003) 409–439.

[4] P.W. Bates, A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal. 150 (1999) 281–305.[5] P.W. Bates, P.C. Fife, X. Ren, X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal. 138 (1997) 105–136.[6] S.N. Chow, J. Mallet-Paret, W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations 149 (1998) 248–291.[7] C.H. Hsu, S.S. Lin, Existence and multiplicity of traveling waves in a lattice dynamical system, J. Differential Equations 164 (2000) 431–450.[8] J. Mallet-Paret, The fredholm alternative for functional differential equations of mixed type, J. Dynam. Differential Equations 11 (1999) 1–47.[9] J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dynam. Differential Equations 11 (1999) 49–127.[10] J. Mallet-Paret, Travelingwaves in spatially discrete dynamical systems of diffusive type, in: J.W.Macki, P. Zecca (Eds.), Dynamical Systems, in: Lecture

Notes in Mathematics, vol. 1822, Springer, Berlin, 2003, pp. 231–298.[11] X. Chen, Existence, uniqueness and asymptotic stability of traveling waves in non-local evolution equation, Adv. Differential Equations 2 (1997)

125–160.[12] X. Chen, J. Guo, Existence and asymptotic stability of travellingwaves of discrete quasilinearmonostable equations, J. Differential Equations 184 (2002)

549–569.[13] S. Ma, X. Zou, Propagation and its failure in a lattice delayed differential equationwith global interaction, J. Differential Equations 212 (2005) 129–190.[14] S. Ma, X. Zou, Existence, uniqueness and stability of traveling waves in a discrete reaction–diffusion equation with delay, J. Differential Equations 217

(2005) 54–87.[15] B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal. 22 (1991) 1016–1020.[16] R. Carretero-González, D.K. Arrowsmith, F. Vivaldi, One-dimensional dynamics for traveling fronts in coupled map lattices, Phys. Rev. E 61 (2000)

1329–1336.[17] A. Tonnelier, Wave propagation in discrete media, J. Math. Biol. 44 (2002) 87–105.[18] N.F. Britton, Travelling wave front solutions of a differential-difference equation arising in the modelling of myelinated nerve axon, in: B. Sleeman,

R. Jarvis) (Eds.), Ordinary and Partial Differential Equations, in: Lecture Notes in Mathematics, vol. 1151, Springer, Berlin, 1984, pp. 77–89.[19] H. Chi, J. Bell, B. Hassard, Numerial solutions of a nonlinear advanced-delay differential equation from nerve condition theory, J. Math. Biol. 24 (1986)

583-561.[20] A.I. Volpert, V.A. Volpert, V.A. Volpert, Traveling Wave Solutions of Parabolic Systems, in: Translations of Mathematical Monographs, vol. 140, AMS,

Providence, 1994.[21] J.W. Cahn, S.N. Chow, E.S. Van Vleck, Spatially discrete nonlinear diffusions, Rocky Mountain J. Math. 25 (1995) 87–118.[22] J.W. Cahn, J. Mallet-Paret, E.S. Van Vleck, Traveling wave solutions for systems of ODE’s on a two dimensional spatial lattice, SIAM J. Appl. Math. 59

(1998) 455–493.[23] S.N. Chow, Lattice dynamical systems, in: J.W. Macki, P. Zecca (Eds.), Dynamical Systems, in: Lecture Notes in Mathematics, vol. 1822, Springer, Berlin,

2003, pp. 1–102.[24] W. Shen, Traveling waves in time periodic lattice dynamical systems, Nonlinear Anal. TMA 54 (2003) 319–339.[25] B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differential Equations 96 (1992) 1–27.[26] B. Zinner, G. Harris, W. Hudson, Traveling wave fronts for the discrete Fisher’s equation, J. Differential Equations 105 (1993) 46–62.[27] X. Chen, J. Guo, Uniqueness and existence of travelling waves for discrete quasilinear monostable dynamics, Math. Ann. 326 (2003) 123–146.[28] J. Wu, X. Zou, Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations, J. Differential

Equations 135 (1997) 315–357.[29] X. Liang, X.Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math. 60 (2007)

1–40.[30] J.K. Hale, Coupled oscillators on a circle: Dynamical phase transitions (Säo Paulo, 1994), Resenhas 1 (1994) 441–457.[31] J. Huang, G. Lu, X. Zou, Existence of traveling wave fronts of delayed lattice differential equations, J. Math. Anal. Appl. 298 (2000) 538–558.[32] S. Ma, P. Weng, X. Zou, Asymptotic speed of propagation and traveling wavefront in a lattice delayed differential equation, Nonlinear Anal. TMA 65

(2006) 1858–1890.[33] H.R. Thieme, X.Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction–diffusion models, J. Differential

Equations 195 (2003) 430–470.[34] K. Zhang, X.Q. Zhao, Spreading speed and travelling waves for a spatially discrete SIS epidemic model, Nonlinearity 21 (2008) 97–112.[35] X. Zou, Traveling wave fronts in spatially discrete reaction–diffusion equations on higher-dimensional lattices, Electron. J. Differ. Equ. Conf., 1,

Southwest Texas State Univ., San Marcos, TX, 1998.[36] E. Renshaw, Modelling Biological Populations in Space and Time, Cambridge University Press, Cambridge, 1991.[37] A.R.A. Anderson, B.D. Sleeman, Wave front propagation and its failure in coupled systems of discrete bistable cells modeled by Fitzhugh–Nagumo

dynamics, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 5 (1995) 63–74.[38] V.I. Nekorkin, V.B. Kazantsev, S. Morfu, J.M. Bibault, P. Marqui, Theoretical and experimental study of two discrete coupled Nagumo chains, Phys. Rev.

E 64 (2001) 036602.[39] J. Huang, G. Lu, Traveling wave solutions in delayed lattice dynamical system, Chinese Ann. Math. 25 (2004) 153–164.[40] J. Huang, G. Lu, S. Ruan, Traveling wave solutions in delayed lattice differential equations with partial monotonicity, Nonlinear Anal. TMA 60 (2005)

1331–1350.[41] G. Lin, W.T. Li, S. Pan, Traveling wavefronts in delayed lattice dynamical systems with global interaction, J. Difference Eqns. Appl. (in press).[42] S. Ai, Travelling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differential Equations 232 (2007) 104–133.[43] S.A. Gourley, J.H.W. So, J. Wu, Non-locality of reaction–diffusion equations induced by delay: biological modeling and nonlinear dynamics, J. Math.

Sci. 124 (2004) 5119–5153.[44] Y. Hosono, Traveling waves for a diffusive Lotka–Volterra competition model, I: Singular perturbation, Discrete Contin. Dyn. Syst. Ser. B 3 (2003)

75–95.[45] J. Huang, X. Zou, Existence of traveling wavefronts of delayed reaction–diffusion systems without monotonicity, Discrete Contin. Dyn. Syst. Ser. B 9

(2003) 925–936.[46] G. Lin,W.T. Li, M. Ma, Travelling wave solutions in delayed reaction diffusion systemswith applications tomulti-species models, Discrete Contin. Dyn.

Syst. Ser. B 13 (2010) 393–414.[47] S. Ma, Traveling wavefronts for delayed reaction–diffusion systems via a fixed point theorem, J. Differential Equations 171 (2001) 294–314.[48] K.W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc. 302 (1987)

587–615.[49] H. Wang, On the existence of traveling waves for delayed reaction–diffusion equations, J. Differential Equations 247 (2009) 887–905.[50] Z. Wang,W.T. Li, S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations 20 (2008) 573–603.[51] Z.C. Wang, W.T. Li, S. Ruan, Traveling wave fronts in reaction–diffusion systems with spatio-temporal delays, J. Differential Equations 222 (2006)

185–232.

G. Lin, W.-T. Li / Nonlinear Analysis: Real World Applications 11 (2010) 3666–3679 3679

[52] Z.C.Wang,W.T. Li, S. Ruan, Existence and stability of travelingwave fronts in reaction advection diffusion equations with nonlocal delay, J. DifferentialEquations 238 (2007) 153–200.

[53] J. Wu, X. Zou, Traveling wave fronts of reaction–diffusion systems with delay, J. Dynam. Differential Equations 13 (2001) 651–687.[54] W.T. Li, G. Lin, S. Ruan, Existence of travelingwave solutions in delayed reaction–diffusion systemswith applications to diffusion-competition systems,

Nonlinearity 19 (2006) 1253–1273.[55] A.W. Leung, Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering, Kluwer Academic Pub, Durdrecht, 1989.[56] C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, 1992.[57] Q. Ye, Z. Li, Introduction to Reaction–Diffusion Equations, Science Press, Beijing, 1990.