Volume 14, Number 1, July 2010 pp. 111–128
TRAVELING PLANE WAVE SOLUTIONS OF DELAYED LATTICE DIFFERENTIAL SYSTEMS IN COMPETITIVE
LOTKA-VOLTERRA TYPE
Cheng-Hsiung Hsu
Department of Mathematics National Central University Chung-Li 32001, Taiwan
Ting-Hui Yang
Department of Mathematics Tamkang University Tamsui, Taipei County 25137, Taiwan
(Communicated by Yuan Lou)
Abstract. In this work we consider the existence of traveling plane wave solutions of systems of delayed lattice differential equations in competitive Lotka-Volterra type. Employing iterative method coupled with the explicit construction of upper and lower solutions in the theory of weak quasi-monotone dynamical systems, we obtain a speed, c∗, and show the existence of traveling
plane wave solutions connecting two different equilibria when the wave speeds are large than c∗.
1. Introduction. The purpose of this work is to investigate the existence of trav-eling plane wave solutions of systems of N delayed 2-dimensional lattice differential equations (2D-LDEs) in competitive Lotka-Volterra type. The nth 2D-LDE in the systems is of the form
d
dtun;i,j(t) = Ln[un;i,j](t) + un;i,j(t)fn ui,j(t), (ui,j)b n t
, (1)
for (i, j) ∈ Z2 and 1 ≤ n ≤ N , where ui,j(t) := (u1;i,j(t), · · · , uN;i,j(t)), (ui,j)bnt(τ1, · · · , τn−1, τn+1, · · · , τN)
:=(u1;i,j(t − τ1), · · · , un−1;i,j(t − τn−1), un+1;i,j(t − τn+1), · · · , uN;i,j(t − τN)), and
Ln[un;i,j](t) = dn,1un;i+1,j(t)+dn,2un;i,j+1(t)
+ dn,3un;i−1,j(t) + dn,4un;i,j−1(t) − dn,0un;i,j(t). Here τi and di,j are positive real constants which represent the time delays and coupling coefficients respectively. Let τ := max{τ1, · · · , τn−1, τn+1, · · · , τN}. All fn are C1 functions from RN× C1([−τ, 0], R)N −1 to R where C1([−τ, 0], R)N −1 is
2000 Mathematics Subject Classification. Primary: 34A33, 34C37, 34K10, 35C07; Secondary: 92B20.
Key words and phrases. Traveling Wave Solution, Delayed Lattice Differential Equations, Up-per And Lower Solutions, Heteroclinic Solutions.
the Banach space of continuous differentiable N − 1 dimensional functions mapping the interval [−τ, 0] into R with supremum norm. For any C1 function u : R1 → RN, (u)bn
t ∈ C1([−τ, 0], R)N −1 means that (u)ntb(s) = u1(t + s1), · · · , un−1(t + sn−1), un+1(t + sn+1), · · · , uN(t + sN)for sn ∈ [−τ, 0] and 1 ≤ n ≤ N . Here the natation “bn” means that in the n-layer there is no delay effect from the same layer. Systems (1) are infinite dimensional, consisting of infinitely many ordinary dif-ferential equations, indexed by points in a three-dimensional lattice which consist of N layers of two-dimensional plane lattice. In the position (i, j) of nth-layer, the state un;i,j is linear coupling with nearest neighbor states, un;i+1,j, un;i,j+1, un;i−1,j, and un;i,j−1. Interactions between different layers are governed by the nonlinear function
Fn (ui,j)t= un;i,j(t)fn ui,j(t), (ui,j)ntb
for 1 ≤ n ≤ N. (2) Such systems arise from the study of dynamics of multi-layer neural networks [28], material science [4], chemical reaction theory [12], image processing and pattern recognition [9, 10, 33, 34], and population dynamics of multiple species in biology [31]. We also refer to the papers [7, 25] for the detailed account of the theory and applications of lattice differential equations.
On the other hand, it is often that when one discretizes some partial differential equations one ends up with a lattice differential equation to solve. For example, if dn,i = 1 for i = 1, · · · , 4, and dn,0 = 4 then the operator Ln represents the discrete two-dimensional Laplacian operator. Thus equations (1) can be viewed as the spatial discretization of the following partial differential equations defined in the plane
∂un(x, t)
∂t = dn∆un(x, t) + un(x, t)fn u(x, t), ut(x) b n, with x ∈ R2 and 1 ≤ n ≤ N . Specifically, if
fn u(x, t), ut(x)bn= (rn− pnun(x, t) − N X m=1,m6=n
sn,mum(x, t − τm)) for some positive constants rn, pn and sn,m, then systems (1) can be viewed as the spatial discretization of the diffusive competitive LotkaVolterra systems of N -species equations with delay effects in the plane. The systems model the interaction among various competing species, has been studied extensively, and various suffi-cient conditions for the coexistence and extinction of the competing species are obtained, cf. [11,22,29,36].
Our aim is to study the existence of traveling plane wave solutions of (1). A traveling plane wave solution of systems (1) is a solution of the form
un;i,j(t) = φn(t − ic cos θ − jc sin θ), n = 1, · · · , N, (3) where 1/c > 0 is the wave speed; θ ∈ [0, π/2] is the direction of waves propagation; φnare continuously differentiable functions. According to (3), the profile equations of systems (1) can be written as
φ′n(t) = Ln[φn](t) + Fn(Φt), Fn(Φt) = φn(t)fn(Φ(t), Φbn
t)
for n = 1, · · · , N, with c1 := c cos θ, c2 := c sin θ, Φ(t) = (φ1(t), · · · , φN(t)), Φbn t ∈ C1([−τ, 0], R)N −1, and
Ln[φn](t) := dn,1φn(t − c1)+dn,2φn(t − c2)
+ dn,3φn(t + c1) + dn,4φn(t + c2) − dn,0φn(t). Typically, traveling wavefront solutions arise from the competition between two equilibria. To find a traveling plane wave solution of (1) connecting two equilibria is equivalent to find a heteroclinic trajectory of (4) with asymptotically boundary conditions. It is obvious that (4) has a trivial solution 0 := (0, · · · , 0). Some sufficient conditions for the uniqueness of positive equilibrium and global asymptotic stability for Lotka-Volterra competition-diffusion systems with discrete time delays were given in [30]. Generalizing the ideas of [30], we will give sufficient conditions to guarantee the existence of a positive equilibrium Φ⋆ of (4) in Section 2. Then we look for the existence of heteroclinic orbits of (4) that satisfies the following asymptotically boundary conditions:
lim
t→−∞Φ(t) = 0 and t→∞lim Φ(t) = Φ
⋆. (5)
Traveling wave solutions for a single lattice differential equation without or with delay have drawn considerable attention in the past decades, see, e.g., [2,3,5,6,8,
13, 17, 18,21, 26, 24, 35, 37,38] and many references cited therein. Particularly, Wu and Zou [35] developed a monotone iterative scheme and used a non-standard ordering, quasi-monotone or exponentially quasi-monotone, in the profile set to prove the existence of traveling wave solutions LDEs with asymptotical boundary conditions of by an upper-lower solution method. This technique was generalized to a delayed LDEs on higher dimensional lattices [38]. From another point of view, to show the existence of traveling wave solutions with asymptotically boundary conditions is equivalent to find a heteroclinic orbit connecting two equilibria of the corresponding profile equation which is a mixed type functional differential equation. Hence, by the same approach, Hsu et al. [17,18] generalized the results of [35] to a general scalar functional differential equation in delay, advance or mixed type with some suitable conditions.
Recently, researchers have started to investigate systems of LDEs [1, 12,20,27,
31]. Huang et al. [20] considered the following systems of two delayed LDEs: dun
dt = m X j=1
aj[g(un+j(t)) − 2g(un(t)) + g(un−j(t))] + f1(un(t), vn(t − τ )), dvn dt = m X j=1 bj[g(vn+j(t)) − 2g(vn(t)) + g(vn−j(t))] + f2(un(t − τ ), vn(t)). (6)
Let the nonlinear reaction terms of (6) satisfy the quasi monotonicity condition [35].
(QM) There exist two positive constants β1 and β2 such that
f1(ψ1, ψ2) − f1(φ1, φ2) + β1[ψ1(0) − φ1(0)] ≥ 2A[g(ψ1(0)) − g(φ1(0))], f2(ψ1, ψ2) − f2(φ1, φ2) + β2[ψ2(0) − φ2(0)] ≥ 2B[g(ψ2(0)) − g(φ2(0))] for (ψ1, ψ2), (φ1, φ2) ∈ C([−τ, 0], R)2 with 0 (φ1(s), φ2(s)) (ψ1(s), ψ2(s)) (k1, k2), for s ∈ [−τ, 0] and some positive constant k1, k2, where A =Pmj=1aj and
B =Pmj=1bj. Then the existence of traveling wave solutions of (6) can be estab-lished by the results of Wu and Zou [35]. Here the notation “≺ ()” denote the stan-dard order in high dimension, that is, Φ = (φ1, · · · , φN) and Ψ = (ψ1, · · · , ψN) ∈ C([−τ, 0], R)N, denote Φ (≺)Ψ if φn(s) ≤ (<)ψn(s) for n = 1, · · · , N with s ∈ [−τ, 0].
However, the nonlinear reaction terms of some important examples from practical problems may not satisfy the condition (QM). Hence the methods used in [35] cannot be applied. Here is a example which is the discrete diffusive predator-prey model with delay for ecological systems:
dun
dt = d1[un+1− 2un+ un−1] + un[1 − un] − aunvn(t − τ1), dvn
dt = d2[vn+1− 2vn+ vn−1] − vn+ bun(t − τ2)vn,
(7)
Hence a modified (QM) condition for the reaction terms called partial quasi mono-tonicity (PQM) was introduced simultaneously in delayed LDEs [20] and in delayed reaction diffusion systems [19].
(PQM) There exist two positive constants β1 and β2 such that
f1(ψ1, ψ2) − f1(φ1, ψ2) + β1[ψ1(0) − φ1(0)] ≥ 2A[g(ψ1(0)) − g(φ1(0))], f1(ψ1, ψ2) − f1(ψ1, φ2) ≤ 0,
f2(ψ1, ψ2) − f2(φ1, φ2) + β2[ψ2(0) − φ2(0)] ≥ 2B[g(ψ2(0)) − g(φ2(0))] for (ψ1, ψ2), (φ1, φ2) ∈ C([−τ, 0], R)2 with 0 (φ1(s), φ2(s)) (ψ1(s), ψ2(s)) (k1, k2), for s ∈ [−τ, 0] and some positive constant k1, k2, where A = Pmj=1aj and B = Pmj=1bj. We remark that the functions f2 satisfies the same monotone condition (QM). A new cross-iteration scheme was given to show the existence of traveling wave solutions of (6) if a pair of upper-lower solutions can be constructed and the condition (PQM) are satisfied for reaction terms. In particular, this results can be applied to (7) to show the existence of traveling wave solutions.
The same story happened to the following two species delayed competition sys-tems.
∂
∂tu1(x, t) = d1 ∂2
∂x2u1(x, t) + r1u1(x, t)[1 − a1u1(x, t) − b1u2(x, t − τ1)], ∂
∂tu2(x, t) = d2 ∂2
∂x2u2(x, t) + r2u2(x, t)[1 − b2u1(x, t − τ2) − a2u2(x, t)], (8)
where di and τi are positive constants. It is easy to check that the reaction terms of (8) satisfies neither the condition (QM) nor the condition (PQM). Thus Li et al. [23] provided a condition on the reaction terms called the weak quasi monotone condition (WQM) stated in the following:
(WQM) There exist β1> 0 and β2> 0 such that
f1(ψ1(0), ψ2(−τ1)) − f1(φ1(0), ψ2(−τ1)) + β1[ψ1(0) − φ1(0)] ≥ 0, f1(ψ1(0), ψ2(−τ1)) − f1(ψ1(0), φ2(−τ1)) ≤ 0,
f2(ψ1(−τ2), ψ2(0)) − f2(ψ1(−τ2), φ2(0)) + β2[ψ2(0) − φ2(0)] ≥ 0, f2(ψ1(−τ1), ψ2(0)) − f2(φ1(−τ2), ψ2(0)) ≤ 0
for (φ1, φ2), (ψ1, ψ2) ∈ C([−τ, 0], R)2 with 0 (φ1(s), φ2(s)) (ψ1(s), ψ2(s)) (M1, M2), and s ∈ [− max{τ1, τ2}, 0]. Based on the assumption (WQM), they
reduced the existence of traveling wave solutions of (8) to the existence of an ad-missible pair of upper and lower solutions (cf. Definition 3.1). By assuming the existence of an admissible pair of upper and lower solutions of (8), they applied the cross-iterative method to establish the existence of traveling wave solutions.
Motivated by the previous works of [23,20,19, 35], we will provide a condition which we still denote it by the same name (WQM) on the nonlinear reaction terms of N -systems of LDEs in Section 2. In the strategy of iterative scheme, the con-struction of lower and upper solutions is nontrivial for any specific model. Following the ideas of [23, 24], we can explicitly construct upper and lower solutions of the corresponding systems of the wave profiles of (1) for some classes of nonlinear re-action functions F satisfying the condition (WQM). Applying the technique of the cross-iterative method and Schauder’s fixed point theorem, we show the existence of traveling plane wave solutions of (1). The results can be applied to many models, e.g. the Lotka-Volterra competition systems with distributive time delays.
The remainder of this paper is organized as follows. In Section 2, some necessary notations and definitions are introduced. Then the existence of strictly positive equilibrium is obtained under suitable assumptions. We also examine the weak quasi monotone properties of the systems (4). In Section 3, with the aid of real roots of the corresponding characteristic function of (4) at the trivial solution, we construct the upper and lower solutions of (4). Based on the results of Sections 2 and 3, we show the existence of traveling plane wave solutions of (4) and (5) in Section 4 by using the cross-iterative method and Schauder’s fixed point theorem. In the last section, we apply our main results to the Lotka-Volterra competition systems with distributive time delays, and obtain the existence of traveling plane wave solutions.
2. Preliminary. In this section, we have three main purposes. The first one is to show the existence of a positive equilibrium of (4) under some sufficient conditions on reaction terms. The second one is to state a general (WQM) condition on N -systems of LDEs. Then some sufficient conditions are imposed on reaction functions to guarantee that the condition (WQM) is satisfied. Finally, we investigate the characteristic equations about the trivial solution 0. The characteristic roots will help us to construct a pair of upper-lower solutions of (4).
From now on in this paper, we just consider the existence of a heteroclinic orbit of (4) with asymptotical boundary condition (5) instead of finding the traveling wavefront solutions of (1). And the nonlinear reaction functions Fn are always of the form (2).
2.1. The existence of a positive equilibrium. We use the notation [Ψ|φn] = [ψ1, · · · , ψN|φn] to denote a vector or a vector function Ψ which the nth component is replaced by φn, that is,
[Ψ|φn] := (ψ1, · · · , ψn−1, φn, ψn+1, · · · , ψN).
And the notation Ψnb = (ψ1, · · · , ψN)nb is denoted a vector or a vector function which is removed the nth component, that is,
Ψbn= (ψ1, · · · , ψn−1, ψn+1, ψN) for 1 ≤ n ≤ N .
To find the equilibrium Φ⋆of (4) satisfying Φ⋆≻ 0, we have to solve the following systems of nonlinear algebraic equations,
( 4 X i=1
dn,i− dn,0)φ∗n+ φ∗nfn(Φ⋆, (Φ⋆)bn) = 0 for n = 1, · · · , N.
For mathematical simplicity, we assume that P4i=1dn,i = dn,0. Then the above equation can be simplified as
fn(Φ⋆, (Φ⋆)nb) = 0 for n = 1, · · · , N.
Lemma 2.1. Let K = (k1, · · · , kN) and L = (ℓ1, · · · , ℓN) be two constant vectors such that 0 ≺ K L. If gn∈ C1(R2N −1, R),
∂gn/∂xm≤ 0 and gn([K|ℓn], Knb) ≤ 0 ≤ gn([L|kn], Lnb),
for 1 ≤ n ≤ N and 1 ≤ m ≤ 2N − 1, then there exist two constant vectors Φ = (φ1, · · · , φN) and Ψ = (ψ1, · · · , ψN) in RN such that K Φ Ψ L and
gn([Φ|ψn], Φbn) = 0 = gn([Ψ|φn], Ψbn) for all 1 ≤ n ≤ N .
Proof. Let Λ := {X = (x1, · · · , xN) ∈ RN : K X L} and Mn be positive constants such that
Mn> max − ∂ ∂xn(xngn(X, X b n)) : X ∈ Λ , (9)
for all n = 1, · · · , N . For any X, Y ∈ Λ, define Gn(X) := xngn(X, Xbn),
Zn(α; X, Y ) := (1 − α)y1, · · · , (1 − α)yn−1, αyn, (1 − α)yn+1, · · · , (1 − α)yN + αx1, · · · , αxn−1, (1 − α)xn, αxn+1, · · · , αxN
= [(1 − α)Y |αyn] + [αX|(1 − α)xn],
for n = 1, · · · , N and α ∈ [0, 1]. Obviously, Zn(α; X, Y ) ∈ Λ if X and Y in Λ. If X Y , by (9) and the Mean Value Theorem, we have
yngn([X|yn], Xbn) − xngn([Y |xn], Ynb) =Gn([X|yn]) − Gn([Y |xn])) =Gn(Zn(1; X, Y )) − Gn(Zn(0; X, Y )) = d dα α=eαGn([(1 − α)Y |αyn] + [αX|(1 − α)xn]) ≤Mn(xn− yn)
for some eα ∈ [0, 1]. Hence,
Mnxn+ xngn([Y |xn], Ynb) ≥ Mnyn+ yngn([X|yn], Xbn). (10) Let Φ(0)= K, Ψ(0)= L, and consider the following iterations:
Mnφ(m) n = Mnφ(m−1)n + φ(m−1)n gn [Ψ(m−1)|φ(m−1)n ], (Ψ(m−1))bn , Mnψ(m) n = Mnψ(m−1)n + ψn(m−1)gn [Φ(m−1)|ψn(m−1)], (Φ(m−1))nb . We claim that K Φ(m−1) Φ(m) Ψ(m) Ψ(m−1) L,
for all m ≥ 1. Once the claim is true, then the assertion of this lemma follows by letting m → ∞.
Now we prove the above claim by using the induction method. From (9) and (10), it is obviously that
K Φ(0) Φ(1) Ψ(1) Ψ(0) L. Suppose that
K Φ(m−1) Φ(m) Ψ(m) Ψ(m−1) L for some m ≥ 1. Applying (10) to the inequalities:
[Ψ(m−1)|φ(m)n ] [Ψ(m)|φ(m−1)n ], Ψ(m) Φ(m) and [Φ(m)|ψn(m−1)] [Φ(m−1)|ψ(m)n ], the claim holds obviously. Hence the proof is complete.
2.2. Weak quasi monotonicity. Now the conditions (WQM) conditions are stated for the nonlinear reaction functions F = (F1, · · · , FN) for general N as follows.
(WQM): there exist N positive constants β1, · · · , βN such that (i) Fn([Ψ|φn]) − Fn Φ≤ 0,
(ii) Fn([Φ|ψn]) − Fn(Φ) + βn ψn(0) − φn(0)≥ 0
for 1 ≤ n ≤ N , Ψ, Φ ∈ C1([−τ, 0], R)N with 0 Φ(s) Ψ(s) M for s ∈ [−τ, 0] and a positive constant vector M .
Now we explore the condition (WQM) on the reaction functions of the form Fn(Φ) =φn(0)fn(Φ(0), Φbn)
=φn(0)fn(φ1(0), · · · , φN(0),
φ1(−τ1), · · · , φn−1(−τn−1), φn+1(−τn+1), · · · , φN(−τN))
(11)
for Φ = (φ1, · · · , φN) ∈ C1([−τ, 0], R)N.
Lemma 2.2. Assume that Φ and Ψ ∈ C1([−τ, 0], R)N such that 0 Φ(s) Ψ(s) M for some constant vector M = (M1, · · · , MN) ≻ 0. If the functions fn of (11) satisfies ∂fn/∂xm≤ 0 for 1 ≤ m ≤ 2N − 1, then reaction function Fn satisfies the condition (WQM).
Proof. (i). It is sufficiently to check that
fn [Ψ|φn](0), Ψbn− fn Φ(0), Φbn≤ 0
since φn is nonnegative. According to the assumptions of fn, it is obviously that fn([Ψ|φn](0), Ψbn) − fn(Φ(0), Φnb) = Dfn(ξ1, ξ2) · ([Ψ − Φ|0n](t), Ψbn− Φnb) ≤ 0, for some ξ1 and ξ2. Hence the results follows.
(ii). Based on the assumptions, we have
ψn(0)fn([Φ|ψn](0), Φbn) − φn(0)fn(Φ(0), Φbn) =ψn(0)fn([Φ|ψn](0), Φbn) − ψn(0)fn(Φ(0), Φbn) + ψn(0)fn(Φ(0), Φnb) − φn(0)fn(Φ(0), Φnb) =ψn(0)Df (ξ1, ξ2) · ([0|ψn− φn](0), 0) + ψn(0) − φn(0)fn(Φ(0), Φnb) ≥ ψn(0) − φn(0)(−Mn max 0ξ1M,0ξ2Mnb kDf (ξ1, ξ2)k + fn(M, Mbn),
for some ξ1 and ξ2. Thus, the assertion of this part follows by taking βn> Mn max
0η1M,0η2Mnb
kDf (η1, η2)k − fn(M, Mbn), i = 1, · · · , N. The proof is complete.
According to Lemma 2.2, we can define operators H = (H1, · · · , HN), G = (G1, · · · , GN) : C1([−τ, 0], R)N → C1(R, RN) by Hn(Φ)(t) := Ln[φn](t) + φn(t)fn(Φ(t), Φbn t) + βnφn(t), Gn(Φ)(t) := e−βnt Z t −∞ eβnsHn(Φ)(s)ds, t ∈ R, i = 1, · · · , N.
Then the profile equation (4) can be represented as φ′
n(t) + βnφn(t) − Hn(Φ)(t) = 0, n = 1, · · · , N,
and a fixed point of G is equivalent to the solutions of (4). By Lemma2.2, Gn and Hn have the following properties.
Lemma 2.3. Assume Φ and Ψ satisfy the assumptions of Lemma 2.2. Then (1) Hn([Ψ|φn]) ≤ Hn(Φ) ≤ Hn([Φ|ψn]), for n = 1, · · · , N .
(2) Gn([Ψ|φn]) ≤ Gn(Φ) ≤ Gn([Φ|ψn]), for n = 1, · · · , N .
Proof. The results follow obviously from Lemma2.2. We omit the details.
By Lemma2.3, it motivates us to use the iterative scheme to obtain the existence of traveling plane wave solutions. But some properties of characteristic equations and characteristic roots should be investigated.
2.3. Characteristic functions and characteristic roots. First, we give the definition of the characteristic functions of (4). The characteristic function arises from the linearized equation of (4) at the equilibrium 0, and its roots play crucial roles in studying the behavior of solutions of (4) near 0.
Definition 2.4. Let c > 0 and θ ∈ [0, π/2]. The characteristic function of (4) at 0 is defined by
∆(λ, c) = QN n=1
∆n(λ, c), where ∆n(λ, c), n = 1, · · · , N , are of the form
∆n(λ, c) = −λ + dn,1e−λc1+ dn,2e−λc2+ dn,3eλc1+ dn,4eλc2− dn,0+ fn(0, 0),
where c1= c cos θ and c2= c sin θ.
Now, we explore some properties of the functions ∆n(λ, c).
Lemma 2.5. Assume that for all n dn,3≥ dn,1> 0, dn,4≥ dn,2> 0, P4i=1dn,i= dn,0, and fn(0, 0) > 0. There exists a c∗> 0 which depends on {dn,i}4
i=1 such that if 0 < c < c∗ we can find two real characteristic roots λn,1(c) and λn,2(c) such that 0 < λn,1(c) < λn,2(c) and ∆n(λ, c) = = 0, if λ = λn,1, λn,2, > 0, if 0 < λ < λn,1(c), < 0, if λn,1(c) < λ < λn,2(c), > 0, if λ > λn,2(c). (12)
Proof. We first define ¯
c = min{˜c, 1/(dn,3− dn,1+ dn,4− dn,2)} where
˜
c := inf{c > 0 : ∆n(λ, c) > 0, for all λ > 0}.
(If dn,1= dn,3 and dn,2= dn,4, then we take 1/(dn,3− dn,1+ dn,4− dn,2) = ∞.) It is clear that ˜c > 0. Hence ¯c > 0. For 0 < c < ¯c and λ > 0, we have
∂∆n(λ, c)
∂c = λ cos θ(dn,3e
λc1− dn,1e−λc1) + λ sin θ(dn,4eλc2− dn,2e−λc2) > 0,
∂∆n(λ, c)
∂λ = −1 + c1(dn,3e
λc1− dn,1e−λc1) + c2(dn,4eλc2− dn,2e−λc2),
∂2∆n(λ, c) ∂λ2 = c
2
1(dn,3eλc1+ dn,1e−λc1) + c22(dn,4eλc2+ dn,2e−λc2) > 0. Then we yield
∂∆n(0, c)
∂λ = −1 + c1(dn,3− dn,1) + c2(dn,4− dn,2) < 0, ∆n(0, c) = dn,1+ dn,2+ dn,3+ dn,4− dn,0+ fn(0, 0) > 0.
Thus there exists a λ∗(c) > 0 such that ∆n(λ, c) attains its global minimum at λ = λ∗(c). Moreover, λ∗ satisfy the equation
1 = c1(dn,3eλ∗c
1− dn,1e−λ∗c1) + c2(dn,4eλ∗c2− dn,2e−λ∗c2), (13)
and hence we have dλ ∗
dc < 0 by implicit differentiation of (13).
Now we study the behavior of the curve λ∗(c). From (13) it is easy to see that λ∗(c) → ∞ as c → 0+, and λ∗(c) → 0 as c → ∞. These imply that
lim c→0+∆n(λ
∗(c), c) = −∞ and lim c→∞∆n(λ
∗(c), c) > 0.
Furthermore, ∆n(λ∗(c), c) is monotone increasing with respect to c. Since if c1> c2, then
∆n(λ∗(c1), c1) > ∆n(λ∗(c1), c2) ≥ ∆n(λ∗(c2), c2).
Note that ∆n(λ∗(c2), c2) is a global minimum for such fixed c2. Hence we can find a particular c∗ ∈ (0, ˜c) such that the statement of lemma is true. The proof is complete.
Summarize the above results, we make the following assumptions on reaction terms and coupling coefficients of (4).
(A1) There exist K = (k1, · · · , kN) and L = (ℓ1, · · · , ℓN) in RN such that 0 ≺ K L and fn([K|ℓn], Knb) ≤ 0 ≤ fn([L|kn], Lbn) for all n.
(A2) The functions fn are C1 functions from RN × C1([−τ, 0], R)N −1 to R and ∂fn/∂xm≤ 0 for 1 ≤ m ≤ 2N − 1.
(A3) Assume that for all n dn,3 ≥ dn,1 > 0, dn,4 ≥ dn,2 > 0 , P4i=1dn,i = dn,0, and fn(0, 0) > 0.
Remark 1. (i) Let us reexamine (4) with the assumptions (A1)∼(A3). The assumption (A1) implies that the systems (4) have a positive equilibrium. The conditions (WQM) is satisfied for the reaction terms if (A2) is hold. The assumption (A3) help us to understand the linear behavior of the heteroclinic solution near the trivial solution 0. This is crucial to construct a pair of upper-lower solutions in next section.
(ii) The reaction terms of many classical and typical models have the form Fn in (11) and satisfy the assumption (A2). Here we list some continuous and discrete delayed reaction diffusion equations which we know:
• The logistic scalar equation. ∂u(x, t)
∂t = D ∂2
∂x2u(x, t) + ru(x, t)[1 − au(x, t)].
• A diffusive delay equation which can be used to model the growth of the population of Daphnia. [32,16] ∂u(x, t) ∂t = D ∂2u(x, t) ∂2x + ru(x, t) 1 − au(x, t) 1 + bu(x, t) • The Belousov-Zhabotinskii reaction model with delay.
∂u(x, t)
∂t = D
∂2
∂x2u(x, t) + u(x, t)[1 − u(x, t) − rv(x, t − τ1)], ∂v(x, t)
∂t = D
∂2
∂x2v(x, t) − bu(x, t − τ2)v(x, t).
• The Lotka-Volterra competition-diffusion systems of N -species equations in the plane. ∂un(x, t) ∂t =dn∆un(x, t) + un(x, t) rn− pnun(x, t) − N X m=1,m6=n sn,mum(x, t − τm), where x ∈ R2 , and rn, pn and sn,mare nonnegative constants.
3. Construction of upper and lower solutions. This section is devoted to the construction of upper and lower solutions of (4). First, we give the definition. Definition 3.1. Assume Φ = φ1, · · · , φN and Ψ = ψ1, · · · , ψN belong to C(R, R)N such that 0 Φ, Ψ M = (M1, · · · , MN) 0. Then Ψ and Φ are called an upper solution and a lower solution of (4) respectively, if they are differ-entiable almost everywhere and satisfy
(1) φ′
n(t) ≤ Ln[φn](t) + φn(t)fn([Ψ|φn](t), Ψbnt), n = 1, · · · , N , a.e.; (2) ψ′
n(t) ≥ Ln[ψn](t) + ψn(t)fn([Φ|ψn](t), Φntb), n = 1, · · · , N , a.e..
Now we construct a pair of upper-lower solutions of (4). First, let η be the number satisfying 1 < η < minn λn,2 λn,1, λn,1+ λm,1 λn,1 m, n = 1, · · · , No≤ 2. (14) For δ > 1, we define functions hn(t) by
hn(t) := eλn,1t− δeηλn,1t, n = 1, · · · , N. (15)
Then it is easy to see that lim
t→−∞hn(t) = 0, limt→∞hn(t) = −∞ and h ′
n(t) = λn,1(eλn,1t− δηeηλn,1t). Thus there exists a unique t⋆
n(δ) < 0 such that h⋆n:= hn(t⋆n) = max
t∈R hn(t) > 0 and δ→∞lim t ⋆
If δ is large enough then hn(0) < 0 and there exists σn> 1 such that the length of the intervals In:= {t| hn(t) ≥ h⋆
n/σn, t ∈ R} is equal to max{c∗, τ } for n = 1, · · · , N. Denote σ := max{σn| n = 1, · · · , N } and tn(δ) by
tn(δ) := max{t| hn(t) = h⋆
n/σ}, n = 1, · · · , N, then limδ→∞tn(δ) = −∞ and h⋆
n < φ⋆n for all n when δ is large enough. Further-more, for any γ > 0, let εn > 0, n = 1, · · · , N be such that
hn(tn) = φ⋆n− εne−γtn, (or εn= (φ⋆n− h⋆n/σ)eγtn), (16) then is easy to see that hn(t) > hn(tn) for tn− c∗< t < tn.
Next, we assume that there exist bεn > 0, n = 1, · · · , N satisfying the following assumption: (A4) εn∂fn ∂xn(X, Y ) < N X m=1,m6=n b εm ∂fn ∂xm(X, Y ) + ∂fn ∂ym(X, Y ) , b εn∂fn ∂xn(X, Y ) < N X m=1,m6=n εm ∂fn ∂xm(X, Y ) + ∂fn ∂ym(X, Y ) , for X, Y ∈ RN × RN −1. We further define the numbers btn, n = 1, · · · , N by
φ⋆n+ bεne−γbtn= eλn,1btn, n = 1, · · · , N. (17) Then it is obvious that min{btn| n = 1, · · · , N } > τ + max{tn| n = 1, · · · , N } if δ is large enough, since btn is bounded below by (17).
Finally, we define the functions φ−
n, φ+n, n = 1, · · · , N by φ− n(t) := eλn,1t− δeηλn,1t, t ≤ tn, φ⋆ n− εne−γt, t > tn, φ+n(t) := eλn,1t, t ≤ btn, φ⋆ n+ bεne−γt, t > btn, and Φ−:= (φ− 1, · · · , φ−N) and Φ+:= (φ + 1, · · · , φ+N). Then lim t→−∞Φ −(t) = lim t→−∞Φ +(t) = 0 and lim t→∞Φ −(t) = lim t→∞Φ +(t) = Φ⋆, (18) and Φ−(t) < Φ+(t) for t ≤ tn or t ≥ btn, see Figure1. On the interval [tn, btn], φ−
n(t) and φ+
n(t) are concave downwards and concave upwards respectively. Consequently, if (φ−
n(tn))′ ≤ (φ+n(tn))′ then φ−n(t) ≤ φ+n(t) for all t ≥ tn by taking γ satisfying 0 < γ ≤ min
1≤n≤N{λn,1e
λn,1tn/εn}.
Lemma 3.2. Assume that δ is large enough, 0 < γ ≤ min1≤n≤N{λn,1eλn,1tn/εn}
small enough, and there exists positive numbers {bεn}N
n=1 satisfying (A4). Then Φ− and Φ+ are lower and upper solutions of (4) respectively.
Proof. We only have to show that Φ− and Φ+satisfy the differential inequalities of Definition3.1 for t ∈ R \ {tn, btn| n = 1, · · · , N }. To simplify the computations, we introduce the notation
( ¯dn,0, ¯dn,1, ¯dn,2, ¯dn,3, ¯dn,4) := (−dn,0, dn,1, dn,2, dn,3, dn,4), (α0, α1, α2, α3, α4) := (0, −c1, −c2, c1, c2).
Then the function ∆n(λ, c) in the characteristic equations can be rewritten as ∆n(λn,1, c) = −λn,1+Pidn,ie¯ λn,1αi+ fn(0, 0) = 0. (19)
tn btn 0 Φ⋆ eλn,1t− δeηλn,1t φ⋆ n− εne−γt eλn,1t φ⋆ n+ bεne−γt
Figure 1. Graphs of a pair of upper-lower solutions Φ+ and Φ−.
Note that
∆n(ηλn,1, c) = −ηλn,1+Pidn,ie¯ ηλn,1αi+ fn(0, 0) < 0. (20)
Now we start the proof of the first differential inequality of Definition3.1. If t ≤ tn, then (φ−
n)′(t) = λn,1eλn,1t− δηλn,1eηλn,1t. According to equations (19), (20) and the Mean Value Theorem, we have
Ln[φ− n](t) + φ−n(t)fn([Φ +|φ− n](t), (Φ + t)bn) ≥X i ¯ dn,i(eλn,1(t+αi)− δeηλn,1(t+αi)) + φ− n(t)fn([Φ+|φ−n](t), (Φ+t)bn) =(φ−n)′(t) − δeηλn,1t∆(ηλn,1, c) + φ−n(t) fn([Φ+|φ−n](t), (Φ+t)bn) − fn(0, 0) =(φ−n)′(t) − δeηλn,1t∆(ηλn,1, c) + φn−(t)Dfn(Ψ1, Ψ2) · [Φ+|φ−n](t), (Φ+t)bn , (21)
for some Ψ1 and Ψ2. By (14), we have −δeηλn,1t∆(ηλn,1, c) = O(eηλn,1t) and
φ−n(t)Dfn(Ψ1, Ψ2) · ([Φ+|φ−n](t), (Φ+t)bn) =φ−n(t) ∂fn ∂xn(Ψ1, Ψ2)φ − n(t) + X m6=n (∂fn ∂xm(Ψ1, Ψ2)e λm,1t+ ∂fn ∂ym(Ψ1, Ψ2)e λm,1(t−τm)) =O(e2λn,1t) + O(e(λn,1+λm,1)t),
as t → −∞. Since ∆(ηλn,1, c) < 0, the summation in equation (21) is positive if tn is small enough. If t > tn then Ln[φ− n](t) + φ−n(t)fn([Φ+|φ−n](t), (Φ + t)nb) − (φ−n)′(t) ≥ X i ¯
dn,i(φ⋆n− εne−γ(t+αi)) + φ−n(t)fn([Φ+|φ−n](t), (Φ+t)bn) − γεne−γt = −εne−γtI(γ) + φ−n(t) fn([Φ+|φ−n](t), (Φ+t)bn) − fn(Φ⋆, (Φ⋆)bn)
where I(γ) := Pidn,ie−γαi+ fn(Φ⋆, (Φ⋆)bn) + γ. Note that I(γ) → 0 as γ → 0+. For tn < t ≤ btn, we have fn([Φ+|φ− n](t), (Φ + t)bn) − fn(Φ⋆, (Φ⋆)bn) = Dfn(Ψ1, Ψ2) · ([Φ+|φ−n](t) − Φ⋆, (Φt+)nb− (Φ⋆)bn) ≥ −εne−γbtn∂fn ∂xn(Ψ1, Ψ2) + X m6=n ∂fn ∂xm(Ψ1, Ψ2)bεme γbtn+ X m6=n ∂fn ∂ym(Ψ1, Ψ2)bεme γbtn = e−γbtn − εn∂fn ∂xn(Ψ1, Ψ2) + X m6=n ∂fn ∂xm(Ψ1, Ψ2)bεm+ X m6=n ∂fn ∂ym(Ψ1, Ψ2)bεm > 0.
Similarly, for t ∈ (btn, ∞) we have
fn([Φ+|φ−n](t), (Φ+t)nb) − fn(Φ⋆, (Φ⋆)bn) = Dfn(Ψ1, Ψ2) · ([Φ+|φ− n](t) − Φ⋆, (Φ+t)nb− (Φ⋆)bn) = e−γt − εn∂fn ∂xn(Ψ1, Ψ2) + X m6=n ∂fn ∂xm(Ψ1, Ψ2)bεm+ X m6=n ∂fn ∂ym(Ψ1, Ψ2)bεme γτm) > 0.
Combining the above discussions, if γ is small enough then we obtain the first differential inequality of Definition3.1.
Next, we prove of the second differential inequality of Definition3.1. If t ≤ btn, then (φ+ n)′(t) = λn,1eλn,1tand Ln[φ+n](t) + φ+n(t)fn([Φ−|φ+n](t), (Φ−t)bn) ≤ X i ¯ dn,ieλn,1(t+αi)+ eλn,1tfn([Φ−|φ+ n](t), (Φ−t)bn) = (φ+n)′(t) + eλn,1t fn([Φ−|φ+ n](t), (Φ − t)bn) − fn(0, 0bn) ≤ (φ+n)′(t) for n = 1, · · · , N . On the other hand, if t > btn, then (φ+
n)′(t) = −γbεne−γt and Ln[φ+n](t) + φ+n(t)fn([Φ−|φ+n](t), (Φ−t )nb) − (φ+n)′(t) ≤ X i ¯ dn,i(φ⋆ n+ bεne −γ(t+αi)) + φ+ n(t)fn([Φ−|φ + n](t), (Φ−t)bn) + γbεne−γt = εneb −γtI(γ) + φ+ n(t) fn([Φ−|φ + n](t), (Φ−t)nb) − fn(Φ⋆, (Φ⋆)bn) . Similar to previous estimation, I(γ) → 0 as γ → 0+ and
fn([Φ−|φ+ n](t), (Φ−t)bn) − fn(Φ⋆, (Φ⋆)bn) = Dfn(Ψ1, Ψ2) · ([Φ−|φ+n](t) − Φ⋆, (Φt−)bn− (Φ⋆)bn) = e−γt εnb ∂fn ∂xn(Ψ1, Ψ2) − X m6=n ∂fn ∂xm(Ψ1, Ψ2)εm− X m6=n ∂fn ∂ym(Ψ1, Ψ2)εme γτm) < 0.
Therefore, the second differential inequality holds when γ is small enough. The proof is complete.
4. Existence of traveling plane waves. After constructing the upper and lower solutions of (4) in previous section, now we start to show the existence of travel-ing plane wave solutions by ustravel-ing the iterative method and Schauder’s fixed point theorem.
Denote CM(R, RN) := {(u1, · · · , uN)|un∈ C(R, R), 0 ≤ un≤ M, n = 1, · · · , N } where M = maxt∈R,1≤n≤Nφ+n, and subspace Γ of CM(R, RN) by
Γ := {Φ ∈ CM(R, RN) | Φ− Φ Φ+}. (22) Then Γ is closed, convex and bounded under the supremum norm. To apply the Schauder’s fixed point theorem for the existence of traveling plane wave solutions, we need the following properties of the operator G on the space Γ.
Lemma 4.1.
(i) G is a continuous operator from CM(R, RN) to C(R, RN). (ii) G is an invariant and compact operator on Γ.
Proof. (i) First, we show that G maps CM(R, RN) into C(R, RN). By the definitions of Gn, for any t ∈ R and h > 0, we have
|Gn(Φ)(t + h) − Gn(Φ)(t)| = |e−βn(t+h) Z t+h −∞ eβnsHn(Φ)(s)ds − e−βnt Z t −∞ eβnsHn(Φ)(s)ds| ≤ (1 − e−βnh) Z t −∞ eβn(s−t)|Hn(Φ)(s)|ds + Z t+h t eβn(s−t−h)|Hn(Φ)(s)|ds. Therefore, lim h→0+|Gn(Φ)(t + h) − Gn(Φ)(t)| = 0, uniformly for t ∈ R.
The above result holds similarly for h < 0. Hence G(Φ) ∈ C(R, RN). Moreover, {G(Φ)|Φ ∈ CM(R, RN)} is uniformly equicontinuous.Next, by the assumption (A2) of fn, if Φ, Ψ ∈ CM(R, RN) then there exists a constant Lf(M ) > 0 such that
|φnfn(Φ, Ψnb t) − ψnfn(Ψ, Ψbnt)| ≤ Lf N X n=1 kφn− ψnk. Hence |Gn(Φ)(t) − Gn(Ψ)(t)| ≤ Z t −∞ eβn(s−t) φnfn(Φ, Φnb t) − ψnfn(Ψ, Ψbnt) +βn|φn− ψn| + |Ln[φn] − Ln[ψn]|(s)ds ≤ (Lf+ βf+ 2 4 X n=0 dn,i)( N X n=1 kφn− ψnk) Z t −∞ eβn(s−t)ds ≤ (Lf+ βf+ 2 4 X n=0 dn,i)( N X n=1 kφn− ψnk)/βn (23) for all n = 1, · · · , N , where βf := max1≤n≤Nβn. Since the right hand side of above inequality is independent on t, this implies that for any ε > 0 there exists δ > 0 such that if kΦ − Ψk < δ then
Therefore the assertion of part (1) follows.(ii) By Lemma2.3and the properties of upper and lower solutions, if t ∈ R \ {tn, btn}N
n=1 then
φ−n(t) ≤ Gn([Φ+|φ−n])(t) ≤ Gn(Φ−)(t) ≤ Gn([Φ−|φ+n])(t) ≤ φ+n(t). (24) Following the same arguments, we have Φ− ≤ G(Φ) ≤ Φ+ for any Φ ∈ Γ. Hence, G is invariant on Γ.
The proof of compactness property for the iterations can be found in [23], so we omit it. The proof is complete.
By Lemma4.1, we obtain the following main results.
Main Theorem. Assume (A1)-(A3) and the same assumptions of Lemma 3.2
Then for any 0 < c < c∗, there exists traveling plane wave solutions of (4) and (5). 5. Applications. In this section we will apply the main theorem to show the exis-tence of traveling plane wave solutions of the Lotka-Volterra N -species competition systems on two-dimensional lattices with discrete diffusion and distributive time de-lays. The dynamics of this competition model is governed by the following systems:
dun;i,j dt =Ln[un;i,j] + un;i,jrn− pnun;i,j− N X m=1,m6=n sn,m Z 0 −τn kn(s)um;i,j(t + s)ds, (25)
for (i, j) ∈ Z2, 1 ≤ n ≤ N , where Ln[un;i,j](t) are defined as before in Section 1, rn are the natural growth rates; pn account for self-regulation of each spicy; sn,mare the competing rates; kn(s) ∈ C([−τn, 0], (0, ∞)) are delay kernels and normalized
such that Z
0 −τn
kn(s)ds = 1, for n = 1, · · · , N.
The systems (25) model the interaction among various competing species, has been studied extensively, and various sufficient conditions for the coexistence and extinc-tion of the competing species are obtained, cf. [11, 22, 29, 36]. The coefficients rn, pn, sn,m play a fundamental row in its asymptotic behavior. In particular, if N = 2 and dn,i = 0 for all n and i, that is no diffusion term, Gopalsamy studied the stability of the equilibrium of systems (25). For the ecological significance of (25), one can refer to [14, 15] and the references cited therein.
To find a strictly positive stationary solution Φ⋆ = (φ⋆
1, · · · , φ⋆N) of (25), we should solve the following systems of linear algebraic equations.
0 = dn+ rn− pnφ⋆ n− N X m=1,m6=n sn,mφ⋆ m, (26)
where dn:=P4i=1dn,i− dn,0. Denote the matrices S, P and the vector r, d by P = (pn,m) with pn,n= pn and pn,m= 0, if n 6= m;
S = (sn,m) with sn,n= 0, n, m = 1, · · · , N ; r= (r1, · · · , rN)T and d = (d1, · · · , dN)T. In matrix form, we have to solve a linear systems
It is obvious that the solution Φ⋆ exists and is unique if and only if P + S is nonsingular. More additional conditions on the parameters should be imposed to ensure the strictly positivity of Φ⋆. Hence we assume the following conditions hold for systems (25).
(LV1) assume dn,3≥ dn,1> 0, dn,4≥ dn,2> 0 and dn+ rn> 0 for all n;
(LV2) assume the matrix P +S is nonsingular, and there exists two constant vectors L K ≻ 0 such that
P K + SL d + r P L + SK.
Next, let we check assumptions (A1)∼(A4). According the conditions (LV1) and (LV2), conditions (A1)∼(A3) hold obviously. The following we illustrate some specific examples which satisfy (A4).
Example. Assume N ≥ 2, rn = r, dn = d, r + d > 0, pn = 1 for n = 1, · · · , N , sn,m = α > 0 for all n 6= m and sn,n = 0 for all n. Let α be small enough, e.g. (N − 1)α < 1, then there exists a unique positive equilibrium Φ⋆ = (φ⋆
1, · · · , φ⋆N) with
φ⋆n =
rn+ dn
1 + (N − 1)α, n = 1, · · · , N. The condition (A4) is equivalent to
αX
m6=n
εm< bεn< εn
(N − 1)α, n = 1, · · · , N.
Let us fix any numbers ε1, · · · , εN. If α is small enough then it is easy to see that there exist positive numbers bε1, · · · , bεN satisfying the above conditions.
Therefore, we have the following results.
Theorem 5.1. Assume the systems (25) satisfies the assumptions (LV1)∼(LV2) and (A4). Then there exists c∗ > 0 such that for any 0 < c < c∗, (25) has a traveling plane wave solution satisfying the asymptotically boundary conditions (5). Acknowledgments. The authors thank the referees for their valuable comments and suggestions that help the improvement of the manuscript.
This work is supported in part by the National Science Council of Taiwan and the National Center for Theoretical Sciences of Taiwan.
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Received March 2009; revised January 2010.
E-mail address: chhsu@math.ncu.edu.tw E-mail address: thyang@mail.tku.edu.tw
