## Abstract

A non-monotone time-delayed lattice system with global interaction is considered. The spreading speed, including the upward convergence, is established by comparison arguments and a fluctuation method. The existence of travelling waves is obtained by Schauder’s fixed-point theorem and a limiting process. It turns out that the minimal wave speed of travelling waves coincides with the spreading speed.

## 1. Introduction

Spreading speed (short for the asymptotic speed of spread/propagation), as an important notion in biological invasions, was first introduced by Aronson & Weinberger (1975, 1978) for reaction–diffusion equations. Since then, quite a few works have shown the coincidence of the spreading speed with the minimal speed for travelling waves under appropriate assumptions for various evolution systems (e.g. Arson & Weinberger 1975; Thieme & Zhao 2003; Fang *et al*. 2008; Hsu & Zhao 2008; and references therein). A general theory of spreading speeds and travelling waves has been developed for monotone semiflows (Weinberger 1982; Lui 1989; Li *et al.* 2005; Liang *et al*. 2006; Liang & Zhao 2007).

Weng *et al*. (2003) derived the following monostable time-delayed lattice system with global interaction to describe the growth of mature population of a single species in a patchy environment:
1.1
where *β*(*k*)≥0 with , the birth function *b* has the property that *b*(0)=0 and *b*(*w*)−*dw*=0 has a positive solution *w**, such that *b*(*w*)−*dw*>0,∀*w*∈(0,*w**). In the case where *b*(*w*) is monotone in *w*∈[0,*w**], they obtained the spreading speed and its coincidence with the minimal wave speed of travelling waves under some conditions, which were weakened by Liang & Zhao (2007) in the application part. Ma & Zou (2005) then established the uniqueness (up to translation) and stability of travelling waves for the local case (i.e. *β*(*k*)=0 for all *k*≠0) of equation (1.1) with monotone birth functions. However, birth functions, such as logistic type and Ricker type, are not monotone in general. So it is worthy to study system (1.1) with non-monotone birth functions, e.g. Ma *et al*. (2006, theorems 2.1 and 2.2), on spreading speed for equation (1.1) in a weak sense. The purpose of the current work is to address the spreading speed (with the upward convergence) and travelling waves in the non-monotone case.

It is well known that travelling-wave profiles of the reaction–diffusion equation ∂_{t}*u*=*DΔu*+*f*(*u*) are monotone. But when the nonlinear term appears in a non-local way or involves time delay, travelling-wave profiles may lose monotonicity if *f* is not monotone. This phenomenon has been observed both numerically and analytically (e.g. Gourley 2000; Kyrychko *et al*. 2005; Faria & Trofimchuk 2006). More precisely, if the nonlinear term is quasi-monotone (e.g. Martin & Smith 1990), then the solution semiflow is still monotone, and hence, the theory developed in Liang & Zhao (2007) can be applied to obtain monotone travelling waves. Note that system (1.1) does not satisfy the quasi-monotone condition when *b* is not monotone. Thus, it is difficult to show that the solution semiflow or each wave profile is monotone in such a case.

Recently, there have been increasing efforts on the existence of travelling waves for non-monotone delayed evolution equations (Wu & Zou 2001; Faria & Trofimchuk 2006; Faria *et al*. 2006; Ou & Wu 2007), in which these authors showed the existence for sufficiently large wave speeds or small delays. Wu & Zou (2001) used the idea of exponential ordering for delayed differential equations to obtain some existence results for travelling waves (see also Huang & Zou 2003). Hsu & Zhao (2008) employed a comparison argument and Schauder’s fixed-point theorem to study spreading speeds and travelling waves for an integro-difference equation. Ma (2007) also applied Schauder’s fixed-point theorem to study travelling waves for a time-delayed and non-local reaction–diffusion equation.

The rest of this paper is organized as follows. In §2, we present preliminaries. In §3, we use comparison arguments and a fluctuation method to establish the spreading speed *c** and the upward convergence. As a consequence of the spreading speed, we then get the non-existence of travelling waves with the wave speed less than *c**. In §4, we first employ Schauder’s fixed-point theorem to obtain the existence of travelling waves *u*(*j*+*ct*) with , and *c*>*c**. Then, we use a limiting argument to get a limiting wave-profile that satisfies the wave-profile equation with *c*=*c**. With the help of a basic property of wave profiles, we finally show that the limiting wave profile *u* satisfies and .

## 2. Preliminaries

In this section, we present some necessary notations, assumptions and preliminary results. We start with the definitions of spreading speeds, upward convergence and travelling waves for equation (1.1).

## Definition 2.1.

A number *c**>0 is called the spreading speed for a function if for every *c*>*c**, and if there exists some *ϵ*>0 such that for every *c*∈(0,*c**).

From this definition, we see that *c** and *ϵ* depend on the function *u*. However, we will show that the solutions *w*_{j}(*t*) of equation (1.1) with initial functions having compact supports share the same *c** and *ϵ*. So, we call such *c** the spreading speed of equation (1.1). By the upward convergence, we mean , where *w** is the positive equilibrium.

## Definition 2.2

A pair (*u*,*c*) is said to be a travelling wave of (1.1 ) if is a non-trivial and bounded solution of equation (1.1 ) having the form *w*_{j}(*t*)=*u*(*j*+*ct*). The function *u* is called the wave profile and the number *c* is called the wave speed.

Define
where *c* is regarded as a parameter. We call *Δ*(*c*,*λ*)=0 the characteristic equation, and its solutions are eigenvalues. We impose the following assumption on *β*(*k*):

(K) , and there exists

*λ*^{♯}>0 such that is convergent when*λ*∈[0,*λ*^{♯}) and , where*λ*^{♯}may be .

Next, we present some properties of the characteristic equation.

## Lemma 2.3.

*Assume b′(0)>d and assumption (K) hold. Then, Δ(c,λ) has the following properties:*

*for each c>0,Δ(c,λ) is a concave function of λ≥0,**the system*2.1*admits a unique positive solution*(c**,λ**),*and**for each c>c**,*there are exactly two positive eigenvalues λ*_{i}*=λ*_{i}*(c),i=1,2, with λ*_{1}*<λ*_{2}*and Δ(c,λ)>0 for λ∈(λ*_{1}*,λ*_{2}).

## Proof.

By direct computations, one can obtain the result (Weng *et al*. 2003). ▪

In order to establish the spreading speed and its coincidence with the minimal wave speed, we first recall some known results for equation (1.1) with monotone birth function *b*. Assume that

(E) the function

*b*is locally Lipschitz continuous from to and has the following properties:*b*(0)=0,*b*′(0)>*d*,*b*′′(0) exists and*b*(*w*)≤*b*′(0)*w*for*w*≥0,*b*(*w*)=*dw*has the smallest positive solution*w**, and*b*is non-decreasing on [0,*w**].

Define and . Let be the space of all bounded and continuous functions from to . We use the standard pointwise ordering in , and . For a given number *η*>0, define , and .

The following result comes from Weng *et al*. (2003, theorems 3.1 and 5.1) and Liang & Zhao (2007, theorems 5.3 and 5.4), which shows that the spreading speed coincides with the minimal wave speed when *b* is monotone.

## Theorem 2.4.

*Suppose ( E) and (K) hold. Let c* be defined as in lemma 2.3. Then, for any given function* ,

*equation (1.1 ) has a unique global solution*,

*with*

*w*(*t*)=*ϕ*(*t*) for −*r*≤*t*≤0 and 0≤*w*(*t*)≤*w** for*t*≥−*r*. Furthermore, the following statements are valid:*if**, ,*w*_{j}(*t*)=0 for*t*∈[−*r*,0] and*j*outside a bounded interval, then, for any*c*>*c**if**, ,*w*(*t*)≢0 for*t*∈[−*r*,0], then for any*c*<*c**and**for any*,*c*≥*c**, equation (1.1 ) has a travelling-wave solution (*u*,*c*), such that*u*(*x*) is continuous and non-decreasing in*and*.*Moreover, for any**.*c*<*c**, equation (1.1 ) has no travelling wave connecting 0 and*w*

To extend the above result to the case where the birth function *b* is not monotone, we modify the assumption (E) into the following one:

(E′) The function

*b*is locally Lipschitz continuous from to and has the following properties:*b*(0)=0,*b*′(0)>*d*,*b*′′(0) exists and*b*(*w*)≤*b*′(0)*w*for*w*≥0,*b*_{+}(*w*)=*dw*has the smallest positive solution*w**_{+}, andthere is

*η*>0 such that*b*_{±}(*w*)=*b*(*w*) for*w*∈[0,*η*], where*b*_{±}(*w*), respectively, are defined by

It is easy to see that both *b*_{−} and *b*_{+} are non-decreasing and *b*_{−}(*w*)≤*b*(*w*)≤*b*_{+}(*w*) for *w*∈[0,*w**_{+}]. If *b* itself is non-decreasing, then *b*_{±}=*b* and (E′) becomes (E). If *b*(*w*)=*dw* has the smallest positive solution *w** and *b*(*w*)<*dw* for *w*>*w**, then (E′)(2) holds. Also, (E′)(2) ensures that *b*_{−}(*w*)=*dw* and *b*(*w*)=*dw* both have the smallest positive solution in [0,*w**_{+}], namely *w**_{−} and *w**, respectively. Further, (E′)(3) guarantees that *b* and *b*_{±} have the same linearization at zero. A sufficient condition for (E′)(3) to hold is that *b*′(0)>0 and there is *η*>0 such that *b*∈*C*^{1}[0,*η*]. Obviously, *b*_{±} both satisfy (E) with *b*=*b*_{±} if *b* satisfies (E′).

Note that *b*_{±} satisfy (E) with *b*=*b*_{±} and , respectively, and *b*_{±} have the same linearization as that of *b*. Then the results stated in theorem 2.4 with , respectively, are valid for the following two equations:
2.2
and
2.3

To prove the upward convergence, we impose the following assumption on the birth function *b*. Define .

(P) For any

*u*,*v*∈[*w**_{−},*w**_{+}] satisfying*u*≤*w**≤*v*, and , we have*u*=*v*.

By the same argument as in Hsu & Zhao (2008, lemma 2.1), it follows that either of the following two conditions is sufficient for (P) to hold:

(P1)

*wb*(*w*) is strictly increasing for*w*∈[*w**_{−},*w**_{+}] or(P2)

*b*(*w*) is non-increasing for*w*∈[*w**,*w**_{+}] and is strictly decreasing for*w*∈(0,*w**], where .

We point out that the functions of logistic type *b*(*w*)=*rw*(1−*w*/*k*) and Ricker type *b*(*w*)=*pwe*^{−qw} both satisfy the assumption (E′) and the condition (P), when the parameters are in appropriate ranges (Hsu & Zhao 2008). Take *b*(*w*)=*pwe*^{−qw} with *p*,*q*>0 for example. When *p*/*d*>1, (E′) holds. Moreover, when *p*/*d*∈(1,*e*], *b*(*w*) is monotone for *w*∈[0,*w**]; when *p*/*d*∈(*e*,*e*^{2}], *b* loses monotonicity but satisfies the condition (P2).

## 3. The spreading speed

In this section, we establish the spreading speed, the upward convergence and the non-existence of travelling waves. In what follows, we always assume (K) and (E′) hold. Let *c** be defined as in lemma 2.3. We start with the well-posedness for the initial-value problems of equation (1.1).

## Lemma 3.1.

*For any* *, equation (1.1 ) has a unique global solution* *through ϕ with 0≤w(t)≤w**_{+}*,∀t≥−r.*

Here, we omit the proof since it is essentially the same as in the case where *b* is monotone (Weng *et al*. 2003, theorem 4.1).

The following lemma is the comparison principle for solutions to equations (1.1), (2.2) and (2.3).

## Lemma 3.2.

*For any* *and* *, with ϕ*_{−}*≤ϕ≤ϕ*_{+}*, let w*_{−}*(t,ϕ*_{−}*) be the solution of equation (2.3 ) through ϕ*_{−}*, w*_{+}*(t,ϕ*_{+}*) be the solution of equation (2.2 ) through ϕ*_{+} *and w(t,ϕ) be the solution of equation (1.1 ) through ϕ. Then,* *.*

## Proof.

Let *v*(*t*)=*w*_{−}(*t*)−*w*(*t*). Since *ϕ*_{−}≤*ϕ*, we have *w*_{−}(*t*−*r*)≤*w*(*t*−*r*) for *t*∈[0,*r*]. Note that *b*_{−}(*w*)≤*b*(*w*) for *w*∈[0,*w**_{+}] and *b*_{−}(*w*) is non-decreasing for *w*∈[0,*w**_{+}]. It then follows that, for any *t*∈[0,*r*],
3.1
By using the discrete Fourier transform (Weng *et al*. 2003; Liang & Zhao 2007), we see that, for any *t*∈[0,*r*],
where
By the same argument, we can obtain *v*(*t*)≤0 for *t*∈[*r*,2*r*]. Step by step, we finally obtain *v*(*t*)≤0,∀*t*≥−*r*. Thus, *w*_{−}(*t*,*ϕ*_{−})≤*w*(*t*,*ϕ*). Similarly, *w*(*t*,*ϕ*)≤*w*_{+}(*t*,*ϕ*_{+}). ▪

By the standard comparison arguments as in Thieme & Zhao (2003, theorem 2.1) and Ma *et al*. (2006, theorem 2.1), we can easily obtain the convergence to zero, even for a larger class of initial data than those having compact supports. The following result shows that the number *c** defined in lemma 2.3 is a spreading speed in a strong sense for equation (1.1), and in particular, the upward convergence holds under some appropriate assumptions.

## Theorem 3.3.

*Let* *and w(t) be the unique global solution of equation (1.1 ) through ϕ with 0≤w(t)≤w*_{+}. Then, the following statements are valid*:

*if**, ,*ϕ*_{j}(*t*)=0,∀*t*∈[−*r*,0],|*j*|≥*k*for some*k*>0, then, for any*c*>*c**if**,*ϕ*(*t*)≢0 for*t*∈[−*r*,0], then, for any*c*<*c**and**if*.*b*(*w*)/*w*is strictly decreasing for*w*∈[*w**_{−},*w**_{+}] and the condition (P) holds, then,

## Proof.

For any , define by . It then follows from lemma 3.2 that , which, together with the fact that *c** is the spreading speed of solutions for both equations (2.2) and (2.3), implies that *c** is also the spreading speed of equation (1.1) in terms of statements (i) and (ii).

To prove the upward convergence, we proceed as in Thieme (1979, §3.9). Define by
Clearly, *g*(*u*,*v*) is non-decreasing in *u* and non-increasing in *v*, and *g*(*w*,*w*)=*b*(*w*). For *β*∈(0,*c**), we set
We choose a sequence , such that as and .

Rewrite equation (1.1) in the following way:
It then follows that
By Fatou’s lemma,
Let 0<*β*<*γ*<*c**. It then follows that, for any given and , there holds |*j*−*k*|≤*γ*(*t*_{j}+*s*) when *j* is sufficiently large. Then, we have
and
Thus, the following inequality holds:
Set
Then, we have
and hence,
3.2
Similarly, we have
3.3
By the definition of function *g*, we can find , such that
3.4
It then follows that
3.5
and hence,
This, together with the strict monotonicity of *b*(*w*)/*w* on [*w**_{−},*w**_{+}], implies that *u*≤*w**≤*v*. By equation (3.5) and property (P), we obtain *u*=*v*. It then follows from equation (3.5) that *V*_{*}(*c*,*γ*)=*V**(*c*,*γ*)=*u*=*v*. Thus, we see from equations (3.2) to (3.4) that
By the uniqueness of positive fixed point of on [0,*w**], it follows that *V*_{*}(*c*,*γ*)=*w**. Consequently, *w**=*V*_{*}(*c*,*γ*)≤*W*_{*}(*c*)≤*W**(*c*)≤*V**(*c*,*γ*)=*w**, which means for any *c*∈(0,*c**). ▪

The non-existence of travelling waves with speed *c*<*c** was stated in Ma *et al*. (2006, corollary 2.2) without proof. For completeness, we provide an elementary proof of it below.

## Theorem 3.4.

*For any c<c*, equation (1.1 ) has no travelling-wave solution (u,c), with*

*and*.

## Proof.

Assume, for the sake of contradiction, that for some *c*_{0}∈(0,*c**), equation (1.1) has a travelling wave (*u*,*c*_{0}) with . Let *c*_{0}<*c*_{1}<*c*_{2}<*c** with *c*_{1}<2*c*_{0} and [*c*_{1}*t*] be the integer part of *c*_{1}*t*. Define the function *m*(*t*):=−[*c*_{1}*t*]+*c*_{0}*t* and the intervals *I*_{k}:=[*k*/*c*_{1},(*k*+1)/*c*_{1}),*k*≥0. Then, it follows that *m*(*t*) is continuous and increasing on each *I*_{k}, and , and hence,
Since *u*≢0, we can find such that *u*(*ξ*_{0})>0. Choose such that *t*_{0}:=(*ξ*_{0}+*j*_{0})/*c*_{0}≥0. Define , with *ϕ*_{j}(*θ*)=*u*(*j*+*c*_{0}(*t*_{0}+*θ*)),*θ*∈[−*r*,0]. Then, *ϕ*_{−j0}(0)=*u*(*ξ*_{0})>0, and hence, *ϕ*≢0. Let be the solution of equation (1.1) through *ϕ*. Thus, the uniqueness of solutions of equation (1.1) implies that *w*_{j}(*t*,*ϕ*)≡*u*(*j*+*c*_{0}(*t*+*t*_{0})),*t*≥0. By theorem 3.3(ii), it then follows that
Letting *j*=−[*c*_{1}*t*] in the above inequality, we further have
which is a contradiction. ▪

## 4. Travelling waves

In this section, we prove the existence of travelling waves (*u*,*c*), with and *c*≥*c**, where *c** is the spreading speed of equation (1.1) established in theorem 3.3. Substituting *w*_{j}(*t*)=*u*(*j*+*ct*) into equation (1.1), we get the wave-profile equation
4.1

The existence of travelling waves was studied in Ma *et al*. (2006, theorem 3.1) for system (1.1) with monotone birth function *b* via the upper–lower solution method and the Schauder fixed-point theorem. In order to get the existence of travelling waves for the non-monotone case, we construct two monotone systems to sandwich such a system and then use the Schauder fixed-point theorem. Further, we employ the properties of the spreading speed (theorem 3.3(ii) and (iii)) to obtain the asymptotic behaviour of the wave profile at .

## Theorem 4.1.

*Assume that ( K) and (E′) hold. Then, for any c>c*, equation (1.1 ) has a travelling-wave solution (u,c), with*

*and*

*If, in addition,*.

*b*(*w*)/*w*is strictly decreasing for*w*∈[*w**_{−},*w**_{+}] and condition (*P*) is satisfied, then

## Proof.

Rewrite the wave-profile equation (4.1) as
4.2
where
4.3
and *δ*>0 is chosen so that (*δ*−*d*/*c*−2*D*/*c*)>0. We define *H*_{+} and *H*_{−} by replacing *b* with *b*_{+} and *b*_{−} in (4.3), respectively. Moreover, we know that *H*_{±} are non-decreasing and
For solutions , equation (4.2) is equivalent to . It is thus natural to define an operator by
4.4
Similarly, we can define *T*_{+} and *T*_{−} by replacing *H* with *H*_{+} and *H*_{−} in equation (4.4), respectively. Moreover, *T*_{±} are both non-decreasing, *T*(*w**)=*w**,*T*_{+}(*w**_{+})=*w**_{+},*T*_{−}(*w**_{−})=*w**_{−} and .

For any *c*>*c**, let *λ*_{1}=*λ*_{1}(*c*) be defined as in lemma 2.3. Define
where *ϵ*>0 and *ς* are parameters. Since *T*_{±} are non-decreasing, it then follows that (see the proof of Weng *et al*. (2003, lemma 3.3)) and whenever *ϵ*>0 and *ς*>0 are properly chosen.

For a given *λ*>0, let
and , then (*X*_{λ},∥⋅∥_{λ}) is a Banach space. Note that, for any given *λ*∈(0,*λ*_{1}), and are elements of *X*_{λ}.

Define the subset by . Clearly, *Y* is a convex and closed subset of *X*_{λ} and, for any *ϕ*∈*Y* ,
Thus, .

Next, we show that *T* is compact on *Y* . For any *ϕ*,*ψ*∈*Y* , we have
where *l* is a Lipschitz constant of *b*(*w*) as *w*∈[0,*w**_{+}], and hence, there exists *M*>0 such that
4.5
Thus, *T* is continuous on *Y* .

Note that *H* is bounded on *Y* , that is, for some *K*>0. For any *ϕ*∈*Y* and ,
4.6
Therefore, *T*(*Y*) is a family of uniformly bounded and equi-continuous functions on . Thus, for any given sequence {*ψ*_{n}}_{n≥1} in *T*(*Y*), there exists a subsequence, still denoted by {*ψ*_{n}}_{n≥1}, and such that *ψ*_{n}(*x*)→*ψ*(*x*) uniformly for *x* in any compact subset of . Since , we have , and hence, *ψ*∈*Y* . Now, it remains to show *ψ*_{n}→*ψ* in *X*_{λ}. Note that *ψ*_{n}(*x*)*e*^{−λx}→*ψ*(*x*)*e*^{−λx} uniformly for *x* in any compact subset of and . It then follows that, for any *ϵ*>0, there exist *B*>0 and *N*>1 such that and |*ψ*_{n}(*x*)−*ψ*(*x*)|*e*^{−λx}<*ϵ*,∀|*x*|≤*B*,*n*≥*N*. Thus, ∥*ψ*_{n}−*ψ*∥_{λ}<*ϵ*,∀*n*≥*N*. Therefore, *T*(*Y*) is compact in *X*_{λ}. Now, Schauder’s fixed-point theorem implies that the operator *T* admits a fixed point *u* in *Y* . Clearly, and *u* is non-trivial. Thus, (*u*,*c*) is a travelling-wave solution connecting 0.

Let and . Since is a solution of equation (1.1), by theorem 3.3, we see that
In particular, taking *j*=−[*γt*],∀*γ*∈[*γ*_{1},*γ*_{2}], we have
uniformly for *γ*∈[*γ*_{1},*γ*_{2}]. Setting *ξ*:=*γt* yields
uniformly for *γ*∈[*γ*_{1},*γ*_{2}]. Define . Choose *ξ*_{0}>0 such that (1/*γ*_{2}−1/*γ*_{1})*ξ*_{0}+1/*γ*_{2}<0. Thus, we have
This, together with the fact that *a*(*ξ*,*γ*) is continuous and strictly decreasing in *γ*, implies that
It then follows that .

By the same arguments as in the proof of Hsu & Zhao (2008, theorem 3.1), it follows that is a consequence of the upward convergence. ▪

The following result is about the existence of the travelling wave with speed *c**.

## Theorem 4.2.

*Assume that ( K) and (E′) hold. Then, equation (1.1 ) has a travelling wave (u,c*) with*

*and*.

*If, in addition,*.

*b*(*w*)/*w*is strictly decreasing for*w*∈[*w**_{−},*w**_{+}] and condition (*P*) is satisfied, then

## Proof.

We first show that, for any *α*∈(0,*w**_{−}), there exists a travelling wave (*u*^{α},*c**) such that *u*^{α}(0)=*α*, *u*^{α}(*x*)<*α* for all *x*<0 and . By using a limiting argument (cf. Brown & Carr 1977; Thieme & Zhao 2003), we choose a sequence such that . According to theorem 3.4, there exists a travelling wave (*u*_{j},*c*_{j}) of equation (1.1) for each *j* such that . Since each , is also such a solution, and , we can assume that *u*_{j}(0)=*α*<*w**_{−} and *u*_{j}(*x*)≤*α*,∀*x*<0,*j*≥1. For any , it follows by equation (4.6) that

Thus, {*u*_{j}}_{j≥1} is an equi-continuous and uniformly bounded sequence of functions on . By Ascoli’s theorem and a nested subsequence argument, it follows that there exists a subsequence of {*c*_{j}}, still denoted by {*c*_{j}}, such that *u*_{j}(*x*) converges uniformly on every bounded interval, and hence pointwise on to *u**(*x*). Note that
4.7
Letting in equation (4.7) and using the dominated convergence theorem, we then get and *u*^{α}(0)=*α*, *u*^{α}(*x*)≤*α*,∀*x*<0.

In order to get , we prove a basic property for wave profiles *u*^{α} with small *α*: . Because is a non-increasing function of *y*, it suffices to show that is bounded. Since *b*′(0)>*d* and , we may choose *ϵ*_{0}>0 and *N*>0 such that . For such *ϵ*_{0}, there exists *δ*_{0}>0 such that *b*(*u*)≥(1−*ϵ*_{0})*b*′(0)*u*,∀*u*∈[0,*δ*_{0}]. Define *q*(*u*)(*x*):=*u*(*x*+1)+*u*(*x*−1)−2*u*(*x*). Choosing *α*≤*δ*_{0} and integrating equation (4.1) with *c*=*c** and *u*=*u*^{α} from *y* to with , we have
4.8

Note that *u*^{α}(*x*) is a bounded function on ,
and
It then follows from equation (4.8) that is bounded, and hence, the aforementioned property holds.

Now we are able to show for sufficiently small *α*>0. It suffices to prove exists. Integrating the wave-profile equation (4.1) from *y* to 0, we have
4.9
We consider three terms of the right-hand side of equation (4.9), respectively. The first term has limit when because *u*^{α} is integrable on . Since is convergent for *λ*∈(−*λ*^{♯},*λ*^{♯}), by the same argument as in the proof of Thieme & Zhao (2003, theorem 3.2), we have , and hence, the second term also has the limit as . This is because
where *M*_{1} and *M*_{2} are chosen such that and *u*(*x*)<*M*_{2} for all . Note that the third term is a non-decreasing function of *y* since *b*(*u*)≤*b*′(0)*u* for all *u*≥0, and is also bounded because all other terms in equation (4.9) are bounded. Therefore, the third term has the limit when , and so does *u*^{α}(*y*).

As in the proof of theorem 3.4, the upward convergence implies that . ▪

Finally, we remark that our method also applies to the following general non-local lattice system with distributed time delays: 4.10

## Acknowledgements

This research is supported in part by the Chinese Government Scholarship (for J. F.), the NSF of China (no. 10771045) (for J. W.) and the NSERC of Canada and the MITACS of Canada (for X.-Q. Z.). We are grateful to two anonymous referees for their careful reading and helpful suggestions that led to an improvement of our original manuscript.

## Footnotes

- Received November 1, 2009.
- Accepted December 23, 2009.

- © 2010 The Royal Society