## Abstract

We consider positive travelling fronts, *u*(*t*, *x*)=*ϕ*(*ν*.*x*+*ct*), *ϕ*(−∞)=0, *ϕ*(∞)=*κ*, of the equation *u*_{t}(*t*, *x*)=Δ*u*(*t*, *x*)−*u*(*t*, *x*)+*g*(*u*(*t*−*h*, *x*)), *x*∈^{m}. This equation is assumed to have exactly two non-negative equilibria: *u*_{1}≡0 and *u*_{2}≡*κ*>0, but the birth function *g*∈*C*^{2}(, ) may be non-monotone on [0,*κ*]. We are therefore interested in the so-called monostable case of the time-delayed reaction–diffusion equation. Our main result shows that for every fixed and sufficiently large velocity *c*, the positive travelling front *ϕ*(*ν*.*x*+*ct*) is unique (modulo translations). Note that *ϕ* may be non-monotone. To prove uniqueness, we introduce a small parameter *ϵ*=1/*c* and realize a Lyapunov–Schmidt reduction in a scale of Banach spaces.

## 1. Introduction and main result

In this paper, we consider the time-delayed reaction–diffusion equation(1.1)Equation (1.1) and its non-local versions are widely used to model many physical, chemical, ecological and biological processes (see Faria *et al*. (2006) for further references). The nonlinear *g* is referred to in ecology literature as the *birth function*, and we will suppose that −*s*+*g*(*s*) is of the monostable type. Thus, equation (1.1) has exactly two non-negative equilibria: *u*_{1}≡0 and *u*_{2}≡*κ*>0. For ‖*ν*‖=1, we say that the wave solution, *u*(*x*, *t*)=*ϕ*(*ν*.*x*+*ct*), of (1.1) is a wavefront (or a travelling front) if the profile function *ϕ* satisfies the boundary conditions *ϕ*(−∞)=0 and *ϕ*(+∞)=*κ*. After scaling, such a profile *ϕ* is a positive heteroclinic solution of the delay differential equation(1.2)Observe that *ϕ* needs not be a monotone. In a biological context, *u* is the size of an adult population, so we will consider only positive travelling fronts.

If we take *h*=0 in (1.1), we obtain a monostable reaction–diffusion equation without delay. The problem of existence of travelling fronts for this equation is quite well understood. In particular, for each such equation we can give a positive real number *c*_{*}, such that for every *c*≥*c*_{*} the equation admits exactly one travelling front *u*(*x*, *t*)=*ϕ*(*ν*.*x*+*ct*). Furthermore, equation (1.1) does not have any travelling front propagating at the velocity *c*<*c*_{*}. The profile *ϕ* is necessarily a strictly increasing function. For example, see theorems 8.3(ii), 8.7 and 2.39 in Gilding & Kersner (2004).

However, the situation will change drastically if we take *h*>0. In fact, current results seem a long way from proving similar results concerning the existence, uniqueness and geometric properties of wavefronts for the delayed equation (1.1). This is despite the fact that the existence of travelling fronts for equation (1.1) has been studied intensively in recent years for some specific subclasses of birth functions, e.g. So *et al*. (2001), Wu & Zou (2001), Faria *et al*. (2006), Ma (2007), Trofimchuk & Trofimchuk (2008) and references therein. Clearly, most of the available information is for the so-called monotone case, where *g* is monotone on [0,*κ*]. However, very little is known so far about the number of *positive* wavefronts (modulo translation) for an arbitrary fixed *c*≥*c*_{*}, even for equations with monotone birth functions. In fact, very few theoretical studies are devoted to the uniqueness problem for equation (1.1) and its non-local extensions. To the best of our knowledge, the first uniqueness result for a non-local version of equation (1.1) is due to Thieme & Zhao (2003), who extended an integral-equations approach (see Diekmann (1979); Thieme (1979)) to scalar non-local reaction–diffusion equations with delays. Besides this work, it appears that uniqueness has been established for small delays in Ai (2007) and for a family of unimodal piece-wise linear birth functions (i.e. tent maps) in Trofimchuk *et al*. (2007). Since ‘*asymmetric*’ tent maps mimic the main features of general unimodal birth functions, we believe that the uniqueness of a positive wavefront can be proved for delayed equations with the unimodal birth function satisfying the following assumptions:

The steady state

*y*_{1}(*t*)≡*κ*>0 (*y*_{2}(*t*)≡0) of the equation(1.3)is exponentially stable and globally attractive (hyperbolic) and*g*∈*C*^{1}(_{+},_{+}),*p*≔*g*′(0)>1 and*g*″(*s*) exists and is bounded near 0. We suppose that*g*has exactly two fixed points, 0 and*κ*>0. Set , we assume that*g*(*s*)>0 for*s*∈(0,*ζ*_{2}].

Set *A*≔sup{*a*∈(0,*κ*/2]: *g*′(*s*)>0, *s*∈[0,*a*)}. It should be observed that assumption (G) implies the existence of a positive *ζ*_{1}≤min{*g*(*ζ*_{2}),*A*}, such that . Note that . Without restricting the generality, we may also suppose that .

In this paper, we follow the approach of Faria *et al*. (2006) to prove the uniqueness (up to translations) of positive wavefront for a given (sufficiently fast) speed *c*. In the case of (1.1), this approach essentially relies on the fact that, in ‘good’ spaces and with suitable *g*′(0) *g*′(*κ*), the linear operator is a surjective Fredholm operator. Here, *ψ* is a heteroclinic solution of equation (1.2) considered with *ϵ*=0. Consequently, the Lyapunov–Schmidt reduction can be used to prove the existence of a smooth family of travelling fronts in some neighbourhood of *ψ*. As it has been shown in Faria & Trofimchuk (2006), this family also contains positive solutions. However, an important and natural question about the number of the positive wavefronts still remains unanswered. We solve this problem in the present paper, establishing the following result.

*Assume (H) and* (*G*). *Then there exists a unique positive wavefront (up to translations) of equation* (*1.1*) *for each sufficiently large speed c*.

Note that the wavefront, whose existence and uniqueness is established in theorem 1.1, may be non-monotone. For other results concerning the existence, uniqueness and oscillation properties of a non-monotone wavefront for equation (1.1), see Trofimchuk *et al*. (2007).

In order to apply theorem 1.1, we need to find sufficient conditions to ensure the global attractivity of the positive equilibrium of (1.3). Some results in this direction were found in Liz *et al*. (2005) for nonlinearities satisfying a generalized Yorke condition. The reader can be referred to Röst & Wu (2007) for the case of unimodal nonlinearities, and for further references. On the other hand, Ivanov *et al*. (2002) provide various conditions that are sufficient to guarantee the exponential stability of the positive steady state. The works mentioned above yield the following.

*Let g*∈*C*^{3}(_{+}, _{+}) *be such that*

*the Schwarz derivative**is negative for all x*>0,*x*≠*x*_{M},*g has only one critical point x*_{M}(*global maximum*),*g has exactly two fixed points,*0*and κ*>0.*Moreover,**Γ*_{0}≔*g*′(0)>1,,

*and**either Γ*≔*g*′(*κ*)∈[0,1)*or*

*Then there exists a unique positive wavefront* (*up to translations*) *of equation* (*1.1*) *for each sufficiently large speed c*.

We only need to verify assumptions (H) and (G) of theorem 1.1. Since *g* is *C*^{3}-smooth, it is immediately obvious that (ii) and (iii) imply (G). Next, condition (iv) ensures that the characteristic equation *λ*+1=*g*′(0)exp(−*λh*) has no roots on the imaginary axis; therefore, the trivial steady state is hyperbolic.

For the remainder of the proof, we assume that (i)–(iii) hold. Consequently, if *g*′(*κ*)∈[0,1), then the positive equilibrium is exponentially stable (e.g. see corollary 3.2 in Ivanov *et al*. (2002)) and globally attracting (e.g. see proposition 3.2 in Röst & Wu (2007)). The second line of condition (v) also ensures the exponential stability of *κ* (see theorem 2.9 in Ivanov *et al*. (2002)) and the global attractivity of *κ* (see corollary 2.3 in Liz *et al*. (2005)). Therefore, (i)–(v) imply (H). ▪

Below, we apply theorem 1.1 and corollary 1.2 to two time-delayed reaction–diffusion population models. First, we consider the diffusive Nicholson blowflies equation(1.4)This equation was introduced by So & Yang (1998) and generalizes the famous Nicholson blowflies equationwhich has been intensively studied for the last decade. Equation (1.4) takes into account the spatial distribution of the species, and there is growing interest in understanding the factors that influence the spatial spread of the growing population in that model. Relevant biological discussion can be found in Gourley *et al*. (2004), where various modifications of (1.4) were proposed and studied.

After a linear rescaling of both variables *u* and *t*, we can assume that *δ*=*b*=1. Equation (1.4) can therefore be written in the following normalized form:(1.5)The case of interest is *p*>1, where equation (1.5) has a unique positive steady state *κ*=ln *p*. It is immediate to check that the birth function *g*(*s*)=*ps*e^{−s}, with *s*≥0, satisfies conditions (i)–(iii) of the previous corollary. In this way, the conclusion of corollary 1.2 holds if *Γ*_{0}=*p* and *Γ*=1−ln *p* satisfy conditions (iv) and (v), respectively. It is worth mentioning that (v) trivially holds if *Γ*∈[−1,1) (that is, when 0<ln *p*≤2).

As a second application, let us consider the birth function *g*(*u*)=*pu*/(1+*u*^{n}). This function was proposed in 1977 by Mackey and Glass to model haematopoiesis (blood cell production). The Mackey–Glass equation with non-monotone nonlinearity can be written in the following normalized form:(1.6)The corresponding reaction–diffusion equation with delay is(1.7)Taking *p*>1 in equation (1.7), we find that conditions (ii), (iii) and (G) are satisfied with *κ*=(*p*−1)^{1/n}. Furthermore, if *n*≥2 then the Schwarz derivative of *g*(*u*)=*pu*/(1+*u*^{n}) is negative (see lemma 3 in Gopalsamy *et al*. (1998)). Consequently, the conclusion of corollary 1.2 holds if *n*≥2 and if both *Γ*_{0}=*p* and *Γ*=1−*n*+*n*/*p* satisfy conditions (iv) and (v), respectively. Now, suppose that *n*∈(1,2]. Then, corollary 3.2 in Ivanov *et al*. (2002) (theorem 2 in Gopalsamy *et al*. (1998)) guarantees that the positive steady state of equation (1.6) is exponentially stable (globally attractive). Therefore, if *n*∈(1,2] and , then theorem 1.1 assures the existence of a unique positive wavefront (modulo translations) of equation (1.7) for each sufficiently large speed *c*.

The structure of this paper is as follows, §2 contains preliminary facts and explains some notations. In §3, following the approach of Faria *et al*. (2006), we realize the Lyapunov–Schmidt reduction in a scale of Banach spaces and §4 contains the core lemma of the paper. As an application of this lemma, we obtain an alternative proof of the existence of positive wavefronts, see theorem 4.2. Finally, in §5 we show that there exists exactly one wavefront for each fixed sufficiently fast speed.

## 2. Preliminaries

This section contains several auxiliary results that will be required later.

*Assume* (*G*). *If y*≢0 *is a non-negative solution of equation* (*1.3*), *then*

See Liz *et al*. (2002), theorem 3.6(a). ▪

*Assume* (*G*). *Consider wavefront* *to equation* (*1.1*). *Then there exists a unique τ, such that ϕ*(*τ*)=*A*, *ϕ*′(*s*)>0 *for all s*≤*τ*.

See Trofimchuk *et al*. (2007), proposition 2.1. ▪

*Suppose that p*>1 *and h*>0. *Then the characteristic equation*(2.1)*has only one real root* 0<*λ*<*p*−1. *Moreover, all roots λ*, *λ*_{j}, *j*=2, 3, …, *of* (*2.1*) *are simple and we can enumerate them in such a way that λ*>Re *λ*_{2}=Re *λ*_{3}≥⋯.

See Faria & Trofimchuk (2006), lemma 7. ▪

Everywhere in the following, *λ*_{j} stands for a root of (2.1). Note that we write *λ* instead of *λ*_{1}.

*Assume (H)* and (*G*) *and let λ be as in* *lemma 2.3*. *Then,* (*1.3*) *has a unique positive heteroclinic solution ψ* (*up to translations*). *Moreover,* *for each δ*>0 *and some t*_{0}∈.

See Faria & Trofimchuk (2006), lemma 8. ▪

*Let* {*λ*_{α}(*ϵ*), *α*∈}, *where* ∪{∞}⊂, *denote the* (*countable*) *set of roots to the equation*(2.2)*If p*>1*,* *h*>0, *,* *then* (*2.2*) *has exactly two real roots λ*_{1}(*ϵ*) *and* *λ*_{∞}(*ϵ*)*,* *such that**Moreover,* (i) *there exists an interval* *such that, for every ϵ*∈, *all roots λ*_{α}(*ϵ*), *α*∈ *of* (*2.2*) *are simple and the functions λ*_{α}: → *are continuous,* (ii) *we can enumerate λ*_{j}(*ϵ*), *j*∈, *in such a way that there exists* *for each j*∈, *where λ*_{j}∈ *are the roots of* (*2.1*), *with λ*_{1}=*λ,* *and* (iii) *for all sufficiently small ϵ, every vertical strip ξ*≤Re *z*≤2(*p*−1) *contains only a finite set of m*(*ξ*) *roots* (*if ξ*∉{Re *λ*_{j}, *j*∈}, *then m*(*ξ*) *does not depend on ϵ*) *λ*_{1}(*ϵ*), …, *λ*_{m(ξ)}(*ϵ*) *to* (*2.2*), *while the half-plane* Re *z*>2(*p*−1) *contains only the root λ*_{∞}(*ϵ*).

See Faria & Trofimchuk (2006), lemma 13. ▪

Assume (H) and (G) and let *ψ* be the positive heteroclinic solution from lemma 2.4. For a fixed *μ*≥0, we set and , , . Consider the Banach spaceequipped with the norm |*y*|_{μ}. We will need the operators , where is the Nemitski operator, , andSince and , we obtainObserve that are well defined, e.g. and, for *t*≤*h*,

*Operator families* *are continuous in the operator norm*. *In particular*, *as ϵ*→0.

See Faria & Trofimchuk (2006), lemma 12. ▪

*If (H) holds and* , *then* *is a surjective Fredholm operator and* dim Ker(*I*−)=*d*(*μ*).

First, we establish that (*I*−) is an epimorphism. Take some *f*∈*C*_{μ}() and consider the following integral equation:If we set *z*(*t*)=*y*(*t*)−*f*(*t*), this equation is transformed intoHence, in order to establish the surjectivity of *I*−, it suffices to prove the existence of a *C*_{μ}()-solution of the equation(2.3)First, note that all the solutions of (2.3) are bounded on the positive semi-axis _{+} due to the boundedness of *q*(*t*)*f*(*t*−*h*) and the exponential stability of the homogeneous *ω*-limit equation *z*′(*t*)=−*z*(*t*)+*g*′(*κ*)*z*(*t*−*h*). Here, we use the persistence of exponential stability under small bounded perturbations (e.g. see §5.2 in Chicone & Latushkin (1999)) and the fact that *q*(+∞)=*g*′(*κ*). Furthermore, since every solution *z* of (2.3) satisfies with *ϵ*(+∞)=0, we get . Next, by effecting the change of variables *z*(*t*)=exp(*μt*)*v*(*t*) to equation (2.3), we get a linear inhomogeneous equation of the form(2.4)where and at *t*=−∞. Since the *α*-limit equation to the homogeneous part of (2.4) is hyperbolic, due to the above-mentioned persistence of the property of exponential dichotomy, we again conclude that equation (2.4) also has an exponential dichotomy on _{−}. Thus, (2.4) has a solution , which is bounded on _{−} (while ), so that is a *C*_{μ}()-solution of equation (2.3).

Next, we prove that dim Ker. It is clear that *ϕ*_{j}∈Ker(I−) if and only if *ϕ*_{j} is a *C*_{μ}()-solution of the equation(2.5)We already have seen that every solution of (2.5) satisfies *y*(+∞)=0, thus we only have to show that there exist solutions *ϕ*_{j} with . In fact, we will prove that, for each Re *λ*_{j}>*μ* and , there is with Set *q*(*t*)=*p*+*ϵ*(*t*), then *v*_{j}(*t*) can be chosen as a bounded solution of the equation(2.6)Since at −∞, we get the following *α*-limit form of (2.6)This autonomous equation is exponentially stable since its characteristic equationhas roots *z*_{j}=*λ*_{j}−*λ*−*δ* with Re *z*_{j}=Re *λ*_{j}−*λ*−*δ*<0. Thus, (2.6) has a unique solution *v*_{j} bounded in _{−}. It is clear that *d*(*μ*) solutions {*ϕ*_{j}} are linearly independent; we claim that, in fact, system {*ϕ*_{j}} generates Ker(*I*−). By way of contradiction, suppose that *φ*∈Ker(I−)−〈*ϕ*_{j}〉.

As *φ* solves the equationwe get (e.g. Mallet-Paret (1999), p. 28)where *z*(*t*) is the eigensolution corresponding to the eigenvalues *ζ* with *μ*≤ Re *ζ*<*λ*+*μ*. In this way,(2.7)Now takeSince , we can writeThus, *r*(*t*)≔*φ*(*t*)−*w*(*t*) satisfies *r*(*t*)=*O*(exp(*λ*−*δ*)*t*), *t*→−∞, and solves(2.8)Applying proposition 7.1 from Mallet-Paret (1999), we conclude thatwhere *z*(*t*) is the eigensolution corresponding to the eigenvalues *ζ*, such that *λ*−*δ*≤Re *ζ*<2*λ*−*δ* and, in consequence, *z*(*t*)=*C*_{1}e^{λt}, for some *C*_{1}. Hence,for small *δ*>0. The latter formula improves (2.7), and if we takethen . Since *r*_{1}(*t*) satisfieswe can proceed as before to get *t*→−∞, where *z*_{1}(*t*) is the eigensolution corresponding to the eigenvalues *ζ*, such that *λ*+*δ*≤*ζ*<2*λ*+*δ*. Thus, *z*_{1}(*t*)=0 and *t*→−∞. Iterating this procedure (and subtracting *δ*/2^{k} from the exponent 2*λ*+*δ* on the step *k*), we can conclude that *r*_{1}(*t*)=*O*(exp(*kλt*)), *t*→−∞, *k*≥2. This means that *r* is a small solution of (2.5). However, equation (2.5) cannot have solutions with superexponential decay at −∞ (e.g see Faria & Trofimchuk (2006), p. 9) and thus *r*(*t*)=0. This implies that *φ*∈〈*ϕ*_{j}〉, a contradiction. ▪

Throughout the rest of the paper, we will suppose that the *C*^{1}-smooth function *g* is defined and bounded on the whole real axis . This assumption does not restrict the generality of our framework, since it suffices to take any smooth and bounded extension on _{−} of the nonlinearity *g* described in (G). Note that, since there exists a finite *g*′(0), we have *g*(*s*)=*sγ*(*s*) for a bounded *γ*∈*C*(). Set . As it can be easily checked, , so that actually is well defined. Furthermore, we have lemma 2.8.

*Assume that g*∈*C*^{1}(). *Then* *is Fréchet continuously differentiable on C*_{μ}() *with differential* .

We have that . By the Taylor formula, , *θ*∈[*v*,*v*_{0}]. Fix some *y*_{0}∈*C*_{μ}(). Since functions in *C*_{μ}() are bounded and *g*′ is uniformly continuous on bounded sets of , for any given *δ*>0, there is *σ*>0, such that for we have that and . ▪

## 3. Lyapunov–Schmidt reduction

Being a bounded solution of equation (1.2), each travelling wave *ϕ* should satisfy(3.1)For *C*_{μ}()-solutions, this equation takes the form .

*Assume* (*H*) *and* (*G*). *Let ψ be the positive heteroclinic from* *lemma 2.4*. *Then, for every* *, there are open balls* _{μ}=(−*ϵ*_{μ}, *ϵ*_{μ})*,* *, and the continuous family of heteroclinics* *of equation* (*1.2*) *such that ψ*_{0,0}=*ψ*. *For each* *, the subset* *is a C*^{1}-*manifold of dimension d*(*μ*). *Moreover, there exists a C*_{μ}()-*neighbourhood* *of ψ and ϵ*_{1}>0*, such that every solution* *of equation* (*1.2*) *satisfies* *for some* . *Finally, given a closed subinterval* *, we can choose open sets* *,* *to be constant for μ*∈.

Set and then define by We have that *F*(0, 0)=0. Furthermore, lemmas 2.6 and 2.8 imply that and *F*_{ϕ}(*ϵ*, *ϕ*) is continuous in a neighbourhood of (0, 0). SetThen, *r*_{ϕ}(0, 0)=*F*_{ϕ}(0, 0)−*L*=0. By lemma 2.7, we have that dim *V*<∞ and that *L* is surjective. Thus, *V* has a topological complement *W* in *C*_{μ}() so that *C*_{μ}()=*V*⊕*W* and any *ϕ*∈*C*_{μ}() can be written in the form *ϕ*=*v*+*w*, *v*∈*V* and *w*∈*W*. Recalling that *Lv*=0, we get This suggests the following definition:where is the restriction of *L* to *W*. It is clear that and is continuous in a neighbourhood of (0,0,0). Since is bijective, we have that is continuous from *C*_{μ}() to *W*. As a consequence, we can apply the implicit function theorem (e.g. see theorem 2.3(i) in Ambrosetti & Prodi (1993)) toIn this way, we find neighbourhoods of 0, _{μ}⊂*R*_{μ}, _{μ}⊂*V* and _{μ}⊂*W* and a continuous map , such that for all . Moreover, without restricting the generality, we can suppose that *Φ*(*ϵ*, *v*, *w*)=0 with implies *w*=*γ*(*ϵ*,*v*) (e.g. see theorem 2.3(ii) in Ambrosetti & Prodi (1993)).

Hence, the continuous family contains all solutions of equation (1.2) from the small neighbourhoods of *ψ*, with *ψ*_{0,0}=*ψ*. Since *γ*_{v}(0, 0)=0 and *γ*_{v}(*ϵ*, *v*) is continuous for each fixed *ϵ*∈_{μ}, we conclude that is a *C*^{1}-smooth manifold of dimension *d*(*μ*). Note that (3.1) implies that . Thus, , so that {*ψ*_{ϵ,v}} are heteroclinic solutions of (1.2).

Finally, the last conclusion of the theorem follows from the simple observations that (i) the sets are non-increasing in *μ* and (ii) the function *d*(*t*) is piece-wise constant, with discontinuities at . ▪

## 4. Asymptotic formulae

Throughout this section, we denote by *β*, *γ*, *η*, *b*, *C*, *C*_{j}, *C*_{*}, … some positive constants that are independent of the parameters , where and *Ω*⊂^{q}. We also assume that *h*>0 and *p*>1.

*Let continuous* *satisfy*(4.1)*Suppose further that* , *and that* (*ϵ*, *v*)∈*Λ*_{0}×*Ω*. *Then, given σ*∈(0, *b*), *it holds that**where, with some continuous and bounded* ,*is a finite sum of eigensolutions of* (*4.1*) *associated with the roots* *of* (*2.2*) *and* .

Applying the Laplace transform to equation (4.1), we obtainwhere andSince is bounded, is holomorphic in the open half-plane . Similarly, is holomorphic in . Since *r*_{ϵ,v} is an entire function, the functionis meromorphic in Re *z*>−*b*, with only finitely many poles there.

*Step I*. We claim that there are such that if Re *z*=−*b*+*σ*′, (*ϵ*, *v*)∈*Λ*_{1}×*Ω*. Indeed, take *σ*′∈(0, *σ*) such that the line Re *z*=−*b*+*σ*′ does not contain any eigenvalue and 1−*b*+*σ*′≠0. We haveand

As a bounded solution of (4.1), *y*_{ϵ,v} should satisfy, for all *t*∈,(4.2)where are the roots of *ϵ*^{2}*z*^{2}+*z*−1=0 and . Differentiating (4.2), we obtain(4.3)so thatFix *k*>−*b*+*σ*′ and consider the vertical strip , thenso that .

Set , then and(4.4)

Now, set *y*_{0}=*ηβ* for some *η*>2 satisfying and *ηβ*>*b*−*σ*′. For all *z* such that Re *z*=−*b*+*σ*′ and , we haveThus, , so that(4.5)for all , Re *z*=−*b*+*σ*′ and *ϵ*∈*Λ*_{1}.

Finally, for all , we haveCombining this inequality with (4.4) and (4.5), we prove the main assertion of step I.

*Step II*. Taking *k*>0, in virtue of (4.4) we can use the inversion formula(4.6)By lemma 2.5, *H*_{ϵ,v}(*z*) has only finitely many poles in the strip −*b*<Re *z*≤−*γ*. Also, *H*_{ϵ,v}(*z*)→0 uniformly in the strip −*b*+*σ*′≤Re *z*≤*k*, as and . Thus, we may shift the path of integration in (4.6) to the left, to the line Re *z*=−*b*+*σ*′ and obtain whereBy lemma 2.5, the roots of equation *Χ*(*z*,*ϵ*)=0 are simple for all small *ϵ*. Hence,It is easy to check that *B*_{j}(*ϵ*, *v*) is continuous on its domain of definition (observe here that the continuity of follows from (4.3)). Take *j* such that , then . In addition, if *ϵ*→0 thenHence,if *ϵ*∈*Λ*_{2}, for some small *ϵ*_{2}>0 and *v*∈*Ω*.

*Step III*. Consider and . We haveBy the Plancherel theorem,Hence, is integrable on [0,+∞) and by the Cauchy–Schwarz inequality,

*Step IV*. We claim that there exist real numbers *C*_{9}>0 and *ϵ*_{3}>0, such that for all . In order to prove this, it suffices to show that *v*_{ϵ,v} is uniformly bounded for small *ϵ*∈*Λ*_{3}. Sincewe find that satisfieswhere and is defined byThe variation of constants formula yields(4.7)A direct integration of (4.7) givesAfter changing the order of integration in the iterated integral, we getAdditionally, recalling step II, we find thatAs a consequence, for all small *ϵ* and *v*∈*Ω*, we haveFinally, since , lemma 4.1 is proved. ▪

*In* *theorem 3.1*, *take μ=λ−δ,* *with small δ*>0. *Assume that ψ is the positive heteroclinic of* (*1.3*) *normalized by* *,* *t*→−∞. *Then, we can choose a neighbourhood* *of ψ and a neighbourhood* *of* 0∈^{2} *in such a way that* *is positive and unique in* (*up to translations in t*) *for every fixed ϵ*. *Moreover,* *at t*→−∞ *for some* .

First, we take _{μ}, as in theorem 3.1. It follows from lemma 2.5 and theorem 3.1 that _{μ}⊂ and that we can choose positive *δ* and _{μ} such that for all *ϵ*∈_{μ}. If we set *y*_{ϵ,v}(*t*)=ψ_{ϵ,v}(−*t*), then *y*_{ϵ,v} satisfies (4.1), where

Lemma 4.1 ensures that there are and such thatHere, *B*(0, 0)=1 is continuous and , *t*≥0 for some *C*_{*}>0.

Hence, there are and *T*>0 (independent of *ϵ*,*v*), such that , *t*>*T*, for all . On the other side, uniformly on . Consequently, since *ψ* is bounded from below by a positive constant on [−*T*,∞), we conclude that *y*_{ϵ,v} is positive on , if (*ϵ*, *v*) belongs to a sufficiently small neighbourhood of the origin. Without the loss of the generality, we can assume additionally that for all .

Next, for every fixed , the subset is homeomorphic to . On the other hand, for every *n*>0, the collection of positive heteroclinics is a continuous one-dimensional manifold in *C*_{μ}(). Since we obtain that . Consequently, is unique in (up to shifts in *t*) for every fixed small *ϵ*. ▪

*Set* , *where* _{0} *and* _{0} *are as in* *theorem 3.1*. *Then there exists a neighbourhood* *of* 0 *and C*>0 *such that, for all* *,* *we have*(4.8)*where* *and* *is continuous*.

Let be such that , for all *ϵ*∈′. The last assertion of theorem 3.1 implies that, for some *γ*>0 and *C*_{1}>0,(4.9)If we set , then *y*_{ϵ,v} satisfies (4.1), where

Set *Γ*=sup{*γ*>0 such that (4.9) holds for all (*ϵ*,*v*)∈∩(′×_{0})}. Applying lemma 4.1, we getwhere are continuous and *t*≥0, , for some *C*_{3}>0 and open ″⊂′. Since *Γ*>0 is finite and *y*_{ϵ,v}(*t*)>0, we obtainso that *Γ*≥*λ*, see lemma 2.5. Next, due to lemma 2.4, it holds that *B*(0, 0)>0. Hence, *Γ*=*λ*. ▪

*Given δ*∈(0, *λ*) *and* (*ϵ*_{j}, *v*_{j})∈^{*}, *j*=0, 1, …, *the convergence*

On the contrary, suppose that there is a sequence and *η*>0 such thatIt follows from (4.8) that there exist *C*>0 and *T*<0 such thatThus,

Next, since uniformly on , we can find *j*_{*} such thatBut all this means that for all *j*≥*j*_{*}, a contradiction. ▪

## 5. Proof of theorem 1.1

Everywhere below, all positive wavefronts *ϕ* will be normalized by the conditions *ϕ*(0)=*ζ*_{1}/2 and *ϕ*′(*s*)>0, *s*<0, with *ζ*_{1}≤*A* defined in (G). The possibility of such a normalization was established in lemma 2.2. Let *ψ*, *ψ*(0)=*ζ*_{1}/2, *ψ*(*s*)<*ζ*_{1}/2, *s*<0, be the positive heteroclinic of (1.3) given in lemma 2.4. By theorem 4.2, there exists a neighbourhood of (0, *ψ*), such that, for every fixed *ϵ*∈(−*ϵ*_{0}, *ϵ*_{0}), there is a unique normalized positive wavefront . We claim that, if *ϵ* is sufficiently small, then this *ψ*_{ϵ} will be the unique normalized positive wavefront of equation (1.2). By way of contradiction, let us suppose that we can find a sequence *ϵ*_{j}→0 and normalized positive wavefronts .

*Assume* (*H*) *and* (*G*). *Then* *uniformly on* .

First, we prove the uniform convergence on compact subsets of . Since *g* is a bounded function, we obtain from (3.1) thatHence, by the Ascoli-Arzelà theorem combined with the diagonal method, is precompact in *C*(, ). Thus, every has a subsequence converging in *C*(, ) to some continuous positive bounded function *φ*(*s*), such that *φ*′(*s*)≥0, *s*≤0 and *φ*(0)=*ζ*_{1}/2. Making use of Lebesgue's dominated convergence theorem, we deduce, from equation (3.1), thatTherefore, *φ* is a positive bounded solution of equation (1.3) and, since the equilibrium *κ* of equation (1.3) is globally attractive, it holds that *φ*(+∞)=*κ*. On the other hand, since , we have that *φ*(−∞)=0. Hence, due to lemma 2.4, we obtain that *φ*(*t*)=*ψ*(*t*), *t*∈. Next, if uniformly on , then there exist a subsequence (for short, we will again write instead of ), a sequence {*S*_{j}} and positive numbers *T*, *δ*<*κ*/6, such thatSince converges uniformly on [−2*T*,2*T*] to *ψ* and are monotones increasing on (−∞,0], we can suppose that for all *t*∈(−∞,2*T*] and *n*≥*n*_{0}. In this way, *S*_{j}→+∞ and we can suppose thatConsider the sequence of heteroclinics to equation (1.2). We have that and when *t*∈(*T*−*S*_{j},0). Arguing as above, we find that {*y*_{j}} contains a subsequence converging, on compact subsets of , to some solution *y*_{*}(*t*) of (1.3) satisfying and for all *t*<0. Lemma 2.1 implies that . Since *y*_{*}(0)≠*κ*, we have established the existence of a non-constant positive bounded and separated from 0 solution to (1.3). This contradicts with the global attractivity of *κ*. ▪

*in* .

Since in *C*_{0}(), we have that for some *v*_{j}∈_{0}. Now we can apply corollary 4.4 to find that in . ▪

Lastly, theorem 4.2 and corollary 5.2 imply that , a contradiction that completes the proof of theorem 1.1.

## Acknowledgments

The authors thank Teresa Faria for useful discussions. They also express their gratitude to the anonymous referees whose valuable comments and suggestions helped to improve the original manuscript. This research was supported by FONDECYT (Chile) project 1071053. S.T. was partially supported by CONICYT (Chile) through PBCT programme ACT-05 and by the University of Talca programme ‘Reticulados y Ecuaciones’.

## Footnotes

- Received January 12, 2008.
- Accepted April 22, 2008.

- © 2008 The Royal Society