JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 705197 10.1155/2012/705197 705197 Research Article Existence of Traveling Fronts in a Food-Limited Population Model with Spatiotemporal Delay Zhao Hai-Qin 1 Liu San-Yang 2 Li Wan-Tong 1 College of Mathematics and Information Science Xianyang Normal University Xianyang Shaanxi 712000 China xysfxy.cn 2 Department of Mathematics Xidian University Xi’an Shaanxi 710071 China xidian.edu.cn 2012 19 12 2012 2012 25 08 2012 23 10 2012 2012 Copyright © 2012 Hai-Qin Zhao and San-Yang Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is concerned with the traveling fronts of a diffusive food-limited population model with spatiotemporal delay. Sufficient conditions are established for the existence of traveling wave fronts by choosing different kinds of delay kernels. The approach used here is the upper-lower solution method and monotone iteration technique. Our work extends and/or covers some previous results.

1. Introduction

This paper is concerned with the traveling fronts for the following food-limited model: (1.1)u(x,t)t=2u(x,t)x2+u(x,t)1-au(x,t)-b(g*u)(x,t)1+adu(x,t)+bd(g*u)(x,t),x,t0, where a, b, and d are nonnegative constants, a+b>0, and the kernel g(x,t) is any integrable nonnegative function satisfying g(-x,t)=g(x,t), (1.2)0+-+g(y,s)dyds=1,(g*u)(x,t)=-t-+g(x-y,t-s)u(y,s)dyds, which was first proposed and analyzed by Gourley and So  on a finite domain Ω.

In the case a=0, b=1,  d=β, (1.1) becomes (1.3)u(x,t)t=2u(x,t)x2+u(x,t)1-(g*u)(x,t)1+β(g*u)(x,t).

Recently, many researchers studied the existence of traveling fronts of (1.3) with some specific g(x,t). For the case (1.4)g(x,t)=δ(t-τ)δ(x), where δ(·) is the Dirac delta function, Gourley  showed that, for any c>2, there exists τ*(c)>0 such that, for any τ<τ*(c), (1.3) has a traveling front connecting the equilibria 0 and 1, by using the approach developed by Wu and Zou . For the case (1.5)g(x,t)=1τe-t/τ14πte-x2/4t,

Gourley and Chaplain  proved the existence of traveling fronts for any c2 and sufficient small τ>0, by employing linear chain techniques to recast the traveling wave equations as a finite-dimensional system of ODEs and using Fenichel's geometric singular perturbation theory  and the Fredholm alternative. For the case (1.6)g(x,t)=δ(t-τ)14πte-x2/4t,

Gourley and Chaplain , by using the method of Canosa , obtained some information on the monotonicity of traveling fronts for sufficiently large c. Furthermore, for these cases (1.7)g(x,t)=tτ2e-t/τ14πte-x2/4t,g(x,t)=δ(t-τ)14πte-x2/4t,g(x,t)=δ(t)12ρe-|x|/ρ,ρ>0,g(x,t)=1τ2e-t/τδ(x),τ>0,

Wang and Li  showed that, for any c>2, there exists τ*(c)>0 (or ρ*(c)>0) such that for any τ<τ*(c) (or ρ<ρ*(c)), (1.3) has a traveling front connecting the equilibria 0 and 1.

In this paper, based on the monotone iteration technique as well as the upper and lower solution method developed by Wang et al. , we will establish the existence of traveling fronts of (1.1) with the kernel functions (1.4)–(1.7). More precisely, we shall show that for any c>2, there exists τ*(c)>0 (or ρ*(c)>0) such that, for any τ<τ*(c) (or ρ<ρ*(c)), (1.1) has a traveling front connecting the equilibria 0 and K=1/(a+b) (see Theorems 2.5 and 2.9 and Remark 2.10), which includes, improves, and/or complements a number of existing results in [24, 7, 9, 10].

The rest of the paper is organized as follows. In Section 2, we establish the existence of traveling wave fronts of (1.1) with the kernel functions (1.4)–(1.7). For the sake of convenience, we present in the Appendix some results developed by Wang et al. .

2. Existence of Traveling Fronts

In this section, we will use Theorem A.2 to establish the existence of traveling fronts of (1.1) by choosing different kernel function g, such as (1.4)–(1.7). It is easy to see that (1.1) has two uniform steady states K0=0 and K=1/(a+b).

Let u(x,t)=ϕ(ξ), ξ=x+ct. Then a traveling front ϕ(ξ) of (1.1) satisfies the boundary conditions ϕ(-)=K0 and   ϕ(+)=K, and the following equation: (2.1)ϕ(ξ)-cϕ(ξ)+ϕ(ξ)1-aϕ(ξ)-b(g*ϕ)(ξ)1+adϕ(ξ)+bd(g*ϕ)(ξ)=0,ξ.

For c>2, let Δc(μ)=μ2-cμ+1 and λ=(c-c2-4)/2. Then Δc(λ)=0. Let 0<ε<λ, α>0, M>1 and γ>λ such that (2.2)λ+ε<γ,λ+ε<c+c2-42,α<λ2(γ+λ),12MαM-1. Clearly, Δc(λ+ε)<0. Define ϕ+(ξ)=K/(1+αe-λξ)   and ϕ-(ξ)=max{Keλξ(1-Meεξ),0}. Then we have the following observations.

Lemma 2.1.

(i)  ϕ+(ξ) is increasing in ξ and satisfies ϕ+(-)=K0 and ϕ+(+)=K;

(ii)  ϕ+(ξ)ϕ-(ξ) for all ξ;

(iii)  eγξ[ϕ+(ξ)-ϕ-(ξ)] is increasing and e-γξ[ϕ+(ξ)-ϕ-(ξ)] is decreasing in ξ;

(iv)  eγξ[ϕ+(ξ+η)-ϕ+(ξ)] is increasing and e-γξ[ϕ+(ξ+η)-ϕ+(ξ)] is decreasing in ξ for every η>0.

Clearly, Lemma 2.1 implies that, for γ>λ+ε, ϕ+(ξ)Γ*, ϕ+(ξ)Γ** and supξϕ-(ξ)>0. Now, we show that ϕ+(ξ) and ϕ-(ξ) are lower and upper solutions of (2.1) by choosing different kernel functions g, respectively.

For the sake of convenience, throughout this section, we let (2.3)f(ϕ(ξ),(g*ϕ)(ξ))=ϕ(ξ)1-aϕ(ξ)-b(g*ϕ)(ξ)1+adϕ(ξ)+bd(g*ϕ)(ξ),ξ.

2.1. The Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M84"><mml:mi>g</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mi>x</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mi>t</mml:mi><mml:mo mathvariant="bold">)</mml:mo><mml:mo mathvariant="bold">=</mml:mo><mml:mi>δ</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mi>t</mml:mi><mml:mo mathvariant="bold">)</mml:mo><mml:mo mathvariant="bold">(</mml:mo><mml:mn>1</mml:mn><mml:mo mathvariant="bold">/</mml:mo><mml:mn>2</mml:mn><mml:mi>ρ</mml:mi><mml:mo mathvariant="bold">)</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo mathvariant="bold">-</mml:mo><mml:mo mathvariant="bold">|</mml:mo><mml:mi>x</mml:mi><mml:mo mathvariant="bold">|</mml:mo><mml:mo mathvariant="bold">/</mml:mo><mml:mi>ρ</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M85"><mml:mi>ρ</mml:mi><mml:mo mathvariant="bold">></mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>

Clearly, g(x,t)=δ(t)(1/2ρ)e-|x|/ρ satisfies (H0) and in this case (2.4)(g*ϕ)(ξ)=-+12ρe-|y|/ρϕ(ξ-y)dy.

Lemma 2.2.

For sufficient small ρ>0, f(ϕ(ξ),(g*ϕ)(ξ)) satisfies (H1**).

Proof.

Let A=(a2+2abd+a2d+ab)/(a+b)2 and B=b(d+1)/(a+b)+abd/(a+b)2. Fix γ>A+2B. Let ϕ1,ϕ2C(,) with 0ϕ1(ξ)ϕ2(ξ)K so that eγξ[ϕ2(ξ)-ϕ1(ξ)] is increasing and e-γξ[ϕ2(ξ)-ϕ1(ξ)] is decreasing in ξ. It is easy to see that for any η, eγξ[ϕ2(ξ+η)-ϕ1(ξ+η)] is increasing and e-γξ[ϕ2(ξ+η)-ϕ1(ξ+η)] is decreasing in ξ. For sufficiently small ρ>0 satisfying 1-ργ>1/2, there is (2.5)(g*ϕ2)(ξ)-(g*ϕ1)(ξ)=-+12ρe-|y|/ρ[ϕ2(ξ-y)-ϕ1(ξ-y)]dy=0+12ρe-y/ρ[ϕ2(ξ-y)-ϕ1(ξ-y)]dy+0+12ρe-y/ρ[ϕ2(ξ+y)-ϕ1(ξ+y)]dy=0+12ρe-y/ρeγy{e-γy[ϕ2(ξ-y)-ϕ1(ξ-y)]+e-γy[ϕ2(ξ+y)-ϕ1(ξ+y)]{12}}dy2[ϕ2(ξ)-ϕ1(ξ)]0+12ρe-y/ρeγydy=11-ργ[ϕ2(ξ)-ϕ1(ξ)]2[ϕ2(ξ)-ϕ1(ξ)]. Hence, (2.6)ϕ2(1-aϕ2-bg*ϕ2)(1+adϕ1+bdg*ϕ1)-ϕ1(1-aϕ1-bg*ϕ1)(1+adϕ2+bdg*ϕ2)=(ϕ2-ϕ1)[1+bdg*ϕ1-(a+abdg*ϕ1)(ϕ2+ϕ1)-a2dϕ1ϕ2-bg*ϕ2-b2dg*ϕ1g*ϕ2]+(abdϕ12-bdϕ1-bϕ1-abdϕ1ϕ2)(g*ϕ2-g*ϕ1)-A(ϕ2-ϕ1)-B(g*ϕ2-g*ϕ1)-(A+2B)(ϕ2-ϕ1)>-γ(ϕ2-ϕ1). Therefore, (2.7)f(ϕ2(ξ),(g*ϕ2)(ξ))-f(ϕ1(ξ),(g*ϕ1)(ξ))=ϕ2(ξ)1-aϕ2(ξ)-b(g*ϕ2)(ξ)1+adϕ2(ξ)+bd(g*ϕ2)(ξ)-ϕ1(ξ)1-aϕ1(ξ)-b(g*ϕ1)(ξ)1+adϕ1(ξ)+bd(g*ϕ1)(ξ)-γ[ϕ2(ξ)-ϕ1(ξ)][1+adϕ2(ξ)+bd(g*ϕ2)(ξ)][1+adϕ1(ξ)+bd(g*ϕ1)(ξ)]>-γ[ϕ2(ξ)-ϕ1(ξ)]. This completes the proof.

Lemma 2.3.

Assume that 1-λρ>0. Then for sufficiently large M>1, ϕ-(ξ) is a lower solution of (2.1).

Proof.

For ξξ0=(1/ε)ln(1/M), ϕ-(ξ)=0, then (2.8)ϕ-(ξ)-cϕ-(ξ)+ϕ-(ξ)1-aϕ-(ξ)-b(g*ϕ-)(ξ)1+adϕ-(ξ)+bd(g*ϕ-)(ξ)=0. Let (2.9)M-(d+1)aKΔc(λ+ε)-(d+1)bK(1-ρλ)(1+ρλ)Δc(λ+ε). For ξ<ξ0<0, ϕ-(ξ)=Keλξ(1-Meεξ), since (2.10)(g*ϕ-)(ξ)=-+12ρe-|y|/ρϕ-(ξ-y)dy=ξ-ξ0+12ρe-|y|/ρeλ(ξ-y)K(1-Meε(ξ-y))dyK-+12ρe-|y|/ρeλ(ξ-y)dy=Keλξ(1-ρλ)(1+ρλ), and h(z)=(1-z)/(1+dz)1-(d+1)z for all    z>0, then (2.11)ϕ-(ξ)-cϕ-(ξ)+ϕ-(ξ)1-aϕ-(ξ)-b(g*ϕ-)(ξ)1+adϕ-(ξ)+bd(g*ϕ-)(ξ)ϕ-(ξ)-cϕ-(ξ)+ϕ-(ξ){1-(d+1)[aϕ-(ξ)+b(g*ϕ-)(ξ)]}K[λ2-M(λ+ε)2eεξ]eλξ-Kc[λ-M(λ+ε)eεξ]eλξ+Keλξ(1-Meεξ)-(d+1)aK2e2λξ(1-Meεξ)2-(d+1)bK2e2λξ(1-ρλ)(1+ρλ)Ke(λ+ε)ξ[-MΔc(λ+ε)-a(d+1)K-(d+1)bK(1-ρλ)(1+ρλ)]0. Thus, we showed that ϕ-(ξ) is a lower solution of (2.1). This completes the proof.

Lemma 2.4.

For sufficiently small ρ>0, ϕ+(ξ) is an upper solution of (2.1).

Proof.

Note that (2.12)ϕ+(ξ)=Kαλe-λξ(1+αe-λξ)2,ϕ+′′(ξ)=-Kαλ2e-λξ+Kα2λ2e-2λξ(1+αe-λξ)3. By an argument similar to [7, Lemma  3.5], for ρ>0 such that 1-2ρλ>0, we have (2.13)(g*ϕ+)(ξ)K1+αe-λξ-Kαλ2ρ21-λ2ρ2·e-λξ(1+αe-λξ)2. Then for sufficiently small ρ>0 with 2λ2-Kbλ2ρ2/(1-λ2ρ2)  >0, (2.14)ϕ+(ξ)-cϕ+(ξ)+ϕ+(ξ)1-aϕ+(ξ)-b(g*ϕ+)(ξ)1+adϕ+(ξ)+bd(g*ϕ+)(ξ)ϕ+(ξ)-cϕ+(ξ)+ϕ+(ξ)[1-aϕ+(ξ)-b(g*ϕ+)(ξ)]-Kαλ2e-λξ+Kα2λ2e-2λξ(1+αe-λξ)3-Kcαλe-λξ(1+αe-λξ)2+Kαe-λξ(1+αe-λξ)2+K2bαλ2ρ21-λ2ρ2·e-λξ(1+αe-λξ)3=Kα2(λ2-cλ+1)e-2λξ-Kα(λ2+cλ-1-Kbλ2ρ2/(1-λ2ρ2))e-λξ(1+αe-λξ)3=-Kα(2λ2-Kbλ2ρ2/(1-λ2ρ2))e-λξ(1+αe-λξ)3<0. This completes the proof.

Therefore, by Theorem A.2(ii), we have the following result.

Theorem 2.5.

For any c>2, there exists ρ*(c)>0 such that, for any ρ<ρ*(c), (1.1) has an increasing traveling wave front ϕ(ξ) that satisfies ϕ(-)=0, ϕ(+)=K and limξ-ϕ(ξ)e-λξ=1.

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It is easy to see that g(x,t)=(t/τ2)e-t/τδ(x) satisfies (H0) and in this case (2.15)(g*ϕ)(ξ)=0+sτ2e-s/τϕ(ξ-cs)ds.

The following two lemmas are similar to Lemmas 2.1 and 2.3, and their proofs are omitted.

Lemma 2.6.

For sufficient small τ>0, f(ϕ(ξ),(g*ϕ)(ξ)) satisfies (H1*).

Lemma 2.7.

For sufficiently large M1, ϕ-(ξ) is a lower solution of (2.1).

Lemma 2.8.

For sufficiently small τ>0, ϕ+(ξ) is an upper solution of (2.1).

Proof.

Note that, for τ>0 such that 1-2λcτ>0, (2.16)(g*ϕ+)(ξ)=0+sτ2e-s/τK1+αe-λ(ξ-cs)ds=0+1τe-s/τK1+αe-λ(ξ-cs)ds-αλce-λξ0+sτe-s/τKe(λc-1/τ)s[1+αe-λ(ξ-cs)]2dsK1+αe-λξ-Kαλcτe-λξ(1-λcτ)(1+αe-λξ)2-Kαλcτe-λξ(1-λcτ)2(1+αe-λξ)2=K1+αe-λξ-Kαλcτe-λξ(2-λcτ)(1-λcτ)2(1+αe-λξ)2. Then for sufficiently small τ>0 with 2λ2-bKλcτ(2-λcτ)/(1-λcτ)2>0, (2.17)ϕ+(ξ)-cϕ+(ξ)+ϕ+(ξ)1-aϕ+(ξ)-b(g*ϕ+)(ξ)1+adϕ+(ξ)+bd(g*ϕ+)(ξ)ϕ+(ξ)-cϕ+(ξ)+ϕ+(ξ)[1-aϕ+(ξ)-b(g*ϕ+)(ξ){12}]-Kαλ2e-λξ+Kα2λ2e-2λξ(1+αe-λξ)3-Kcαλe-λξ(1+αe-λξ)2+Kαe-λξ(1+αe-λξ)2+K2bαλcτe-λξ(2-λcτ)(1-λcτ)2(1+αe-λξ)3=Kα2(λ2-cλ+1)e-2λξ-Kα(λ2+cλ-1-bKλcτ(2-λcτ)/(1-λcτ)2)e-λξ(1+αe-λξ)3=-Kα(2λ2-bKλcτ(2-λcτ)/(1-λcτ)2)e-λξ(1+αe-λξ)3<0. This completes the proof.

Now, by Theorem A.2(i), we have the following result.

Theorem 2.9.

For any c>2, there exists τ*(c)>0 such that, for any τ<τ*(c), (1.1) has an increasing traveling wave front ϕ(ξ) that satisfies ϕ(-)=0, ϕ(+)=K and limξ-ϕ(ξ)e-λξ=1.

Remark 2.10.

Being a careful observation, for these cases where (2.18)g(x,t)=δ(t-τ)δ(x),g(x,t)=1τe-t/τδ(x),g(x,t)=1τe-t/τ14πte-x2/4t,g(x,t)=tτ2e-t/τ14πte-x2/4t,g(x,t)=δ(t-τ)14πte-x2/4t by using the above method, we can get similar results, respectively.

Remark 2.11.

In the case a=0, b=1, d=0, (1.1) reduces to (2.19)u(x,t)t=2u(x,t)x2+u(x,t)[1-(g*u)(x,t)], which has been studied by many researchers, for example, Ashwin et al. , Gourley , and Wu and Zou  and references therein. It is easy to see that our results include and complement those of Ashwin et al. , Gourley , and Wu and Zou .

Remark 2.12.

We mention that Ou and Wu  obtained the persistence of traveling fronts of delayed nonlocal reaction-diffusion equations. Their abstract results could be applied to the model (1.1) to obtain the existence of traveling fronts. But, their results cannot prove the precise asymptotic behavior of the traveling fronts.

Appendix

In this appendix, we present some general results developed by Wang et al. . Consider the following reaction-diffusion system with spatiotemporal delays:(A.1)u(x,t)t=D2u(x,t)x2+f(u(x,t),(g1*u)(x,t),,(gm*u)(x,t)), where x, t0, D=diag(d1,,dn), di>0, i=1,,n, n; fC((m+1)n,n), u(x,t)=(u1(x,t),,un(x,t))T, and (A.2)(gj*u)(x,t)=-t-+gj(x-y,t-s)u(y,s)dyds,j=1,,m,m, and the kernel gj(x,t) is any integrable nonnegative function satisfying gj(-x,t)=gj(x,t), 0+-+gj(y,s)dyds=1, and the following assumption:

-+gj(x,t)dx is uniformly convergent for t[0,a], a>0, j=1,,m. In other words, if given ε>0, then there exists M>0 such that M+gj(x,t)dx<ε for any t[0,a].

Assume u(x,t)=ϕ(ξ) and ξ=x+ct, and then we can write (A.1) in the following form: (A.3)-Dϕ(ξ)+cϕ(ξ)=f(ϕ(ξ),(g1*ϕ)(ξ),,(gm*ϕ)(ξ)),ξ.

A traveling wave front with a wave speed c>0 to (A.1) is a function ϕBC2(,n) and a number c>0 which satisfy (A.3) and the following boundary condition: (A.4)ϕ(-)=0,ϕ(+)=K=(K1,,Kn)Twith  Ki>0,i=1,,n.

In order to tackle the existence of traveling fronts, we need the following monotonicity conditions and assumptions.

There exists a matrix γ=diag(γ1,...,γn) with γi>0, i=1,...,n, such that (A.5)f(ψ(ξ),(g1*ψ)(ξ),,(gm*ψ)(ξ))+γψ(ξ)f(ϕ(ξ),(g1*ϕ)(ξ),,(gm*ϕ)(ξ))+γϕ(ξ), where ϕ, ψC(,n) satisfy 0ϕ(ξ)ψ(ξ)K in ξ and eγt[ψ(ξ)-ϕ(ξ)] is increasing in ξ.

There exists a matrix γ=diag(γ1,...,γn) with γi>0, i=1,...,n, such that (A.6)f(ψ(ξ),(g1*ψ)(ξ),,(gm*ψ)(ξ))+γψ(ξ)f(ϕ(ξ),(g1*ϕ)(ξ),,(gm*ϕ)(ξ))+γϕ(ξ), where ϕ, ψC(,n) satisfy 0ϕ(ξ)ψ(ξ)K in ξ, eγξ[ψ(ξ)-ϕ(ξ)] is increasing in ξ, and e-γξ[ψ(ξ)-ϕ(ξ)] is decreasing in ξ.

f(μ1,,μn)0 for 0<μ<K.

f(μ1,,μn)=0 when μ=0 or μ=K.

Let BC[0,K]={xBC(,n):0x(t)K,  t}, Y={ϕBC(,n):ϕ,ϕL(,n)} and (A.7)Γ*={ϕY:(i)ϕ(ξ)  is  nondecreasing  in  ξ;(ii)0limξ-ϕ(ξ)<K,limξ+ϕ(ξ)=K;(iii)eγξ[ϕ(ξ+η)-ϕ(ξ)]is  increasing  in  ξfor  every  η>0},(A.8)Γ**={ϕY:(i)ϕ(ξ)  is  nondecreasing  in  ξ;(ii)0limξ-ϕ(ξ)<K,limξ+ϕ(ξ)=K;(iii)eγξ[ϕ(ξ+η)-ϕ(ξ)]  is  increasing  in  ξande-γξ[ϕ(ξ+η)-ϕ(ξ)]  is  decreasing  in  ξfor    every    η>0}. Define an operator F:BC[0,K]BC(,n) by (A.9)F(ϕ)(ξ)=f(ϕ(ξ),(g1*ϕ)(ξ),,(gm*ϕ)(ξ)),ξ.

Now we give definitions of the lower and upper solutions of (A.3) as follows.

Definition A.1.

A continuous function ϕ: is called an upper solution of (A.3) if ϕ and ϕ′′ exist almost everywhere in and are essentially bounded on , and if ϕ satisfies, (A.10)-Dϕ(ξ)+cϕ(ξ)f(ϕ(ξ),(g1*ϕ)(ξ),,(gm*ϕ)(ξ)),a.e.    in    . A lower solution of (A.3) is defined in a similar way by reversing the inequality in (A.10).

Theorem A.2.

Assume that (H2), (H3), and (H0) hold. Also assume that ϕ and ψ, where ϕBC[0,K]Y with ϕ0^, limξ-ϕ(ξ)=0 and ϕψ, are lower and upper solutions of (A.3), respectively. Then

if (H1*) holds, ψΓ* and eγξ[ψ(ξ)-ϕ(ξ)] is increasing in ξ, then for c>1-min{γidi,i=1,n}, (A.1) has a traveling wave front ϕ* such that (A.4) holds with ϕϕ*ψ and for a, b with a<b, (A.11)ψm-ϕ*C([a,b],n)0, where (A.12)-D(ψm)+c(ψm)+γψm=Fψm-1+γψm-1(mN),ϕϕ*ψmψ1ψ0=ψ,

if (H1**) holds, ψΓ**, eγξ[ψ(ξ)-ϕ(ξ)] is increasing in ξ and e-γξ[ψ(ξ)-ϕ(ξ)] is decreasing in ξ, where min{γidi,i=1,,n}-1>0, then for 0<c<min{γidi,i=1,,n}-1, (A.1) has a traveling wave front ϕ* such that (A.4) holds with ϕϕ*ψ and for a, b with a<b, and (A.11) and (A.12) hold.

In particular, if limξ-ψ(ξ)=0, then ||ψm-ϕ*||0.

Acknowledgments

The authors are very grateful to the anonymous referees for careful reading and helpful suggestions. H.-Q. Zhao is supported by the Scientific Research Program Funded by Shaanxi Provincial Education Department (no. 12JK0860) and the Specialized Research Fund of Xianyang Normal University (11XSYK202), and S.-Y. Liu is supported by the NSF of China (60974082).

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