# Existence of Traveling Fronts in a Food-Limited Population Model with Spatiotemporal Delay

Authors: Hai-Qin Zhao; San-Yang Liu
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

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.


---

## Body

## 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∈ℝ,t≥0,
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 [1] on a finite domain Ω.In the casea=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 [2] 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 [3]. For the case
(1.5)g(x,t)=1τe-t/τ14πte-x2/4t,Gourley and Chaplain [4] proved the existence of traveling fronts for any c≥2 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 [5] and the Fredholm alternative. For the case
(1.6)g(x,t)=δ(t-τ)14πte-x2/4t,Gourley and Chaplain [4], by using the method of Canosa [6], 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 [7] 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. [8], 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 [2–4, 7, 9, 10].The rest of the paper is organized as follows. In Section2, 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. [8].


## 2. Existence of Traveling Fronts

In this section, we will use TheoremA.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).Letu(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∈ℝ.Forc>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),12≤Mα≤M-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∈ℝ and satisfies ϕ+(-∞)=K0 and ϕ+(+∞)=K;
(ii)ϕ+(ξ)≥ϕ-(ξ) for all ξ∈ℝ;
(iii)eγξ[ϕ+(ξ)-ϕ-(ξ)] is increasing and e-γξ[ϕ+(ξ)-ϕ-(ξ)] is decreasing in ξ∈ℝ;
(iv)eγξ[ϕ+(ξ+η)-ϕ+(ξ)] is increasing and e-γξ[ϕ+(ξ+η)-ϕ+(ξ)] is decreasing in ξ∈ℝ for every η>0.Clearly, Lemma2.1 implies that, for γ>λ+ε, ϕ+(ξ)∈Γ*, ϕ+(ξ)∈Γ** and supξ∈ℝϕ-(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∈ℝ.


### 2.1. The Caseg(x,t)=δ(t)(1/2ρ)e-|x|/ρ, ρ>0

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 smallρ>0, f(ϕ(ξ),(g*ϕ)(ξ)) satisfies (H1**).Proof.
LetA=(a2+2abd+a2d+ab)/(a+b)2 and B=b(d+1)/(a+b)+abd/(a+b)2. Fix γ>A+2B. Let ϕ1,ϕ2∈C(ℝ,ℝ) with 0≤ϕ1(ξ)≤ϕ2(ξ)≤K so that eγξ[ϕ2(ξ)-ϕ1(ξ)] is increasing and e-γξ[ϕ2(ξ)-ϕ1(ξ)] is decreasing in ξ∈ℝ. It is easy to see that for any η∈ℝ, eγξ[ϕ2(ξ+η)-ϕ1(ξ+η)] is increasing and e-γξ[ϕ2(ξ+η)-ϕ1(ξ+η)] is decreasing in ξ∈ℝ. 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}}dy≤2[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 that1-λρ>0. Then for sufficiently large M>1, ϕ-(ξ) is a lower solution of (2.1).Proof.
Forξ≥ξ0=(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))dy≤K∫-∞+∞‍12ρe-|y|/ρeλ(ξ-y)dy=Keλξ(1-ρλ)(1+ρλ),
and h(z)=(1-z)/(1+dz)≥1-(d+1)z for allz>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 smallρ>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 TheoremA.2(ii), we have the following result.Theorem 2.5.
For anyc>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.


### 2.2. The Caseg(x,t)=(t/τ2)e-t/τδ(x), τ>0

It is easy to see thatg(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 Lemmas2.1 and 2.3, and their proofs are omitted.Lemma 2.6.
For sufficient smallτ>0, f(ϕ(ξ),(g*ϕ)(ξ)) satisfies (H1*).Lemma 2.7.
For sufficiently largeM≥1, ϕ-(ξ) is a lower solution of (2.1).Lemma 2.8.
For sufficiently smallτ>0, ϕ+(ξ) is an upper solution of (2.1).Proof.
Note that, forτ>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-λξ∫0+∞‍sτe-s/τKe(λc-1/τ)s[1+αe-λ(ξ-cs)]2ds≥K1+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 TheoremA.2(i), we have the following result.Theorem 2.9.
For anyc>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 casea=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. [9], Gourley [10], and Wu and Zou [3] and references therein. It is easy to see that our results include and complement those of Ashwin et al. [9], Gourley [10], and Wu and Zou [3].Remark 2.12.
We mention that Ou and Wu [11] 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.


## 2.1. The Caseg(x,t)=δ(t)(1/2ρ)e-|x|/ρ, ρ>0

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 smallρ>0, f(ϕ(ξ),(g*ϕ)(ξ)) satisfies (H1**).Proof.
LetA=(a2+2abd+a2d+ab)/(a+b)2 and B=b(d+1)/(a+b)+abd/(a+b)2. Fix γ>A+2B. Let ϕ1,ϕ2∈C(ℝ,ℝ) with 0≤ϕ1(ξ)≤ϕ2(ξ)≤K so that eγξ[ϕ2(ξ)-ϕ1(ξ)] is increasing and e-γξ[ϕ2(ξ)-ϕ1(ξ)] is decreasing in ξ∈ℝ. It is easy to see that for any η∈ℝ, eγξ[ϕ2(ξ+η)-ϕ1(ξ+η)] is increasing and e-γξ[ϕ2(ξ+η)-ϕ1(ξ+η)] is decreasing in ξ∈ℝ. 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}}dy≤2[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 that1-λρ>0. Then for sufficiently large M>1, ϕ-(ξ) is a lower solution of (2.1).Proof.
Forξ≥ξ0=(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))dy≤K∫-∞+∞‍12ρe-|y|/ρeλ(ξ-y)dy=Keλξ(1-ρλ)(1+ρλ),
and h(z)=(1-z)/(1+dz)≥1-(d+1)z for allz>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 smallρ>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 TheoremA.2(ii), we have the following result.Theorem 2.5.
For anyc>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.


## 2.2. The Caseg(x,t)=(t/τ2)e-t/τδ(x), τ>0

It is easy to see thatg(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 Lemmas2.1 and 2.3, and their proofs are omitted.Lemma 2.6.
For sufficient smallτ>0, f(ϕ(ξ),(g*ϕ)(ξ)) satisfies (H1*).Lemma 2.7.
For sufficiently largeM≥1, ϕ-(ξ) is a lower solution of (2.1).Lemma 2.8.
For sufficiently smallτ>0, ϕ+(ξ) is an upper solution of (2.1).Proof.
Note that, forτ>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-λξ∫0+∞‍sτe-s/τKe(λc-1/τ)s[1+αe-λ(ξ-cs)]2ds≥K1+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 TheoremA.2(i), we have the following result.Theorem 2.9.
For anyc>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 casea=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. [9], Gourley [10], and Wu and Zou [3] and references therein. It is easy to see that our results include and complement those of Ashwin et al. [9], Gourley [10], and Wu and Zou [3].Remark 2.12.
We mention that Ou and Wu [11] 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.



---

*Source: 705197-2012-12-20.xml*