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

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.

1. Introduction

This paper is concerned with the traveling fronts for the following food-limited model: 𝜕𝑢(𝑥,𝑡)=𝜕𝜕𝑡²𝑢(𝑥,𝑡)𝜕𝑥²()())+𝑢𝑥,𝑡1−𝑎𝑢𝑥,𝑡−𝑏𝑔∗𝑢(𝑥,𝑡1+𝑎𝑑𝑢(𝑥,𝑡)+𝑏𝑑𝑔∗𝑢(𝑥,𝑡),𝑥∈ℝ,𝑡≥0, (1.1) 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), ₀^+∞+∞_−∞𝑔𝑦,𝑠𝑑𝑦𝑑𝑠=1,𝑔∗𝑢(𝑥,𝑡)=^𝑡_−∞^+∞_−∞𝑔𝑥−𝑦,𝑡−𝑠𝑢𝑦,𝑠𝑑𝑦𝑑𝑠, (1.2) which was first proposed and analyzed by Gourley and So [1] on a finite domain Ω.

In the case 𝑎=0, 𝑏=1,𝑑=𝛽, (1.1) becomes 𝜕𝑢(𝑥,𝑡)=𝜕𝜕𝑡²𝑢(𝑥,𝑡)𝜕𝑥²())+𝑢𝑥,𝑡1−𝑔∗𝑢(𝑥,𝑡().1+𝛽𝑔∗𝑢𝑥,𝑡 (1.3)

Recently, many researchers studied the existence of traveling fronts of (1.3) with some specific 𝑔(𝑥,𝑡). For the case 𝑔(𝑥,𝑡)=𝛿(𝑡−𝜏)𝛿(𝑥), (1.4) 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 𝑔()=1𝑥,𝑡𝜏𝑒^−𝑡/𝜏1√𝑒4𝜋𝑡^{−𝑥2/4𝑡}, (1.5)

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 𝑔()()1𝑥,𝑡=𝛿𝑡−𝜏√𝑒4𝜋𝑡^{−𝑥2/4𝑡}, (1.6)

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 𝑔()=𝑡𝑥,𝑡𝜏²𝑒^−𝑡/𝜏1√𝑒4𝜋𝑡^{−𝑥2/4𝑡}()()1,𝑔𝑥,𝑡=𝛿𝑡−𝜏√𝑒4𝜋𝑡^{−𝑥2/4𝑡},𝑔()(𝑡)1𝑥,𝑡=𝛿𝑒2𝜌^{−|𝑥|/𝜌}()=1,𝜌>0,𝑔𝑥,𝑡𝜏²𝑒^−𝑡/𝜏𝛿(𝑥),𝜏>0, (1.7)

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 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. [8].

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: 𝜙^′′(𝜉)−𝑐𝜙^′(𝜉)(𝜉)(𝜉)(𝜉)+𝜙1−𝑎𝜙−𝑏𝑔∗𝜙1+𝑎𝑑𝜙(𝜉)+𝑏𝑑𝑔∗𝜙(𝜉)=0,𝜉∈ℝ. (2.1)

For 𝑐>2, let Δ_𝑐(𝜇)=𝜇²−𝑐𝜇+1 and 𝜆=(𝑐−√𝑐²−4)/2. Then Δ_𝑐(𝜆)=0. Let 0<𝜀<𝜆, 𝛼>0, 𝑀>1 and 𝛾>𝜆 such that √𝜆+𝜀<𝛾,𝜆+𝜀<𝑐+𝑐²−42𝜆,𝛼<2,1𝛾+𝜆2≤𝑀𝛼≤𝑀−1. (2.2) Clearly, Δ_𝑐(𝜆+𝜀)<0. Define 𝜙₊(𝜉)=𝐾/(1+𝛼𝑒^−𝜆𝜉) and 𝜙₋(𝜉)=max{𝐾𝑒^𝜆𝜉(1−𝑀𝑒^𝜀𝜉),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) ?𝑒^𝛾𝜉[𝜙₊(𝜉)−𝜙₋(𝜉)] is increasing and 𝑒^−𝛾𝜉[𝜙₊(𝜉)−𝜙₋(𝜉)] is decreasing in 𝜉∈ℝ;
(iv) ?𝑒^𝛾𝜉[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, Lemma 2.1 implies that, for 𝛾>𝜆+𝜀, 𝜙₊(𝜉)∈Γ^∗, 𝜙₊(𝜉)∈Γ^∗∗ and sup_𝜉∈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 𝑓𝜙(𝜉),(𝜉)(𝜉)(𝜉)(𝜉)𝑔∗𝜙=𝜙1−𝑎𝜙−𝑏𝑔∗𝜙1+𝑎𝑑𝜙(𝜉)+𝑏𝑑𝑔∗𝜙(𝜉),𝜉∈ℝ. (2.3)

2.1. The Case 𝑔(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 𝑔∗𝜙(𝜉)=^+∞_−∞1𝑒2𝜌^{−|𝑦|/𝜌}𝜙𝜉−𝑦𝑑𝑦. (2.4)

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,𝜙₂∈𝐶(ℝ,ℝ) with 0≤𝜙₁(𝜉)≤𝜙₂(𝜉)≤𝐾 so that 𝑒^𝛾𝜉[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, 𝑒^𝛾𝜉[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 𝑔∗𝜙₂(𝜉)−𝑔∗𝜙₁(𝜉)=^+∞_−∞1𝑒2𝜌^{−|𝑦|/𝜌}𝜙₂𝜉−𝑦−𝜙₁=𝜉−𝑦𝑑𝑦_0+∞1𝑒2𝜌^−𝑦/𝜌𝜙₂𝜉−𝑦−𝜙₁𝜉−𝑦𝑑𝑦+₀^+∞1𝑒2𝜌^−𝑦/𝜌𝜙₂𝜉+𝑦−𝜙₁=𝜉+𝑦𝑑𝑦_0+∞1𝑒2𝜌^−𝑦/𝜌𝑒^𝛾𝑦𝑒^−𝛾𝑦𝜙₂𝜉−𝑦−𝜙₁𝜉−𝑦+𝑒^−𝛾𝑦𝜙₂𝜉+𝑦−𝜙₁𝜙𝜉+𝑦𝑑𝑦≤2₂(𝜉)−𝜙₁(𝜉)₀^+∞1𝑒2𝜌^−𝑦/𝜌𝑒^𝛾𝑦=1𝑑𝑦𝜙1−𝜌𝛾²(𝜉)−𝜙₁(𝜉)≤2𝜙₂(𝜉)−𝜙₁(𝜉). (2.5) Hence, 𝜙₂1−𝑎𝜙₂−𝑏𝑔∗𝜙₂1+𝑎𝑑𝜙₁+𝑏𝑑𝑔∗𝜙₁−𝜙₁1−𝑎𝜙₁−𝑏𝑔∗𝜙₁1+𝑎𝑑𝜙₂+𝑏𝑑𝑔∗𝜙₂=𝜙₂−𝜙₁1+𝑏𝑑𝑔∗𝜙₁−𝑎+𝑎𝑏𝑑𝑔∗𝜙₁𝜙₂+𝜙₁−𝑎²𝑑𝜙₁𝜙₂−𝑏𝑔∗𝜙₂−𝑏²𝑑𝑔∗𝜙₁𝑔∗𝜙₂+𝑎𝑏𝑑𝜙²₁−𝑏𝑑𝜙₁−𝑏𝜙₁−𝑎𝑏𝑑𝜙₁𝜙₂𝑔∗𝜙₂−𝑔∗𝜙₁≥−𝐴𝜙₂−𝜙₁−𝐵𝑔∗𝜙₂−𝑔∗𝜙₁≥−(𝐴+2𝐵)𝜙₂−𝜙₁>−𝛾𝜙₂−𝜙₁. (2.6) Therefore, 𝑓𝜙₂(𝜉),𝑔∗𝜙₂(𝜉)−𝑓𝜙₁(𝜉),𝑔∗𝜙₁(𝜉)=𝜙₂(𝜉)1−𝑎𝜙₂(𝜉)−𝑏𝑔∗𝜙₂(𝜉)1+𝑎𝑑𝜙₂(𝜉)+𝑏𝑑𝑔∗𝜙₂(𝜉)−𝜙₁(𝜉)1−𝑎𝜙₁(𝜉)−𝑏𝑔∗𝜙₁(𝜉)1+𝑎𝑑𝜙₁(𝜉)+𝑏𝑑𝑔∗𝜙₁(𝜉)≥−𝛾𝜙₂(𝜉)−𝜙₁(𝜉)1+𝑎𝑑𝜙₂(𝜉)+𝑏𝑑𝑔∗𝜙₂(𝜉)1+𝑎𝑑𝜙₁(𝜉)+𝑏𝑑𝑔∗𝜙₁(𝜉𝜙)>−𝛾₂(𝜉)−𝜙₁(𝜉). (2.7) 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 𝜉≥𝜉₀=(1/𝜀)ln(1/𝑀), 𝜙₋(𝜉)=0, then 𝜙₋^′′(𝜉)−𝑐𝜙^′₋(𝜉)+𝜙₋(𝜉)1−𝑎𝜙₋(𝜉)−𝑏𝑔∗𝜙₋(𝜉)1+𝑎𝑑𝜙₋(𝜉)+𝑏𝑑𝑔∗𝜙₋(𝜉)=0. (2.8) Let ()𝑀≥−𝑑+1𝑎𝐾Δ_𝑐()−()𝜆+𝜀𝑑+1𝑏𝐾1−𝜌𝜆1+𝜌𝜆Δ_𝑐().𝜆+𝜀 (2.9) For 𝜉<𝜉₀<0, 𝜙₋(𝜉)=𝐾𝑒^𝜆𝜉(1−𝑀𝑒^𝜀𝜉), since 𝑔∗𝜙₋(𝜉)=^+∞_−∞1𝑒2𝜌^{−|𝑦|/𝜌}𝜙₋=𝜉−𝑦𝑑𝑦_{+∞𝜉−𝜉0}1𝑒2𝜌^{−|𝑦|/𝜌}𝑒^{𝜆(𝜉−𝑦)}𝐾1−𝑀𝑒^{𝜀(𝜉−𝑦)}𝑑𝑦≤𝐾^+∞_−∞1𝑒2𝜌^{−|𝑦|/𝜌}𝑒^{𝜆(𝜉−𝑦)}𝑑𝑦=𝐾𝑒^𝜆𝜉,1−𝜌𝜆1+𝜌𝜆 (2.10) and ℎ(𝑧)=(1−𝑧)/(1+𝑑𝑧)≥1−(𝑑+1)𝑧 for?all𝑧>0, then 𝜙₋^′′(𝜉)−𝑐𝜙^′₋(𝜉)+𝜙₋(𝜉)1−𝑎𝜙₋(𝜉)−𝑏𝑔∗𝜙₋(𝜉)1+𝑎𝑑𝜙₋(𝜉)+𝑏𝑑𝑔∗𝜙₋(𝜉)≥𝜙₋^′′(𝜉)−𝑐𝜙^′₋(𝜉)+𝜙₋(𝜉)1−(𝑑+1)𝑎𝜙₋(𝜉)+𝑏𝑔∗𝜙₋(𝜉𝜆)≥𝐾²−𝑀(𝜆+𝜀)²𝑒^𝜀𝜉𝑒^𝜆𝜉−𝐾𝑐𝜆−𝑀(𝜆+𝜀)𝑒^𝜀𝜉𝑒^𝜆𝜉+𝐾𝑒^𝜆𝜉1−𝑀𝑒^𝜀𝜉−(𝑑+1)𝑎𝐾²𝑒^2𝜆𝜉1−𝑀𝑒^𝜀𝜉2−(𝑑+1)𝑏𝐾²𝑒^2𝜆𝜉1−𝜌𝜆1+𝜌𝜆≥𝐾𝑒^{(𝜆+𝜀)𝜉}−𝑀Δ_𝑐()()()𝜆+𝜀−𝑎𝑑+1𝐾−𝑑+1𝑏𝐾1−𝜌𝜆1+𝜌𝜆≥0. (2.11) 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 𝜙^′₊(𝜉)=𝐾𝛼𝜆𝑒^−𝜆𝜉1+𝛼𝑒^−𝜆𝜉2,𝜙₊^′′(𝜉)=−𝐾𝛼𝜆²𝑒^−𝜆𝜉+𝐾𝛼²𝜆²𝑒^−2𝜆𝜉1+𝛼𝑒^−𝜆𝜉3. (2.12) By an argument similar to [7, Lemma??3.5], for 𝜌>0 such that 1−2𝜌𝜆>0, we have 𝑔∗𝜙₊(𝜉)≥𝐾1+𝛼𝑒^−𝜆𝜉−𝐾𝛼𝜆²𝜌²1−𝜆²𝜌²⋅𝑒^−𝜆𝜉1+𝛼𝑒^−𝜆𝜉2. (2.13) Then for sufficiently small 𝜌>0 with 2𝜆²−𝐾𝑏𝜆²𝜌²/(1−𝜆²𝜌²)>0, 𝜙₊^′′(𝜉)−𝑐𝜙^′₊(𝜉)+𝜙₊(𝜉)1−𝑎𝜙₊(𝜉)−𝑏𝑔∗𝜙₊(𝜉)1+𝑎𝑑𝜙₊(𝜉)+𝑏𝑑𝑔∗𝜙₊(𝜉)≤𝜙₊^′′(𝜉)−𝑐𝜙^′₊(𝜉)+𝜙₊(𝜉)1−𝑎𝜙₊(𝜉)−𝑏𝑔∗𝜙₊(𝜉≤)−𝐾𝛼𝜆²𝑒^−𝜆𝜉+𝐾𝛼²𝜆²𝑒^−2𝜆𝜉1+𝛼𝑒^−𝜆𝜉3−𝐾𝑐𝛼𝜆𝑒^−𝜆𝜉1+𝛼𝑒^−𝜆𝜉2+𝐾𝛼𝑒^−𝜆𝜉1+𝛼𝑒^−𝜆𝜉2+𝐾²𝑏𝛼𝜆²𝜌²1−𝜆²𝜌²⋅𝑒^−𝜆𝜉1+𝛼𝑒^−𝜆𝜉3=𝐾𝛼²𝜆²−𝑐𝜆+1𝑒^−2𝜆𝜉−𝐾𝛼𝜆²+𝑐𝜆−1−𝐾𝑏𝜆²𝜌²/1−𝜆²𝜌²𝑒^−𝜆𝜉1+𝛼𝑒^−𝜆𝜉3=−𝐾𝛼2𝜆²−𝐾𝑏𝜆²𝜌²/1−𝜆²𝜌²𝑒^−𝜆𝜉1+𝛼𝑒^−𝜆𝜉3<0. (2.14) 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 lim_{𝜉→−∞}𝜙(𝜉)𝑒^−𝜆𝜉=1.

2.2. The Case 𝑔(𝑥,𝑡)=(𝑡/𝜏²)𝑒^−𝑡/𝜏𝛿(𝑥), 𝜏>0

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

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 (𝐻^∗₁).

Lemma 2.7. For sufficiently large 𝑀≥1, 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, 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)2}≥𝐾𝑑𝑠1+𝛼𝑒^−𝜆𝜉−𝐾𝛼𝜆𝑐𝜏𝑒^−𝜆𝜉(1−𝜆𝑐𝜏)1+𝛼𝑒^−𝜆𝜉2−𝐾𝛼𝜆𝑐𝜏𝑒^−𝜆𝜉(1−𝜆𝑐𝜏)²1+𝛼𝑒^−𝜆𝜉2=𝐾1+𝛼𝑒^−𝜆𝜉−𝐾𝛼𝜆𝑐𝜏𝑒^−𝜆𝜉(2−𝜆𝑐𝜏)(1−𝜆𝑐𝜏)²1+𝛼𝑒^−𝜆𝜉2. (2.16) Then for sufficiently small 𝜏>0 with 2𝜆²−𝑏𝐾𝜆𝑐𝜏(2−𝜆𝑐𝜏)/(1−𝜆𝑐𝜏)²>0, 𝜙₊^′′(𝜉)−𝑐𝜙^′₊(𝜉)+𝜙₊(𝜉)1−𝑎𝜙₊(𝜉)−𝑏𝑔∗𝜙₊(𝜉)1+𝑎𝑑𝜙₊(𝜉)+𝑏𝑑𝑔∗𝜙₊(𝜉)≤𝜙₊^′′(𝜉)−𝑐𝜙^′₊(𝜉)+𝜙₊(𝜉)1−𝑎𝜙₊(𝜉)−𝑏𝑔∗𝜙₊𝜉≤()−𝐾𝛼𝜆²𝑒^−𝜆𝜉+𝐾𝛼²𝜆²𝑒^−2𝜆𝜉1+𝛼𝑒^−𝜆𝜉3−𝐾𝑐𝛼𝜆𝑒^−𝜆𝜉1+𝛼𝑒^−𝜆𝜉2+𝐾𝛼𝑒^−𝜆𝜉1+𝛼𝑒^−𝜆𝜉2+𝐾²𝑏𝛼𝜆𝑐𝜏𝑒^−𝜆𝜉(2−𝜆𝑐𝜏)(1−𝜆𝑐𝜏)²1+𝛼𝑒^−𝜆𝜉3=𝐾𝛼²𝜆²−𝑐𝜆+1𝑒^−2𝜆𝜉−𝐾𝛼𝜆²+𝑐𝜆−1−𝑏𝐾𝜆𝑐𝜏(2−𝜆𝑐𝜏)/(1−𝜆𝑐𝜏)²𝑒^−𝜆𝜉1+𝛼𝑒^−𝜆𝜉3=−𝐾𝛼2𝜆²−𝑏𝐾𝜆𝑐𝜏(2−𝜆𝑐𝜏)/(1−𝜆𝑐𝜏)²𝑒^−𝜆𝜉1+𝛼𝑒^−𝜆𝜉3<0. (2.17) 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 lim_{𝜉→−∞}𝜙(𝜉)𝑒^−𝜆𝜉=1.

Remark 2.10. Being a careful observation, for these cases where 𝑔()()𝛿(𝑥)()=1𝑥,𝑡=𝛿𝑡−𝜏,𝑔𝑥,𝑡𝜏𝑒^−𝑡/𝜏𝛿(𝑥)()=1,𝑔𝑥,𝑡𝜏𝑒^−𝑡/𝜏1√𝑒4𝜋𝑡^{−𝑥2/4𝑡},𝑔()=𝑡𝑥,𝑡𝜏²𝑒^−𝑡/𝜏1√𝑒4𝜋𝑡^{−𝑥2/4𝑡}()()1,𝑔𝑥,𝑡=𝛿𝑡−𝜏√𝑒4𝜋𝑡^{−𝑥2/4𝑡} (2.18) 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 𝜕𝑢(𝑥,𝑡)=𝜕𝜕𝑡²𝑢(𝑥,𝑡)𝜕𝑥²+𝑢(𝑥,𝑡)1−𝑔∗𝑢(𝑥,𝑡), (2.19) 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.

Appendix

In this appendix, we present some general results developed by Wang et al. [8]. Consider the following reaction-diffusion system with spatiotemporal delays:𝜕𝑢(𝑥,𝑡)𝜕𝜕𝑡=𝐷²𝑢(𝑥,𝑡)𝜕𝑥²+𝑓𝑢(𝑥,𝑡),𝑔₁∗𝑢(𝑥,𝑡),…,𝑔_𝑚∗𝑢(𝑥,𝑡), (A.1) where 𝑥∈ℝ, 𝑡≥0, 𝐷=diag(𝑑₁,…,𝑑_𝑛), 𝑑_𝑖>0, 𝑖=1,…,𝑛, 𝑛∈ℕ; 𝑓∈𝐶((ℝ^𝑚+1)^𝑛,ℝ^𝑛), 𝑢(𝑥,𝑡)=(𝑢₁(𝑥,𝑡),…,𝑢_𝑛(𝑥,𝑡))^𝑇, and 𝑔_𝑗∗𝑢(𝑥,𝑡)=^𝑡_−∞^+∞_−∞𝑔_𝑗𝑥−𝑦,𝑡−𝑠𝑢𝑦,𝑠𝑑𝑦𝑑𝑠,𝑗=1,…,𝑚,𝑚∈ℕ, (A.2) and the kernel 𝑔_𝑗(𝑥,𝑡) is any integrable nonnegative function satisfying 𝑔_𝑗(−𝑥,𝑡)=𝑔_𝑗(𝑥,𝑡), ∫₀^+∞∫^+∞_−∞𝑔_𝑗(𝑦,𝑠)𝑑𝑦𝑑𝑠=1, and the following assumption:(𝐻₀)?∫^+∞_−∞𝑔_𝑗(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^+∞𝑔_𝑗(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: −𝐷𝜙^′′(𝜉)+𝑐𝜙^′(𝜉)=𝑓𝜙(𝜉),𝑔₁∗𝜙(𝜉),…,𝑔_𝑚∗𝜙(𝜉),𝜉∈ℝ. (A.3)

A traveling wave front with a wave speed 𝑐>0 to (A.1) is a function 𝜙∈𝐵𝐶²(ℝ,ℝ^𝑛) and a number 𝑐>0 which satisfy (A.3) and the following boundary condition: 𝜙(−∞)=𝟎,𝜙(+∞)=𝐊=𝐾₁,…,𝐾_𝑛^𝑇with𝐾_𝑖>0,𝑖=1,…,𝑛. (A.4)

In order to tackle the existence of traveling fronts, we need the following monotonicity conditions and assumptions. (𝐻^∗₁)?There exists a matrix 𝛾=diag(𝛾₁,...,𝛾_𝑛) with 𝛾_𝑖>0, 𝑖=1,...,𝑛, such that 𝑓𝜓(𝜉),𝑔₁∗𝜓(𝜉),…,𝑔_𝑚(𝜉)(𝜉)𝜙(𝜉),𝑔∗𝜓+𝛾𝜓≥𝑓₁∗𝜙(𝜉),…,𝑔_𝑚∗𝜙(𝜉)+𝛾𝜙(𝜉), (A.5) where 𝜙, 𝜓∈𝐶(ℝ,ℝ^𝑛) satisfy 𝟎≤𝜙(𝜉)≤𝜓(𝜉)≤𝐊 in 𝜉∈ℝ and 𝑒^𝛾𝑡[𝜓(𝜉)−𝜙(𝜉)] is increasing in 𝜉∈ℝ.(𝐻₁^∗∗)?There exists a matrix 𝛾=diag(𝛾₁,...,𝛾_𝑛) with 𝛾_𝑖>0, 𝑖=1,...,𝑛, such that 𝑓𝜓(𝜉),𝑔₁∗𝜓(𝜉),…,𝑔_𝑚(𝜉)(𝜉)𝜙(𝜉),𝑔∗𝜓+𝛾𝜓≥𝑓₁∗𝜙(𝜉),…,𝑔_𝑚∗𝜙(𝜉)+𝛾𝜙(𝜉), (A.6) where 𝜙, 𝜓∈𝐶(ℝ,ℝ^𝑛) satisfy 𝟎≤𝜙(𝜉)≤𝜓(𝜉)≤𝐊 in 𝜉∈ℝ, 𝑒^𝛾𝜉[𝜓(𝜉)−𝜙(𝜉)] is increasing in 𝜉∈ℝ, and 𝑒^−𝛾𝜉[𝜓(𝜉)−𝜙(𝜉)] is decreasing in 𝜉∈ℝ.(𝐻₂)?𝑓(𝜇₁,…,𝜇_𝑛)≠0 for 𝟎<𝜇<𝐊.(𝐻₃)?𝑓(𝜇₁,…,𝜇_𝑛)=𝟎 when 𝜇=𝟎 or 𝜇=𝐊.

Let?𝐵𝐶[𝟎,𝐊]={𝑥∈𝐵𝐶(ℝ,ℝ^𝑛)∶𝟎≤𝑥(𝑡)≤𝐊,𝑡∈ℝ}, 𝑌={𝜙∈𝐵𝐶(ℝ,ℝ^𝑛)∶𝜙^′,𝜙^′′∈𝐿^∞(ℝ,ℝ^𝑛)} and Γ^∗=(i)𝜙(𝜉)()𝜙∈𝑌:isnondecreasingin𝜉∈ℝ;ii𝟎≤lim_{𝜉→−∞}𝜙(𝜉)<𝐊,lim_𝜉→+∞𝜙(𝜉)()𝑒=𝐊;iii^𝛾𝜉𝜙(𝜉Γ𝜉+𝜂−𝜙)isincreasingin𝜉∈ℝforevery𝜂>0_∗∗,=(i)𝜙(𝜉)()𝜙∈𝑌:isnondecreasingin𝜉∈ℝ;ii𝟎≤lim_{𝜉→−∞}𝜙(𝜉)<𝐊,lim_𝜉→+∞𝜙(𝜉)()𝑒=𝐊;iii^𝛾𝜉𝜙(𝜉𝑒𝜉+𝜂−𝜙)isincreasingin𝜉∈ℝand_−𝛾𝜉𝜙𝜉+𝜂−𝜙(𝜉)isdecreasingin𝜉∈ℝforevery𝜂>0. (A.7)(A.8) Define an operator 𝐹∶𝐵𝐶[𝟎,𝐊]→𝐵𝐶(ℝ,ℝ^𝑛) by 𝐹𝜙(𝜉)=𝑓𝜙(𝜉),𝑔₁∗𝜙(𝜉),…,𝑔_𝑚∗𝜙(𝜉),𝜉∈ℝ. (A.9)

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

Definition A.1. A continuous function 𝜙∶ℝ→ℝ is called an upper solution of (A.3) if 𝜙^′ and 𝜙^′′ exist almost everywhere in ℝ and are essentially bounded on ℝ, and if 𝜙 satisfies, −𝐷𝜙^′′(𝜉)+𝑐𝜙^′(𝜉)≥𝑓𝜙(𝜉),𝑔₁∗𝜙(𝜉),…,𝑔_𝑚∗𝜙(𝜉),a.e.inℝ. (A.10) 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 𝜙∈𝐵𝐶[𝟎,𝐊]∩𝑌 with 𝜙≠𝟎, lim_{𝜉→−∞}𝜙(𝜉)=𝟎 and 𝜙≤𝜓, are lower and upper solutions of (A.3), respectively. Then(i)?if (𝐻^∗₁) holds, 𝜓∈Γ^∗ and 𝑒^𝛾𝜉[𝜓(𝜉)−𝜙(𝜉)] is increasing in 𝜉∈ℝ, then for 𝑐>1−min{𝛾_𝑖𝑑_𝑖,𝑖=1,…𝑛}, (A.1) has a traveling wave front 𝜙^∗ such that (A.4) holds with 𝜙≤𝜙^∗≤𝜓 and for 𝑎, 𝑏∈ℝ with 𝑎<𝑏, 𝜓^𝑚−𝜙^∗_{𝐶([𝑎,𝑏],ℝ𝑛)}→0, (A.11) where −𝐷𝜓^𝑚′′+𝑐𝜓^𝑚′+𝛾𝜓^𝑚=𝐹𝜓^𝑚−1+𝛾𝜓^𝑚−1(𝑚∈𝑁),𝜙≤𝜙^∗≤⋯≤𝜓^𝑚≤⋯≤𝜓¹≤𝜓⁰=𝜓, (A.12)(ii)?if (𝐻₁^∗∗) holds, 𝜓∈Γ^∗∗, 𝑒^𝛾𝜉[𝜓(𝜉)−𝜙(𝜉)] is increasing in 𝜉∈ℝ and 𝑒^−𝛾𝜉[𝜓(𝜉)−𝜙(𝜉)] is decreasing in 𝜉∈ℝ, where min{𝛾_𝑖𝑑_𝑖,𝑖=1,…,𝑛}−1>0, then for 0<𝑐<min{𝛾_𝑖𝑑_𝑖,𝑖=1,…,𝑛}−1, (A.1) has a traveling wave front 𝜙^∗ such that (A.4) holds with 𝜙≤𝜙^∗≤𝜓 and for 𝑎, 𝑏∈ℝ with 𝑎<𝑏, and (A.11) and (A.12) hold.
In particular, if lim_{𝜉→−∞}𝜓(𝜉)=𝟎, then ||𝜓^𝑚−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).