Travelling Waves of an n-Species Food Chain Model with Spatial Diffusion and Time Delays

Abstract

We investigate an n-species food chain model with spatial diffusion and time delays. By using Schauder’s fixed point theorem, we obtain the result about the existence of the travelling wave solutions of the food chain model with reaction term satisfying the partial quasimonotonicity conditions.

1. Introduction

In the past few decades, the dynamic relationship between predators and preys has been investigated intensively due to its universal existence and importance in both ecology and mathematical ecology [1–3].

Recently, traveling waves for reaction-diffusion systems (see, e.g., [4–7]) have received considerable attention since they determine the long-term behavior of other solutions in many situations. By now, many powerful methods have been used to study the travelling wave solutions for reaction-diffusion systems, like phase plane techniques in [8], degree theory methods [9, 10], the shooting methods [11], the monotone iteration [1], and so on [12, 13].

Although the existence of travelling wave solutions to reaction-diffusion systems without delay has been widely studied [14–16], delayed reaction-diffusion systems which are more realistic in population dynamic and biological models are much more complicated than ordinary systems. Recently, a number of researchers have studied the existence of travelling wave solutions in delayed reaction-diffusion systems. In [17, 18], Wu and Zou considered delayed reaction-diffusion systems with reaction terms satisfying the so-called quasimonotonicity or exponential quasimonotonicity conditions. In [19], Huang and Zou employed the upper-lower solution technique and the monotone to study the existence of travelling wave solutions for a class of diffusion cooperative Lotka-Volterra systems with delays. In [20], Ma used Schauder's fixed point theorem to study the existence of travelling wave solutions to reaction-diffusion systems with quasimonotonicity reaction terms. In 2010, Gan et al. [21] investigated the existence of travelling wave solutions to the following three-species food chain models: 𝜕𝑢𝜕𝑡=𝐷₁𝜕²𝑢𝜕𝑥²+𝑢(𝑡,𝑥)𝑟₁−𝑎₁₁𝑢(𝑡,𝑥)−𝑎₁₂𝑣(𝑡,𝑥),𝜕𝑣𝜕𝑡=𝐷₂𝜕²𝑣𝜕𝑥²+𝑣(𝑡,𝑥)−𝑟₂+𝑎₂₁𝑣(𝑡−𝜏,𝑥)−𝑎₂₂𝑣(𝑡,𝑥)−𝑎₂₃𝑤(𝑡,𝑥),𝜕𝑤𝜕𝑡=𝐷₃𝜕²𝑤𝜕𝑥²+𝑤(𝑡,𝑥)−𝑟₃+𝑎₃₂𝑣(𝑡−𝜏,𝑥)−𝑎₃₃𝑤(𝑡,𝑥). (1)

They obtain the existence of traveling waves by using Schauder’s fixed point theorem with the reaction term satisfying the partial quasimonotonicity conditions instead of the quasimonotonicity conditions.

Motivated by the above papers, in this paper, we investigate an n-species food chain model with spatial diffusion and time delays: 𝜕𝑢_𝑖(𝑡,𝑥)𝜕𝑡=𝐷_𝑖𝜕²𝑢_𝑖(𝑡,𝑥)𝜕𝑥²−𝑎_𝑖𝑢_𝑖(𝑡,𝑥)+^𝑛_𝑗=1𝑤_𝑖𝑗𝑓_𝑗𝑢_𝑗()+𝑡,𝑥_𝑛𝑗=1ℎ_𝑖𝑗𝑔_𝑗𝑢_𝑗𝑡−𝜏_𝑗(𝑡),𝑥, (2) where 𝐷_𝑖, 𝑎_𝑖, 𝑤_𝑖𝑗,  andℎ_𝑖𝑗 are positive constants and 𝑖=1,2,…,𝑛.

The main propose of this paper is to obtain the sufficient condition for the existence of the travelling wave solutions of system (2) by employing Schauder’s fixed point theorem. This paper is organized as follows. In Section 2, some definition and lemmas are given. And the main results of the paper are established in the last section.

2. Preliminary

On substituting 𝑢_𝑖(𝑡,𝑥)=𝜙_𝑖(𝑥+𝑐𝑡) and denoting the traveling wave coordinate 𝑥+𝑐𝑡 still by 𝑡, we derive from (2) that 𝐷_𝑖𝜙_𝑖^′′(𝑡)−𝑐𝜙^′_𝑖(𝑡)−𝑎_𝑖𝜙_𝑖(𝑡)+^𝑛_𝑗=1𝑤_𝑖𝑗𝑓_𝑗𝜙_𝑗(𝑡+_𝑛𝑗=1)ℎ_𝑖𝑗𝑔_𝑗𝜙_𝑗𝑡−𝜏_𝑗(𝑡)=0. (3) If for some 𝑐>0, system (2) has a solution defined on 𝑅^𝑛 satisfying lim_{𝑡→−∞}𝜙_𝑖(𝑡)=𝜙⁻_𝑖,lim_𝑡→∞𝜙_𝑖(𝑡)=𝜙⁺_𝑖, (4) where (𝜙⁻₁,𝜙⁻₂,…,𝜙⁻_𝑛) and (𝜙⁺₁,𝜙⁺₂,…,𝜙⁺_𝑛) are steady states of (2), then 𝑢_𝑖(𝑡,𝑥)=𝜙_𝑖(𝑥+𝑐𝑡) is called a traveling wave solution of (2) with speed 𝑐. Without loss of generality, we assume that (𝜙⁻₁,𝜙⁻₂,…,𝜙⁻_𝑛)=(0,0,…,0) and (𝜙⁺₁,𝜙⁺₂,…,𝜙⁺_𝑛)=(𝑘₁,𝑘₂,…,𝑘_𝑛).

Rewrite model (3) as 𝐷_𝑖𝜙_𝑖^′′(𝑡)−𝑐𝜙^′_𝑖(𝑡)−𝑎_𝑖𝜙_𝑖(𝑡)+𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)=0, (5) where 𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)=^𝑛_𝑗=1𝑤_𝑖𝑗𝑓_𝑗𝜙_𝑗(𝑡)+^𝑛_𝑗=1ℎ_𝑖𝑗𝑔_𝑗𝜙_𝑗𝑡−𝜏_𝑗(𝑡). (6) In the following parts, we assume that the nonlinear reaction terms 𝐻_𝑖(𝜙₁,𝜙₂,…,𝜙_𝑛)(𝑡)(𝑖=1,2,…,𝑛) satisfy the partial quasimonotonicity conditions (PQM):(1)𝐻_𝑛 is nondecreasing. That is, for 0≤𝜙_𝑗≤𝜙_𝑗≤𝜙_𝑗≤𝑘_𝑗(𝑗=1,2,…,𝑛), 𝐻_𝑛𝜙₁,𝜙₂,…,𝜙_𝑛≤𝐻_𝑛𝜙₁,𝜙₂,…,𝜙_𝑛≤𝐻_𝑛𝜙₁,𝜙₂,…,𝜙_𝑛. (7)(2)For 𝑡∈𝑅, and 0≤𝜙_𝑗≤𝜙_𝑗≤𝜙_𝑗≤𝑘_𝑗(𝑗=1,2,…,𝑛), 𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑖,𝜙_𝑖+1,𝜙_𝑖+2,…,𝜙_𝑛(𝑡)≤𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)≤𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑖,𝜙_𝑖+1,𝜙_𝑖+2,…,𝜙_𝑛(𝑡),𝑖=1,2,…,𝑛−1. (8)

Let 𝐶_𝑘(𝑅,𝑅^𝑛)={(𝜙₁,𝜙₂,…,𝜙_𝑛)∈𝐶(𝑅,𝑅^𝑛)∶0≤𝜙_𝑖≤𝑘_𝑖,𝑖=1,2,…,𝑛}, and for (𝜙₁,𝜙₂,…,𝜙_𝑛)∈𝐶_𝑘(𝑅,𝑅^𝑛), define 𝐹=(𝐹₁,𝐹₂,…,𝐹_𝑛)∶𝐶_𝑘(𝑅,𝑅^𝑛)→𝐶_𝑘(𝑅,𝑅^𝑛) by 𝐹_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)=1𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1×^𝑡_−∞𝑒^{𝜆𝑖1(𝑡−𝑠)}𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑠)+𝑑𝑠_∞𝑡𝑒^{𝜆𝑖2(𝑡−𝑠)}𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑠)𝑑𝑠, (9) where 𝜆_𝑖1=𝑐−𝑐²+4𝐷_𝑖𝑎_𝑖2𝐷_𝑖,𝜆_𝑖2=𝑐+𝑐²+4𝐷_𝑖𝑎_𝑖2𝐷_𝑖. (10) It is easy to see that 𝐹_𝑖(𝜙₁,𝜙₂,…,𝜙_𝑛) satisfy 𝐷_𝑖𝐹_𝑖^′′𝜙₁,𝜙₂,…,𝜙_𝑛−𝑐𝐹^′_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛−𝑎_𝑖𝐹_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛+𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)=0. (11)

Throughout this paper, we always assume that the following assumptions hold. (H₁) ^𝑛_𝑗=1𝑤_𝑖𝑗𝑓_𝑗(0)+^𝑛_𝑗=1ℎ_𝑖𝑗𝑔_𝑗(0)=^𝑛_𝑗=1𝑤_𝑖𝑗𝑓_𝑗𝑘_𝑗+^𝑛_𝑗=1ℎ_𝑖𝑗𝑔_𝑗𝑘_𝑗=0,𝑖=1,2,…,𝑛. (12) (H₂) For any 0≤𝜙¹_𝑖, 𝜙²_𝑖≤𝑘_𝑖(𝑖=1,2,…,𝑛), there exist some positive constants 𝐿_𝑖,𝐾_𝑖>0 such that 𝑓_𝑖𝜙¹_𝑖−𝑓_𝑖𝜙²_𝑖≤𝐿_𝑖𝜙¹_𝑖−𝜙²_𝑖,𝑔_𝑖𝜙¹_𝑖−𝑔_𝑖𝜙²_𝑖≤𝐾_𝑖𝜙¹_𝑖−𝜙²_𝑖. (13)

Let 𝑢>0 and equip Φ=(𝜙₁,𝜙₂,…,𝜙_𝑛)∈𝐶(𝑅,𝑅^𝑛) with the exponential decay norm defined by |Φ|_𝜇=sup_𝑡∈𝑅𝑒^{−𝜇|𝑡|}|Φ(𝑡)|_𝑅^𝑛. And define 𝐵_𝜇(𝑅,𝑅^𝑛)={Φ∈𝐶(𝑅,𝑅^𝑛)∶|Φ|_𝜇<∞}; then it is easy to see that (𝐵_𝜇(𝑅,𝑅^𝑛),|⋅|_𝜇) is a Banach space.

We will study traveling wave solution to system (2) in the following profile set: Γ𝜙₁,𝜙₂,…,𝜙_𝑛,𝜙₁,𝜙₂,…,𝜙_𝑛=(1)𝜙_𝑛(2)𝜙isnondecreasingin𝑅;_𝑖≤𝜙_𝑖≤𝜙_𝑖,𝑖=1,2,…,𝑛. (14) It is easy to check that Γ((𝜙₁,𝜙₂,…,𝜙_𝑛),(𝜙₁,𝜙₂,…,𝜙_𝑛)) is nonempty, convex, closed, and bounded.

Definition 1. Function Φ=(𝜙₁,𝜙₂,…,𝜙_𝑛) is called upper solution of system (2) if Φ is twice differentiable almost everywhere in 𝑅, and there hold 𝐷_𝑖𝜙_𝑖^′′(𝑡)−𝑐𝜙^′_𝑖(𝑡)−𝑎_𝑖𝜙_𝑖(𝑡)+𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑖,𝜙_𝑖+1,𝜙_𝑖+2,…,𝜙_𝑛(𝑡)𝐷≤0,a.e.in𝑅,_𝑛𝜙_𝑛^′′(𝑡)−𝑐𝜙^′_𝑛(𝑡)−𝑎_𝑖𝜙_𝑛(𝑡)+𝐻_𝑛𝜙₁,𝜙₂,…,𝜙_𝑛≤0,a.e.in𝑅. (15)Reversing the direction of above inequalities, we can get the lower solution.

In this paper, we assume that the upper-lower solutions of system (2) Φ=(𝜙₁,𝜙₂,…,𝜙_𝑛) and Φ=(𝜙₁,𝜙₂,…,𝜙_𝑛) satisfy (P₁) (0,0,…,0)≤(𝜙₁,𝜙₂,…,𝜙_𝑛)≤(𝜙₁,𝜙₂,…,𝜙_𝑛)≤(𝑘₁,𝑘₂,…,𝑘_𝑛), 𝑡∈𝑅,  (P₂) lim_{𝑡→−∞}(𝜙₁,𝜙₂,…,𝜙_𝑛)=(0,0,…,0), and lim_𝑡→∞(𝜙₁,𝜙₂,…,𝜙_𝑛)=(𝑘₁,𝑘₂,…,𝑘_𝑛).

Lemma 2. Assume that (PQM) holds; then one has (1)𝐹_𝑛(𝜙₁,𝜙₂,…,𝜙_𝑛)(𝑡) is nondecreasing for 𝑡∈𝑅.(2)For 𝑡∈𝑅,𝑖=1,2,…,𝑛−1 and 0≤𝜙_𝑗≤𝜙_𝑗≤𝑘_𝑗(𝑗=1,2,…,𝑛), there has 𝐹_𝑖𝜙_𝑖,𝜙₂,…,𝜙_𝑖,𝜙_𝑖+1,𝜙_𝑖+2,…,𝜙_𝑛(𝑡)≤𝐹_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)≤𝐹_𝑖𝜙_𝑖,𝜙₂,…,𝜙_𝑖,𝜙_𝑖+1,𝜙_𝑖+2,…,𝜙_𝑛(𝑡). (16)

Lemma 2 is easy to prove, so we omit it.

Lemma 3. Assume that (H₂) holds; then 𝐹_𝑖(𝜙₁,𝜙₂,…,𝜙_𝑛)(𝑡) is continuous with respect to the norm |⋅| in 𝐵_𝜇(𝑅,𝑅^𝑛).

Proof. For any fixed 𝜖>0, choose 𝛿<𝜖/(∑^𝑛_𝑗=1𝑤_𝑖𝑗𝐿_𝑗+∑^𝑛_𝑗=1ℎ_𝑖𝑗𝐾_𝑗); a direct calculation shows that |𝜙¹_𝑗−𝜙²_𝑗|_𝜇<𝛿, |𝜙¹_𝑗(𝑡−𝜏_𝑗(𝑡))−𝜙²_𝑗(𝑡−𝜏_𝑗(𝑡))|<𝛿; then there exists𝐻_𝑖𝜙¹₁,𝜙¹₂,…,𝜙¹_𝑛(𝑡)−𝐻_𝑖𝜙²₁,𝜙²₂,…,𝜙²_𝑛(𝑡)𝑒^{−𝜇|𝑡|}=^𝑛_𝑗=1𝑤_𝑖𝑗𝑓_𝑗𝜙¹_𝑗(𝑡)+^𝑛_𝑗=1ℎ_𝑖𝑗𝑔_𝑗𝜙_𝑗𝑡−𝜏_𝑗(𝑡)−^𝑛_𝑗=1𝑤_𝑖𝑗𝑓_𝑗𝜙²_𝑗(𝑡)−^𝑛_𝑗=1ℎ_𝑖𝑗𝑔_𝑗𝜙²_𝑗𝑡−𝜏_𝑗(𝑡)𝑒^{−𝜇|𝑡|}≤^𝑛_𝑗=1𝑤_𝑖𝑗𝑓_𝑗𝜙¹_𝑗(𝑡)−𝑓_𝑗𝜙²_𝑗(𝑡)𝑒^{−𝜇|𝑡|}+^𝑛_𝑗=1ℎ_𝑖𝑗𝑔_𝑗𝜙¹_𝑗𝑡−𝜏_𝑗(𝑡)−𝑔_𝑗𝜙²_𝑗𝑡−𝜏_𝑗(𝑡)𝑒^{−𝜇|𝑡|}≤^𝑛_𝑗=1𝑤_𝑖𝑗𝐿_𝑗𝛿+^𝑛_𝑗=1ℎ_𝑖𝑗𝐾_𝑗𝛿≤𝜖. (17) For 𝑡>0, we can see that 𝐹_𝑖𝜙¹₁,𝜙¹₂,…,𝜙¹_𝑛−𝐹_𝑖𝜙²₁,𝜙²₂,…,𝜙²_𝑛𝑒^{−𝜇|𝑡|}≤1𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1×^𝑡_−∞𝑒^{𝜆𝑖1(𝑡−𝑠)}×𝐻_𝑖𝜙¹₁,𝜙¹₂,…,𝜙¹_𝑛(𝑠)−𝐻_𝑖𝜙²₁,𝜙²₂,…,𝜙²_𝑛(𝑠)𝑒^{−𝜇|𝑠|}𝑒^𝜇|𝑠|+𝑑𝑠_𝑡+∞𝑒^{𝜆𝑖2(𝑡−𝑠)}×𝐻_𝑖𝜙¹₁,𝜙¹₂,…,𝜙¹_𝑛(𝑠)−𝐻_𝑖𝜙²₁,𝜙²₂,…,𝜙²_𝑛(𝑠)×𝑒^{−𝜇|𝑠|}𝑒^𝜇|𝑠|𝑑𝑠𝑒^−𝜇𝑡≤𝜖𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1^𝑡_−∞𝑒^{𝜆𝑖1(𝑡−𝑠)}𝑒^𝜇|𝑠|+𝑑𝑠_𝑡+∞𝑒^{𝜆𝑖2(𝑡−𝑠)}𝑒^𝜇|𝑠|𝑑𝑠𝑒^−𝜇𝑡=𝜖𝐷_𝑖𝜆_𝑖2−𝜆_𝑖12𝜇𝜆²_𝑖1−𝜇²𝑒^{(𝜆𝑖1−𝜇)𝑡}+𝜆_𝑖2−𝜆_𝑖1𝜇−𝜆_𝑖1𝜆_𝑖2≤𝜖−𝜇𝐷_𝑖𝜆_𝑖2−𝜆_𝑖12𝜇𝜆²_𝑖1−𝜇²+𝜆_𝑖2−𝜆_𝑖1𝜇−𝜆_𝑖1𝜆_𝑖2.−𝜇 (18) For 𝑡<0, we have 𝐹_𝑖𝜙¹₁,𝜙¹₂,…,𝜙¹_𝑛−𝐹_𝑖𝜙²₁,𝜙²₂,…,𝜙²_𝑛𝑒^{−𝜇|𝑡|}≤1𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1×^𝑡_−∞𝑒^{𝜆𝑖1(𝑡−𝑠)}𝐻_𝑖𝜙¹₁,𝜙¹₂,…,𝜙¹_𝑛(𝑠)−𝐻_𝑖𝜙²₁,𝜙²₂,…,𝜙²_𝑛(𝑠)𝑒^{−𝜇|𝑠|}𝑒^𝜇|𝑠|+𝑑𝑠_𝑡+∞𝑒^{𝜆𝑖2(𝑡−𝑠)}𝐻_𝑖𝜙¹₁,𝜙¹₂,…,𝜙¹_𝑛(𝑠)−𝐻_𝑖𝜙²₁,𝜙²₂,…,𝜙²_𝑛(𝑠)×𝑒^{−𝜇|𝑠|}𝑒^𝜇|𝑠|𝑑𝑠𝑒^𝜇𝑡≤𝜖𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1^𝑡_−∞𝑒^{𝜆𝑖1(𝑡−𝑠)}𝑒^𝜇|𝑠|+𝑑𝑠_𝑡+∞𝑒^{𝜆𝑖2(𝑡−𝑠)}𝑒^𝜇|𝑠|𝑑𝑠𝑒^𝜇𝑡≤𝜖𝐷_𝑖𝜆_𝑖2−𝜆_𝑖12𝜇𝜆²_𝑖2−𝜇²−𝜆_𝑖2−𝜆_𝑖1𝜇+𝜆_𝑖1𝜆_𝑖2.+𝜇 (19) So, 𝐹_𝑖∶𝐵_𝜇(𝑅,𝑅^𝑛)→𝐵_𝜇(𝑅,𝑅^𝑛) is continuous with respect to the norm |⋅| in 𝐵_𝜇(𝑅,𝑅^𝑛).

Lemma 4. Assume that (H₁) and (PQM) hold; then 𝐹Γ𝜙₁,𝜙₂,…,𝜙_𝑛,𝜙₁,𝜙₂,…,𝜙_𝑛⊂Γ𝜙₁,𝜙₂,…,𝜙_𝑛,𝜙₁,𝜙₂,…,𝜙_𝑛. (20)

Proof. According to Lemma 2, for any (𝜙₁,𝜙₂,…,𝜙_𝑛) with (𝜙₁,𝜙₂,…,𝜙_𝑛)≤(𝜙₁,𝜙₂,…,𝜙_𝑛)≤(𝜙₁,𝜙₂,…,𝜙_𝑛), there exists 𝐹_𝑖𝜙_𝑖,𝜙₂,…,𝜙_𝑖,𝜙_𝑖+1,𝜙_𝑖+2,…,𝜙_𝑛(𝑡)≤𝐹_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)≤𝐹_𝑖𝜙_𝑖,𝜙₂,…,𝜙_𝑖,𝜙_𝑖+1,𝜙_𝑖+2,…,𝜙_𝑛𝑡);𝐹_𝑛(𝜙₁,𝜙₂,…,𝜙_𝑛≤𝐹_𝑛𝜙₁,𝜙₂,…,𝜙_𝑛≤𝐹_𝑛𝜙₁,𝜙₂,…,𝜙_𝑛. (21) By the definition of upper-lower solutions, we have 𝐷_𝑖𝜙_𝑖^′′(𝑡)−𝑐𝜙^′_𝑖(𝑡)−𝑎_𝑖𝜙_𝑖(𝑡)+𝐻_𝑖𝜙_𝑖,𝜙₂,…,𝜙_𝑖,𝜙_𝑖+1,𝜙_𝑖+2,…,𝜙_𝑛(𝑡)≤0. (22) Choosing (𝜙₁,𝜙₂,…,𝜙_𝑛)=(𝜙_𝑖,𝜙₂,…,𝜙_𝑖,𝜙_𝑖+1,𝜙_𝑖+2,…,𝜙_𝑛) in (9) and denoting 𝜙_𝑖=𝐹_𝑖(𝜙_𝑖,𝜙₂,…,𝜙_𝑖,𝜙_𝑖+1,𝜙_𝑖+2,…,𝜙_𝑛), we get 𝐷_𝑖𝜙_𝑖^′′(𝑡)−𝑐𝜙^′_𝑖(𝑡)−𝑎_𝑖𝜙_𝑖(𝑡)+𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑖,𝜙_𝑖+1,𝜙_𝑖+2,…,𝜙_𝑛(𝑡)=0. (23) Setting 𝑤_𝑖(𝑡)=𝜙_𝑖−𝜙_𝑖 and combining (22) and (23), we have 𝐷_𝑖𝑤_𝑖^′′−𝑐𝑤^′_𝑖−𝑎_𝑖≥0. (24) Repeating the proof of Lemma  3.3 in Wu and Zou [17], we obtain 𝑤_𝑖(𝑡)≤0, which implies that 𝐹_𝑖(𝜙₁,𝜙₂,…,𝜙_𝑛)≤𝜙_𝑖(𝑖=1,2,…,𝑛−1).
Next, choosing (𝜙₁,𝜙₂,…,𝜙_𝑛)=(𝜙_𝑖,𝜙₂,…,𝜙_𝑛) in (9), we can get 𝐹_𝑛(𝜙₁,𝜙₂,…,𝜙_𝑛)≤𝜙_𝑛. Then, by a similar argument, we know that 𝐹_𝑖(𝜙₁,𝜙₂,…,𝜙_𝑛)≥𝜙_𝑖, (𝑖=1,2,…,𝑛−1), and 𝐹_𝑛(𝜙₁,𝜙₂,…,𝜙_𝑛)≥𝜙_𝑛; then 𝐹Γ𝜙₁,𝜙₂,…,𝜙_𝑛,𝜙₁,𝜙₂,…,𝜙_𝑛⊂Γ𝜙₁,𝜙₂,…,𝜙_𝑛,𝜙₁,𝜙₂,…,𝜙_𝑛. (25) This completes the proof.

Lemma 5. The above defined function 𝐹 is compact.

Proof. By the definition of 𝐹, we have 𝐹^′_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛𝑡)=𝜆_𝑖1(𝑒^{𝜆𝑖1𝑡}𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1^𝑡_−∞𝑒^{−𝜆𝑖1𝑠}𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑠)+𝜆𝑑𝑠_𝑖2𝑒^{𝜆𝑖2𝑡}𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1^∞_𝑡𝑒^{−𝜆𝑖2𝑠}𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑠)𝑑𝑠. (26) Thus, 𝐹^′_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)_𝜇=sup_𝑡∈𝑅𝜆_𝑖1𝑒^{𝜆𝑖1𝑡}𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1×^𝑡_−∞𝑒^{−𝜆𝑖1𝑠}𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑠)+𝜆𝑑𝑠_𝑖2𝑒^{𝜆𝑖2𝑡}𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1×^∞_𝑡𝑒^{−𝜆𝑖2𝑠}𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑠)𝑑𝑠𝑒^{−𝜇|𝑡|}≤−𝜆_𝑖1𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1×sup_𝑡∈𝑅𝑒^{𝜆𝑖1𝑡−𝜇|𝑡|𝑡}_−∞𝑒^{−𝜆𝑖1𝑠}𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑠+𝜆)𝑑𝑠_𝑖2𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1×sup_𝑡∈𝑅𝑒^{𝜆𝑖2𝑡−𝜇|𝑡|∞}_𝑡𝑒^{−𝜆𝑖2𝑠}𝐻_𝑖𝜙²₁,𝜙²₂,…,𝜙²_𝑛𝑠≤()𝑑𝑠−𝜆_𝑖1𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1𝐻_𝑖𝜙²₁,𝜙²₂,…,𝜙²_𝑛(𝑡)_𝜇×sup_𝑡∈𝑅𝑒^{𝜆𝑖1𝑡−𝜇|𝑡|𝑡}_−∞𝑒^{−𝜆𝑖1𝑠}𝑒^𝜇|𝑠|+𝜆𝑑𝑠_𝑖2𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1𝐻_𝑖𝜙²₁,𝜙²₂,…,𝜙²_𝑛(𝑡)_𝜇×sup_𝑡∈𝑅𝑒^{𝜆𝑖2𝑡−𝜇|𝑡|∞}_𝑡𝑒^{−𝜆𝑖2𝑠}𝑒^𝜇|𝑠|𝑑𝑠. (27) We will complete the proof by two cases as follows:
(i) case 𝑡>0𝐹^′_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)_𝜇≤−𝜆_𝑖1𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1×sup_𝑡∈𝑅𝑒^{(𝜆𝑖1−𝜇)𝑡0}_−∞𝑒^{−(𝜆𝑖1+𝜇)𝑠}𝑑𝑠+^𝑡₀𝑒^{(𝜇−𝜆𝑖1)𝑠}×𝐻𝑑𝑠_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)_𝜇+𝜆_𝑖2𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1×sup_𝑡∈𝑅𝑒^{(𝜆𝑖2−𝜇)𝑡∞}_𝑡𝑒^{(𝜇−𝜆𝑖2)𝑠}𝑑𝑠𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)_𝜇≤𝜆_𝑖1𝐷_𝑖𝜇+𝜆_𝑖1𝜆_𝑖2−𝜆_𝑖1𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)_𝜇+𝜆_𝑖2𝐷_𝑖𝜆_𝑖2−𝜇𝜆_𝑖2−𝜆_𝑖1𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)_𝜇=1𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1𝜆_𝑖1𝜇+𝜆_𝑖1+𝜆_𝑖2𝜆_𝑖2×𝐻−𝜇_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)_𝜇; (28)
(ii) case 𝑡<0𝐹^′_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)_𝜇≤−𝜆_𝑖1𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1×sup_𝑡∈𝑅𝑒^{(𝜆𝑖1+𝜇)𝑡𝑡}_−∞𝑒^{−(𝜆𝑖1+𝜇)𝑠}×𝐻𝑑𝑠_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)_𝜇+𝜆_𝑖2𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1×sup_𝑡∈𝑅𝑒^{(𝜆𝑖2+𝜇)𝑡0}_𝑡𝑒^{(−𝜆𝑖2−𝜇)𝑠}𝑑𝑠+^∞₀𝑒^{(𝜇−𝜆𝑖2)𝑠}×𝐻𝑑𝑠_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)_𝜇≤𝜆_𝑖1𝐷_𝑖𝜇+𝜆_𝑖1𝜆_𝑖2−𝜆_𝑖1𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)_𝜇+𝜆_𝑖2𝐷_𝑖𝜆_𝑖2−𝜇𝜆_𝑖2−𝜆_𝑖1𝐻_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)_𝜇=1𝐷_𝑖𝜆_𝑖2−𝜆_𝑖1𝜆_𝑖1𝜇+𝜆_𝑖1+𝜆_𝑖2𝜆_𝑖2×𝐻−𝜇_𝑖𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)_𝜇. (29) According to the conditions in (PQM), we get that |𝐻_𝑖(𝜙₁,𝜙₂,…,𝜙_𝑛)(𝑡)|_𝜇 is bounded by a positive number. Therefore, |𝐹^′_𝑖(𝜙₁,𝜙₂,…,𝜙_𝑛)(𝑡)|_𝜇 is bounded. The above estimate for 𝐹^′ shows that 𝐹(Γ((𝜙₁,𝜙₂,…,𝜙_𝑛), (𝜙₁,𝜙₂,…,𝜙_𝑛))) is equicontinuous. It follows from Lemma 4 that 𝐹(Γ((𝜙₁,𝜙₂,…,𝜙_𝑛), (𝜙₁,𝜙₂,…,𝜙_𝑛))) is uniformly bounded.
Next, we define 𝐹^𝑚𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)=(1)𝐹𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)[];(2)𝐹𝜙,𝑡∈−𝑚,𝑚₁,𝜙₂,…,𝜙_𝑛(𝑚)();(3)𝐹𝜙,𝑡∈𝑚,∞₁,𝜙₂,…,𝜙_𝑛(−𝑚),𝑡∈(−∞,−𝑚). (30) Then, for each 𝑚≥1, 𝐹^𝑚(𝜙₁,𝜙₂,…,𝜙_𝑛)(𝑡) is also equicontinuous and uniformly bounded. In the interval [−𝑚,𝑚], it follows from Ascoli-Arzela theorem that 𝐹^𝑚 is compact. On the other hand, 𝐹^𝑚→𝐹 in 𝐵_𝜇(𝑅,𝑅^𝑛) as 𝑚→∞, since sup_𝑡∈𝑅𝐹^𝑚𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)−𝐹𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)𝑒^{−𝜇|𝑡|}=sup_{()∪()𝑡∈−∞,−𝑚𝑚,∞}𝐹^𝑚𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)−𝐹𝜙₁,𝜙₂,…,𝜙_𝑛(𝑡)𝑒^{−𝜇|𝑡|}≤2𝐾𝑒^−𝜇𝑚⟶0,𝑚⟶∞. (31) By Proposition  2.12 in [22], we have that 𝐹∶Γ((𝜙₁,𝜙₂,…,𝜙_𝑛), (𝜙₁,𝜙₂,…,𝜙_𝑛))→Γ((𝜙₁,𝜙₂,…,𝜙_𝑛), (𝜙₁,𝜙₂,…,𝜙_𝑛)) is compact. This completes the proof.

3. Main Results

Theorem 6. Assume that (H₁), (H₂), and (PQM) hold. Moreover, suppose that there is a pair of upper-lower solutions Φ=(𝜙₁,𝜙₂,…,𝜙_𝑛) and Φ=(𝜙₁,𝜙₂,…,𝜙_𝑛) for (2) satisfying (P₁) and (P₂). Then, system (2) has a travelling wave solution.

Proof. Following Lemmas 2 to 5, we see that all the condition in Schauder's fixed point theorem hold. Then we know that there exists a fixed point (𝜙^∗₁,𝜙^∗₂,…,𝜙^∗_𝑛) of 𝐹 in Γ((𝜙₁,𝜙₂,…,𝜙_𝑛),(𝜙₁,𝜙₂,…,𝜙_𝑛)). Now we need to verify the asymptotic boundary conditions. By (P₂) we notice the fact that (0,0,…,0)≤𝜙₁,𝜙₂,…,𝜙_𝑛≤𝜙^∗₁,𝜙^∗₂,…,𝜙^∗_𝑛≤𝜙₁,𝜙₂,…,𝜙_𝑛≤𝑘₁,𝑘₂,…,𝑘_𝑛; (32) we get that lim_{𝑡→−∞}𝜙^∗₁,𝜙^∗₂,…,𝜙^∗_𝑛=(0,0,…,0),lim_𝑡→∞𝜙^∗₁,𝜙^∗₂,…,𝜙^∗_𝑛=𝑘₁,𝑘₂,…,𝑘_𝑛. (33) Therefore, the fixed point (𝜙^∗₁,𝜙^∗₂,…,𝜙^∗_𝑛) satisfies the asymptotic boundary conditions. This completes the proof.

4. Conclusion

In this paper, we studied an n-species food chain model with spatial diffusion and time delays. By using Schauder’s fixed point theorem and cross-iteration methods, we reduced the existence of the travelling wave solutions to the existence of a pair of upper-lower solutions. Finally, we proved that the system (2) has a travelling wave solution. However, in order to investigate the specific form of the travelling wave solution of (2), we still have a lot of work to do in the future.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the NNSF of China under Grant (no. 11271261), Natural Science Foundation of Shanghai (no. 13ZR1430100), Shanghai municipal education commission (no. 14ZZ12), and the Slovenian Research Agency, and a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Programme, FP7-PEOPLE-2012-IRSES-316338.