Abstract and Applied Analysis

Volume 2014 (2014), Article ID 613648, 7 pages

http://dx.doi.org/10.1155/2014/613648

Research Article

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

Fei Hu, Yuyin Xu, Z. Wang, and Wei Ding

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received 21 February 2014; Accepted 28 March 2014; Published 20 May 2014

Academic Editor: Yonghui Xia

Copyright © 2014 Fei Hu et al. 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.

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:

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:

where , , ,  and are positive constants and .

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 and denoting the traveling wave coordinate still by , we derive from (2) that

If for some , system (2) has a solution defined on satisfying

where and are steady states of (2), then is called a traveling wave solution of (2) with speed . Without loss of generality, we assume that and .

Rewrite model (3) as

where

In the following parts, we assume that the nonlinear reaction terms satisfy the partial quasimonotonicity conditions (PQM):

(1) is nondecreasing. That is, for ,

(2)For , and ,

Let , and for , define by

where

It is easy to see that satisfy

Throughout this paper, we always assume that the following assumptions hold.

 (H₁)

 (H₂) For any , , there exist some positive constants such that

Let and equip with the exponential decay norm defined by . And define ; then it is easy to see that is a Banach space.

We will study traveling wave solution to system (2) in the following profile set:

It is easy to check that is nonempty, convex, closed, and bounded.

Definition 1. Function is called upper solution of system (2) if is twice differentiable almost everywhere in , and there hold

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) and satisfy

 (P₁) , ,  (P₂) , and .

Lemma 2. Assume that (PQM) holds; then one has

(1) is nondecreasing for .(2)For , and , there has

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

Lemma 3. Assume that (H₂) holds; then is continuous with respect to the norm in .

Proof. For any fixed , choose ; a direct calculation shows that , ; then there exists

For , we can see that

For , we have

So, is continuous with respect to the norm in .

Lemma 4. Assume that (H₁) and (PQM) hold; then

Proof. According to Lemma 2, for any with , there exists

By the definition of upper-lower solutions, we have

Choosing in (9) and denoting , we get

Setting and combining (22) and (23), we have

Repeating the proof of Lemma  3.3 in Wu and Zou [17], we obtain , which implies that .

Next, choosing in (9), we can get . Then, by a similar argument, we know that , , and ; then

This completes the proof.

Lemma 5. The above defined function is compact.

Proof. By the definition of , we have

Thus,

We will complete the proof by two cases as follows:

(i) case

(ii) case

According to the conditions in (PQM), we get that is bounded by a positive number. Therefore, is bounded. The above estimate for shows that , is equicontinuous. It follows from Lemma 4 that , is uniformly bounded.

Next, we define

Then, for each , is also equicontinuous and uniformly bounded. In the interval , it follows from Ascoli-Arzela theorem that is compact. On the other hand, in as , since

By Proposition  2.12 in [22], we have that , , 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 and 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 of in . Now we need to verify the asymptotic boundary conditions. By (P₂) we notice the fact that

we get that

Therefore, the fixed point 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.

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