Journal of Applied Mathematics

Volume 2012 (2012), Article ID 705197, 11 pages

doi:10.1155/2012/705197

Research Article

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

Hai-Qin Zhao¹ and San-Yang Liu²

¹College of Mathematics and Information Science, Xianyang Normal University, Xianyang, Shaanxi 712000, China

²Department of Mathematics, Xidian University, Xi’an, Shaanxi 710071, China

Received 25 August 2012; Accepted 23 October 2012

Academic Editor: Wan-Tong Li

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.

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:

where , , and are nonnegative constants, , and the kernel is any integrable nonnegative function satisfying ,

which was first proposed and analyzed by Gourley and So [1] on a finite domain .

In the case , ,, (1.1) becomes

Recently, many researchers studied the existence of traveling fronts of (1.3) with some specific . For the case

where is the Dirac delta function, Gourley [2] showed that, for any , there exists such that, for any , (1.3) has a traveling front connecting the equilibria and , by using the approach developed by Wu and Zou [3]. For the case

Gourley and Chaplain [4] proved the existence of traveling fronts for any and sufficient small , 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

Gourley and Chaplain [4], by using the method of Canosa [6], obtained some information on the monotonicity of traveling fronts for sufficiently large . Furthermore, for these cases

Wang and Li [7] showed that, for any , there exists (or ) such that for any (or ), (1.3) has a traveling front connecting the equilibria and .

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 , there exists (or ) such that, for any (or ), (1.1) has a traveling front connecting the equilibria and (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 , such as (1.4)–(1.7). It is easy to see that (1.1) has two uniform steady states and .

Let , . Then a traveling front of (1.1) satisfies the boundary conditions and , and the following equation:

For , let and . Then . Let , , and such that

Clearly, . Define and . Then we have the following observations.

Lemma 2.1. (i)   is increasing in and satisfies and ;

(ii)   for all ;

(iii)   is increasing and is decreasing in ;

(iv)   is increasing and is decreasing in for every .

Clearly, Lemma 2.1 implies that, for , , and . Now, we show that and are lower and upper solutions of (2.1) by choosing different kernel functions , respectively.

For the sake of convenience, throughout this section, we let

2.1. The Case ,

Clearly, satisfies and in this case

Lemma 2.2. For sufficient small , satisfies .

Proof. Let and . Fix . Let with so that is increasing and is decreasing in . It is easy to see that for any , is increasing and is decreasing in . For sufficiently small satisfying , there is

Hence,

Therefore,

This completes the proof.

Lemma 2.3. Assume that . Then for sufficiently large , is a lower solution of (2.1).

Proof. For , , then

Let

For , , since

and for all, then

Thus, we showed that is a lower solution of (2.1). This completes the proof.

Lemma 2.4. For sufficiently small , is an upper solution of (2.1).

Proof. Note that

By an argument similar to [7, Lemma  3.5], for such that , we have

Then for sufficiently small with ,

This completes the proof.

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

Theorem 2.5. For any , there exists such that, for any , (1.1) has an increasing traveling wave front that satisfies , and .

2.2. The Case ,

It is easy to see that satisfies and in this case

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

Lemma 2.6. For sufficient small , satisfies .

Lemma 2.7. For sufficiently large , is a lower solution of (2.1).

Lemma 2.8. For sufficiently small , is an upper solution of (2.1).

Proof. Note that, for such that ,

Then for sufficiently small with ,

This completes the proof.

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

Theorem 2.9. For any , there exists such that, for any , (1.1) has an increasing traveling wave front that satisfies , and .

Remark 2.10. Being a careful observation, for these cases where

by using the above method, we can get similar results, respectively.

Remark 2.11. In the case , , , (1.1) reduces to

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:

where , , , , , ; , , and

and the kernel is any integrable nonnegative function satisfying , , and the following assumption:

()  is uniformly convergent for , , . In other words, if given , then there exists such that for any .

Assume and , and then we can write (A.1) in the following form:

A traveling wave front with a wave speed to (A.1) is a function and a number which satisfy (A.3) and the following boundary condition:

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

() There exists a matrix with , , such that

where , satisfy in and is increasing in .() There exists a matrix with , , such that

where , satisfy in , is increasing in , and is decreasing in .()  for .()  when or .

Let , and

Define an operator by

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

Definition A.1. A continuous function is called an upper solution of (A.3) if and exist almost everywhere in and are essentially bounded on , and if satisfies,

A lower solution of (A.3) is defined in a similar way by reversing the inequality in (A.10).

Theorem A.2. Assume that , , and hold. Also assume that and , where with , and , are lower and upper solutions of (A.3), respectively. Then

(i) if holds, and is increasing in , then for , (A.1) has a traveling wave front such that (A.4) holds with and for , with ,

where

(ii) if holds, , is increasing in and is decreasing in , where , then for , (A.1) has a traveling wave front such that (A.4) holds with and for , with , and (A.11) and (A.12) hold.

In particular, if , then .

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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