Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 605254 | 15 pages | https://doi.org/10.1155/2009/605254

Permanence of a Discrete Nonlinear Prey-Competition System with Delays

Academic Editor: Xue-Zhong He
Received03 Jul 2009
Accepted16 Sep 2009
Published27 Oct 2009

Abstract

A discrete nonlinear prey-competition system with m-preys and (n-m)-predators and delays is considered. Two sets of sufficient conditions on the permanence of the system are obtained. One set is delay independent, while the other set is delay dependent.

1. Introduction

In this paper, we investigate the following discrete nonlinear prey-competition system with delays:

where is the density of prey species at th generation, is the density of predator species at th generation. In this system, the competition among predator species and among prey species is simultaneously considered. For more background and biological adjustments of system (1.1), we can see [15] and the references cited therein.

Throughout this paper, we always assume that for all

( are all bounded nonnegative sequences and Here, for any bounded sequence

( are bounded nonnegative integer sequences, and are all positive constants.

By a solution of system (1.1), we mean a sequence which defined for and which satisfies system (1.1) for. Motivated by application of system (1.1) in population dynamics, we assume that solutions of system (1.1) satisfy the following initial conditions:

where . The exponential forms of system (1.1) assure that the solution of system (1.1) with initial conditions (1.2) remains positive.

Recently, Chen et al. in [1] proposed the following nonlinear prey-competition system with delays:

By using Gaines and Mawhins continuation theorem of coincidence degree theory and by constructing an appropriate Lyapunov functional, they obtained a set of sufficient conditions which guarantee the existence and global attractivity of positive periodic solutions of the system (1.3). In addition, sufficient conditions are obtained for the permanence of the system (1.3) in [2].

On the other hand, though most population dynamics are based on continuous models governed by differential equations, the discrete time models are more appropriate than the continuous ones when the size of the population is rarely small or the population has nonoverlapping generations [315]. Therefore, it is reasonable to study discrete time prey-competition models governed by difference equations.

As we know, a more important theme that interested mathematicians as well as biologists is whether all species in a multispecies community would survive in the long run, that is, whether the ecosystems are permanent. In fact, no such work has been done for system (1.1).

The main purpose of this paper is, by developing the analytical technique of [4, 8, 16], to obtain two sets of sufficient conditions which guarantee the permanence of system (1.1).

2. Main Results

Firstly, we introduce a definition and some lemmas which will be useful in the proof of the main results of this section.

Definition 2.1. System (1.1) is said to be permanent, if there are positive constants and such that each positive solution of system (1.1) satisfies

Lemma 2.2 (see [8]). Assume that satisfies and for whereis a positive constant. Then

Lemma 2.3 (see [8]). Assume that satisfies and where is a constant such that . Then

For system (1.1), we will consider two cases, and respectively, and then we obtain Lemmas 2.42.6.

Lemma 2.4. Assume that Then for every positive solution of system (1.1) with initial condition (1.2), one has where

Proof. Let be any positive solution of system (1.1) with initial condition (1.2), for it follows from system (1.1) that thus Let, we can have By applying Lemma 2.2 to (2.10), we obtain so, we immediately get For any small enough, it follows from (2.12) that there exists enough large such that for all and For and , (2.13) combining with the -th equation of system (1.1) leads to thus Similarly, let , we get By using (2.16), for , according to Lemma 2.2, it follows that setting in above inequality, we have then
This completes the proof.

For convenience, we introduce the following notation.

For

For

Lemma 2.5. Assume that and hold. Then for any positive solution of system (1.1) with initial condition (1.2), one has where

Proof. Let be any positive solution of system (1.1) with initial condition (1.2). From Lemma 2.4, we know that there exists , such that for and For and , (2.25) combining with the -th equation of system (1.1) lead to thus let , we can have where According to Lemma 2.3, we obtain where Setting in (2.30) leads to therefore For any small enough, it follows from (2.33) that there exists enough large such that for all and and so, forand , it follows from system (1.1) that where by using (2.35), similarly to the analysis of (2.33), for and therefore, we easily get This ends the proof of Lemma 2.5.

Denote for

For

Lemma 2.6. Assume that and hold. Then for any positive solution of system (1.1) with initial condition (1.2), one has where

Proof. Let be any positive solution of system (1.1) with initial condition (1.2), for it follows from system (1.1) that It follows from (2.44)that and hence which, together with (2.45), produces similar to the analysis of (2.11) and (2.12), for and thus, we immediately get For any small enough, it follows from (2.50) that there exists enough large such that for all and For and , (2.51) combining with the -th equation of system (1.1) lead to from (2.53), similar to the argument of (2.44) and (2.47), for we have substituting (2.54) into (2.53), we get similar to the analysis of (2.18) and (2.19), for then, For any small enough, it follows from (2.51) and (2.57) that there exists enough large such that for all and Hence, for and , it follows from system (1.1) that from(2.59), similar to the argument of (2.44) and (2.47), for we have and this combined with (2.60) gives Similar to the argument of (2.32) and (2.33), for we obtain then For any small enough, it follows from (2.63) that there exists enough large such that for all and and so, for and , it follows from system (1.1) that
Similar to the argument of (2.61) and (2.62), for we have
then This ends the proof of Lemma 2.6.

Denote

or

Our main result in this paper is the following theorem about the permanence of system (1.1).

Theorem 2.7. Assume that and hold, then system (1.1) is permanent.

Proof. Let be any positive solution of system (1.1) with initial condition (1.2). Suppose . By Lemmas 2.42.6, if system (1.1) satisfies and , then we have The proof is completed.

In this paper, we study a discrete nonlinear predator-prey system with -preys and (-)-predators and delays, which can be seen as the modification of the traditional Lotka-Volterra prey-competition model. From our main results, Theorem 2.7 gives two sets of sufficient conditions on the permanence of the system (1.1). One set is delay independent, while the other set is delay dependent.

Acknowledgments

The author is grateful to the Associate Editor, Professor Xue-Zhong He, and two referees for a number of helpful suggestions that have greatly improved her original submission. This research is supported by the Nation Natural Science Foundation of China (no. 10171010), Key Project on Science and Technology of Education Ministry of China (no. 01061), and Innovation Group Program of Liaoning Educational Committee No. 2007T050.

References

  1. F. Chen, X. Xie, and J. Shi, “Existence, uniqueness and stability of positive periodic solution for a nonlinear prey-competition model with delays,” Journal of Computational and Applied Mathematics, vol. 194, no. 2, pp. 368–387, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. A. Huang, “Permanence of a nonlinear prey-competition model with delays,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 372–381, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. X. Liao, S. Zhou, and Y. N. Raffoul, “On the discrete-time multi-species competition-predation system with several delays,” Applied Mathematics Letters, vol. 21, no. 1, pp. 15–22, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  4. X. Liao, S. Zhou, and Y. Chen, “On permanence and global stability in a general Gilpin-Ayala competition predator-prey discrete system,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 500–509, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. F. Chen, L. Wu, and Z. Li, “Permanence and global attractivity of the discrete Gilpin-Ayala type population model,” Computers & Mathematics with Applications, vol. 53, no. 8, pp. 1214–1227, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  6. R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations, vol. 404 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. View at: MathSciNet
  7. R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. View at: MathSciNet
  8. X. Yang, “Uniform persistence and periodic solutions for a discrete predator-prey system with delays,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 161–177, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. F. Chen, “Permanence in a discrete Lotka-Volterra competition model with deviating arguments,” Nonlinear Analysis: Real World Applications, vol. 9, no. 5, pp. 2150–2155, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. W. Yang and X. Li, “Permanence for a delayed discrete ratio-dependent predator-prey model with monotonic functional responses,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 1068–1072, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. Y. Muroya, “Partial survival and extinction of species in discrete nonautonomous Lotka-Volterra systems,” Tokyo Journal of Mathematics, vol. 28, no. 1, pp. 189–200, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. Y. Saito, W. Ma, and T. Hara, “A necessary and sufficient condition for permanence of a Lotka-Volterra discrete system with delays,” Journal of Mathematical Analysis and Applications, vol. 256, no. 1, pp. 162–174, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  13. L. Chen and L. Chen, “Permanence of a discrete periodic volterra model with mutual interference,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 205481, 9 pages, 2009. View at: Publisher Site | Google Scholar
  14. M. Fazly and M. Hesaaraki, “Periodic solutions for a discrete time predator-prey system with monotone functional responses,” Comptes Rendus Mathématique, vol. 345, no. 4, pp. 199–202, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  15. W. Wang and Z. Lu, “Global stability of discrete models of Lotka-Volterra type,” Nonlinear Analysis: Theory, Methods & Applications, vol. 35, no. 8, pp. 1019–1030, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  16. X. Meng and L. Chen, “Periodic solution and almost periodic solution for a nonautonomous Lotka-Volterra dispersal system with infinite delay,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 125–145, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2009 Hongying Lu. 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.


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