Open Access

Permanence and Stable Periodic Solution for a Discrete Competitive System with Multidelays

Advances in Difference Equations20102009:375486

https://doi.org/10.1155/2009/375486

Received: 7 August 2009

Accepted: 15 December 2009

Published: 7 February 2010

Abstract

The permanence and the existence of periodic solution for a discrete nonautonomous competitive system with multidelays are considered. Also the stability of the periodic solution is discussed. Numerical examples are given to confirm the theoretical results.

1. Introduction

In this paper, we consider the permanence and the periodic solution for the following discrete competitive model with -species and several delays:

(1.1)

In model (1.1), is the population density of the species at n th time step (year, month, day), represents the intrinsic growth rate of species at n th time step, and reflects the interspecific or intraspecific competitive intensity of species to species with time delay at n th time step.

As a special case of model (1.1), the following discrete model

(1.2)

has been investigated as the discrete analogue of the well-known continuous Logistic model [14]:

(1.3)

And many complex dynamics, such as periodic cycles and chaotic behavior, were found in model (1.2) [2, 4, 5].

It is well known that the reproduction rate and the carrying capacity are intensively influenced by the environment; therefore the following model with time varying coefficients

(1.4)

was developed from model (1.2) and has been studied recently in [6].

An equivalent version of model (1.4) can be written as

(1.5)

Models (1.2), (1.4), and (1.5) are both considering of ecosystems for single-species. As a result of coupling equations both described by model (1.5), one can write out the following model for -species:

(1.6)

If ( ) is nonnegative sequences, model (1.6) represents the competitive ecosystem of Lotka-Volterra type with -species [7]. When , model (1.6) was introduced in [8] and recently has been studied in [9]. The autonomous case of (1.6) when has been studied in [10] and the following permanent result was obtained [10, Theorem 2].

Lemma 1.1.

If , then (1.6) is permanent.

It is well known that the effect of time delay plays an important role in population dynamics [11]; therefore, model (1.1) can be constructed from model (1.6) while considering the effect of time delays. Obviously, models (1.2), (1.4), and (1.5) are special cases of model (1.1) for single-species. Model (1.6) is also special case of model (1.1) without delays. Some aspects of model (1.1) has been discussed in the literature. For example, the global asymptotical stability of (1.1) with and the permanence of (1.1) with were investigated in [12]. Necessary and sufficient conditions for the permanence of the autonomous case of (1.1) with two-species

(1.7)

were obtained in [13].

In theoretical population dynamics, it is important whether or not all species in multispecies ecosystem can be permanent [14, 15]. Many permanent or persistent results have been obtained for continuous biomathematical models that are governed by differential equation(s). For example, one can refer to [11, 1621] and references cited therein. However, permanent results on the delayed discrete-time competitive model of Lotka-Volterra type are rarely few [13, 22], especially with -species ( ). In this manuscript, first we will obtain new sufficient conditions for the permanence of (1.1) when .

The population densities observed in the field are usually oscillatory. What cause such phenomenon is a purpose to model population interactions [9, 23]. We will further investigate the existence and stability of the periodic solution for model (1.1) under the assumption that the coefficients of model (1.1) are all periodic with a common period.

The results obtained in this paper are complements to those related with model (1.1). We give some examples to show that the results here are not enclosed by other earlier works. The paper is organized as follows. In next section, we give some preliminaries and obtain the sufficient conditions which guarantee the permanence of model (1.1). In Section 3, we prove the existence of the positive periodic solution of model (1.1) and obtain the sufficient conditions for the stability of the periodic solution.

2. Preliminaries and Permanence

Due to the biological backgrounds of model (1.1), throughout this paper we make the following basic assumptions.

  1. (H1)

    and ( ) are sequences bounded from below and from above by positive constants.

     
  2. (H2)

    ( ) and ( ) are nonnegative and bounded sequences.

     
  3. (H3)

    The initial values are given by .

     

Next we give some definitions that will be be used in this paper. We write and , .

Definition 2.1.

We say that is a solution of (1.1) with initial values ( ) if satisfies (1.1) for and .

Under assumptions ( ), ( ), and ( ), solutions of model (1.1) are all consisting of positive sequences; such solution will be called positive solution of (1.1).

Definition 2.2.

Model (1.1) is said to be permanent if there are positive constants and such that
(2.1)

for each positive solution of model (1.1).

Definition 2.3.

System (1.1) is strongly persistent if each positive solution of (1.1) satisfies
(2.2)

Definition 2.4.

If each positive solution of model (1.1) satisfies that as , we say that the solution of (1.1) is globally attractive or globally stable, where is the maximum norm of the Banach space .

From Definitions 2.2 and 2.3, we know that model (1.1) is strongly persistent if model (1.1) is permanent. For the sake of simplicity, we introduce the following notations for any sequences :

(2.3)

Next we will discuss the sufficient conditions which guarantee that system (1.1) with initial conditions ( ) is permanent. In the following, we denote as the solutions of system (1.1) with initial conditions ( ). Clearly, is positive sequence.

Lemma 2.5.

If satisfies
(2.4)
for , where is a positive sequence bounded from below and from above by positive constants and is a positive integer, then there exists positive constant such that
(2.5)

and .

Proof.

The proof of this lemma is similar to that of Lemma 1 in [24]; we omit the details.

Lemma 2.6.

If satisfies
(2.6)
for , where and are positive sequences bounded from below and from above by positive constants, is a positive integer and . Further, assume that and , then
(2.7)

Proof.

The proof of this lemma is similar to that of Lemma 2 in [24]; we omit the details.

In the following, we denote

(2.8)

Theorem 2.7.

Assume that
(2.9)

then model (1.1) is permanent.

Proof.

From model (1.1), we have
(2.10)
for . Therefore, by Lemma 2.5 there exists positive constants such that
(2.11)

Hence,

(2.12)
for all large and ( ). By Lemma 2.6, ( ) provided that
(2.13)
where is a positive constant. Note that for , then
(2.14)

That is, (2.13) is satisfied if (2.9) holds. Moreover, if (2.9) holds, then and ( ).

Next we give an example to show the feasibility of the conditions of Theorem 2.7. This example also shows that Theorem 2.7 is not enclosed by other related works.

Example 2.8.

Let us consider the following competitive model:
(2.15)

where , , .

From (2.15), , , and hence, (2.9) is satisfied. According to Theorem 2.7, system (2.15) is permanent (see Figure 1).
Figure 1

Permanence of model (2.15): the coefficients are given in Example 2.8.

Remark 2.9.

The permanence of system (2.15) was also investigated in [12]. But our conditions which guarantee the permanence of (2.15) are different from that of [12, Lemma 5]. Adopting the same notations as [12, Lemma 5], we have , , and , , , , , , , , , . But , ; that is, the assumptions and of [12, Lemma 5] are not satisfied. Therefore, the permanence of system (2.15) cannot be obtained by [12, Lemma 5].

Remark 2.10.

The global asymptotical stability of model (1.1) is studied in [12] under the assumption that model (1.1) is strongly persistent. But the authors of [12] did not discuss the strong persistence of model (1.1) with -species ( ). Theorem 2.7 in this paper gives sufficient conditions which guarantee the strong persistence of model (1.1).

3. Periodic Solution

In this section, we assume that the coefficients of model (1.1) are periodic with common period , that is,

(3.1)

The aim of this section is to show the existence of positive periodic solution of model (1.1) under assumption (3.1) and further find additional conditions for the global stability of this positive periodic solution.

Theorem 3.1.

Let the assumptions of Theorem 2.7 and (3.1) be satisfied; then there exists a positive periodic solution of model (1.1) with the period .

Proof.

Model (1.1) is permanent by Theorem 2.7. Therefore, model (1.1) is point dissipative. It follows from [25, Theorem 4.3] that there exists a positive periodic solution of model (1.1). Note (3.1), and the coefficients of model (1.1) are all -periodic. Therefore, this solution is -periodic.

Example 3.2.

Consider the following model:
(3.2)
Direct computation shows that the coefficients of model (3.2) satisfy the assumptions of Theorem 2.7. The coefficients of model (3.2) are all periodic with the common period 20. Hence, model (3.2) has a positive periodic solution by Theorem 3.1 (see Figures 2 and 3). Figures 4 and 5 show that the period of the sequences or is 20, respectively. More precisely, the values of sequence within a period are 0.7330, 0.7358, 0.7383, 0.7403, 0.7415, 0.7419, 0.7413, 0.7399, 0.7377, 0.7351, 0.7323, 0.7295, 0.7271, 0.7252, 0.7241, 0.7238, 0.7243, 0.7257, 0.7277, and 0.7302, respectively. The values of sequence within a period are 0.3869, 0.3846, 0.3821, 0.3796, 0.3775, 0.3759, 0.3750, 0.3749, 0.3756, 0.3770, 0.3790, 0.3813, 0.3838, 0.3862, 0.3882, 0.3898, 0.3907, 0.3908, 0.3902, and 0.3889, respectively.
Figure 2

Periodic solution of model (3.2): the coefficients are given in Example 3.2.

Figure 3

Periodic solution of model (3.2): on the phase plane, the coefficients are given in Example 3.2.

Figure 4

Periodic oscillating of species : the coefficients are given in Example 3.2.

Figure 5

Periodic oscillating of species : the coefficients are given in Example 3.2.

Next, we study the global stability of the positive periodic solution obtained in Theorem 3.1.

Theorem 3.3.

Let the assumptions of Theorem 3.1 be satisfied; further, assume that
(3.3)
i , then for every positive solution of model (1.1), one has
(3.4)

where is the positive periodic solution obtained in Theorem 3.1.

Proof.

Let
(3.5)
where ) is the positive periodic solution of model (1.1). Model (1.1) can be rewritten as
(3.6)
Therefore,
(3.7)

where , , , .

In view of (3.3), we can choose such that

(3.8)

And from Theorem 2.7, there exists a positive integer such that

(3.9)

for and given as above ( , ).

Notice that lies between and , and lies between and ( , ), from (3.7), we have

(3.10)

for and .

Denote

(3.11)
We have . Notice that is arbitrarily given; from (3.10), we get
(3.12)
Therefore,
(3.13)
That is,
(3.14)

and (3.4) follows consequently.

Declarations

Acknowledgments

The research is supported by the foundation of Jiangsu Polytechnic University (ZMF09020020). The author would like to thank the editor Professor Jianshe Yu and the referees for their helpful comments and suggestions which greatly improved the presentation of the paper.

Authors’ Affiliations

(1)
Department of Information Science, Jiangsu Polytechnic University

References

  1. Kocic VK, Ladas G: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993.MATHView ArticleGoogle Scholar
  2. May RM: Nonlinear problems in ecology. In Chaotic Behavior of Deterministic Systems. Edited by: Iooss G, Helleman R, Stora R. North-Holland, Amsterdam, The Netherlands; 1983:515–563.Google Scholar
  3. Lakshmikantham V, Trigiante D: Theory of Difference Equations: Numerical Methods and Applications, Mathematics in Science and Engineering. Volume 181. Academic Press, Boston, Mass, USA; 1988:x+242.Google Scholar
  4. Hastings A: Population Biology: Concepts and Models. Springer, New York, NY, USA; 1996.MATHGoogle Scholar
  5. Goh BS: Management and Analysis of Biological Populations. Elsevier, Amsterdam, The Netherlands; 1980.Google Scholar
  6. Zhou Z, Zou X: Stable periodic solutions in a discrete periodic logistic equation. Applied Mathematics Letters 2003,16(2):165–171. 10.1016/S0893-9659(03)80027-7MATHMathSciNetView ArticleGoogle Scholar
  7. Wang W, Lu Z: Global stability of discrete models of Lotka-Volterra type. Nonlinear Analysis: Theory, Methods & Applications 1999,35(8):1019–1030. 10.1016/S0362-546X(98)00112-6MATHMathSciNetView ArticleGoogle Scholar
  8. May RM: Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos. Science 1974,186(4164):645–647. 10.1126/science.186.4164.645View ArticleGoogle Scholar
  9. Chen Y, Zhou Z: Stable periodic solution of a discrete periodic Lotka-Volterra competition system. Journal of Mathematical Analysis and Applications 2003,277(1):358–366. 10.1016/S0022-247X(02)00611-XMATHMathSciNetView ArticleGoogle Scholar
  10. Lu Z, Wang W: Permanence and global attractivity for Lotka-Volterra difference systems. Journal of Mathematical Biology 1999,39(3):269–282. 10.1007/s002850050171MATHMathSciNetView ArticleGoogle Scholar
  11. Kuang Y: Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering. Volume 191. Academic Press, Boston, Mass, USA; 1993:xii+398.Google Scholar
  12. Wang W, Mulone G, Salemi F, Salone V: Global stability of discrete population models with time delays and fluctuating environment. Journal of Mathematical Analysis and Applications 2001,264(1):147–167. 10.1006/jmaa.2001.7666MATHMathSciNetView ArticleGoogle Scholar
  13. Saito Y, Ma W, Hara T: A necessary and sufficient condition for permanence of a Lotka-Volterra discrete system with delays. Journal of Mathematical Analysis and Applications 2001,256(1):162–174. 10.1006/jmaa.2000.7303MATHMathSciNetView ArticleGoogle Scholar
  14. Hutson V, Schmitt K: Permanence and the dynamics of biological systems. Mathematical Biosciences 1992,111(1):1–71. 10.1016/0025-5564(92)90078-BMATHMathSciNetView ArticleGoogle Scholar
  15. Jansen VAA, Sigmund K: Shaken not stirred: on permanence in ecological communities. Theoretical Population Biology 1998,54(3):195–201. 10.1006/tpbi.1998.1384MATHView ArticleGoogle Scholar
  16. Freedman HI, Ruan SG: Uniform persistence in functional-differential equations. Journal of Differential Equations 1995,115(1):173–192. 10.1006/jdeq.1995.1011MATHMathSciNetView ArticleGoogle Scholar
  17. Hofbauer J, Kon R, Saito Y: Qualitative permanence of Lotka-Volterra equations. Journal of Mathematical Biology 2008,57(6):863–881. 10.1007/s00285-008-0192-0MATHMathSciNetView ArticleGoogle Scholar
  18. Cui J, Takeuchi Y, Lin Z: Permanence and extinction for dispersal population systems. Journal of Mathematical Analysis and Applications 2004,298(1):73–93. 10.1016/j.jmaa.2004.02.059MATHMathSciNetView ArticleGoogle Scholar
  19. Takeuchi Y, Cui J, Miyazaki R, Saito Y: Permanence of delayed population model with dispersal loss. Mathematical Biosciences 2006,201(1–2):143–156. 10.1016/j.mbs.2005.12.012MATHMathSciNetView ArticleGoogle Scholar
  20. Zhang L, Teng Z: Permanence for a delayed periodic predator-prey model with prey dispersal in multi-patches and predator density-independent. Journal of Mathematical Analysis and Applications 2008,338(1):175–193. 10.1016/j.jmaa.2007.05.016MATHMathSciNetView ArticleGoogle Scholar
  21. Allen LJS: Persistence and extinction in single-species reaction-diffusion models. Bulletin of Mathematical Biology 1983,45(2):209–227.MATHMathSciNetView ArticleGoogle Scholar
  22. Muroya Y: Global attractivity for discrete models of nonautonomous logistic equations. Computers & Mathematics with Applications 2007,53(7):1059–1073. 10.1016/j.camwa.2006.12.010MATHMathSciNetView ArticleGoogle Scholar
  23. Itokazu T, Hamaya Y: Almost periodic solutions of prey-predator discrete models with delay. Advances in Difference Equations 2009, 2009:-19.Google Scholar
  24. Yang X: Uniform persistence and periodic solutions for a discrete predator-prey system with delays. Journal of Mathematical Analysis and Applications 2006,316(1):161–177. 10.1016/j.jmaa.2005.04.036MATHMathSciNetView ArticleGoogle Scholar
  25. Hale J: Theory of Functional Differential Equations, Applied Mathematical Sciences. 2nd edition. Springer, New York, NY, USA; 1977:x+365.View ArticleGoogle Scholar

Copyright

© Chunqing Wu. 2009

This article is published under license to BioMed Central Ltd. 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.