Research Article  Open Access
Permanence and Global Stability of Positive Periodic Solutions of a Discrete Competitive System
Abstract
We consider the dynamic behaviors of a discrete competitive system. A good understanding of the permanence, existence, and global stability of positive periodic solutions is gained. Numerical simulations are also presented to substantiate the analytical results.
1. Introduction
In biomathematics, one of the most challenging aspects of mathematical biology is competition modeling. Although the mathematical idea is simple [1], this type of modeling is so difficult to carry out in any generality since there are so many ways for a population to compete; many classical competitive models have been established to describe the relationships between species and the outer environment, and the connections between different species. LotkaVolterra competitive model of two species communities is probably the best known model of mathematical ecology. In 1934, Gause [2] found out the competitive exclusion theory, which states that two species that compete for the exact same resources cannot stably coexist. One of the two competitors will always have an ever so slight advantage over the other that leads to extinction of the second competitor in the long run. Since then the competitive model has increasingly won attention as an important and fundament model in biomathematics. The dynamic relationship between species and their competitors has long been one of the dominant theses in both ecology and mathematical ecology. As a consequence, many excellent results concerned with permanence, extinction, stability and hopf bifurcations, and existence and global stability of positive periodic solutions of LotkaVolterra competitive system are obtained (see [3–16]).
Although much progress has been seen for LotkaVolterra competitive systems, such systems are not well studied in the sense that most results are continuous time versions related. Many authors [17–21] have argued that the discretetime models governed by difference equations are more appropriate than the continuous ones when populations have a short life expectancy, nonoverlapping generations in the real word. Discretetime models can provide efficient computational models of continuous models for numerical simulations. So it is reasonable to study discretetime competitive systems governed by difference equations.
In this paper, we will consider the dynamic behavior of a discretetime competitive system. Let us first introduce its continuous time version which is motivated in [22] where are the population densities of two competing species; are the intrinsic growth rates of species; are the rates of intraspecific competition of the first and second species, respectively; are the rates of interspecific competition of the first and second species, respectively. All the coefficients above are continuous and bounded above and below by positive constants.
Following the same idea and method in [21], one can easily derive the discrete analogue of system (1.1), which takes the form of The exponential form of system (1.2) is more biologically reasonable than that directly derived by replacing the differential by difference in system (1.1) because this exponential form can assure if . Here represent the densities of species at the th generation, are the intrinsic growth rates of species at the th generation, measure the intraspecific effects of the th generation of species on own population, and stand for the interspecific effects of the th generation of species on species .
It is well known that, compared to the continuous time systems, the discretetime ones are more difficult to deal with. The principle aim of this paper is to explore the permanence, existence, and global stability of positive periodic solutions of system (1.2). To the best of our knowledge, no work has been done for system (1.2).
For the sake of simplicity and convenience in the following discussion, the notations below will be used through this paper: where is a bounded sequence and is the set of nonnegative integer numbers.
For biological reasons, in system (1.2) we only consider the solution with the initial value .
The organization of this paper is as follows. In the next section, we establish the permanence of system (1.2). In Section 3, we obtain sufficient conditions which ensure the existence and global stability of positive periodic solutions of system (1.2). Numerical simulations are present to illustrate the feasibility of our main results in final section.
2. Permanence
In this section, we will establish sufficient conditions for the permanence of system (1.2).
Definition 2.1. System (1.2) is said to be permanent if there exist positive constants and such that each positive solution of system (1.2) satisfies
Proposition 2.2. Any positive solution of system (1.2) satisfies
Proof. To prove Proposition 2.2, we consider Case 1 and Case 2.
Case 1. Assume that there exists an such that , from the first equation of system (1.2), it follows that
which implies,
Then
where we use the fact that , and is the set of all real numbers. Hence .
We claim that for all . By way of contradiction, assume that there exists a such that , then . Let
that is, and , then . It is easy to obtain that from the above argument, which is a contradiction. Therefore, for all , then . This proves the claim.
Case 2. Suppose that for all . In particular, exists, denoted by , we will prove by way of contradiction as follows. Assume that , taking limit in the first equation of system (1.2), which leads to
however,
which is a contradiction. This proves the claim. By the fact that , we obtain that . Therefore,
Analogously,
This completes the proof of Proposition 2.2.
Proposition 2.3. Suppose that system (1.2) satisfies the following assumptions: Then any positive solution of system (1.2) satisfies
Proof. By Proposition 2.2, since for each , then there exists an such that for . In order to prove Proposition 2.3, there are two cases to be considered as follows.
Case 1. Assume that there exists an such that , by the first equation of system (1.2), it derives that
therefore,
It follows from the inequality (2.11) that
By (2.13) and (2.15) we have
Hence , where .
In the following we will prove for all . By way of contradiction, assume that there exists a such that , then . Let
that is, and , then , the above argument produces that , which is a contradiction. Therefore, , since can be sufficiently small, it gives that
then . This proves the claim.
Case 2. Assume that for a sufficiently large . In this case, exists, denoted by . For the sake of proving , by way of contradiction, assume that , taking limit in the first equation of system (1.2), it follows that
however,
which is a contradiction. It implies that . By the fact that , we obtain that . Thus . Therefore, . Since can be sufficiently small, we have
Analogously, by the second inequality in (2.11), we can obtain that
This completes the proof of Proposition 2.3.
Now, we are in a position to state Theorem 2.4 whose proof is a direct consequence of Propositions 2.2 and 2.3.
Theorem 2.4. If the inequalities in (2.11) hold, then system (1.2) is permanent.
3. Existence and Global Stability of Positive Periodic Solutions
In this section, we suppose system (1.2) is a periodic system, and then we investigate the existence and global stability of positive periodic solutions of such system. To do this, assume that all the coefficients of system (1.2) are periodic, in other words,
Theorem 3.1. If the inequalities in (2.11) hold, then system (1.2) has at least one strictly positive periodic solution, denoted by .
Proof. We know that is an invariant set of system (1.2) from Propositions 2.2 and 2.3. Define the continuous mapping on Obviously, depends continuously on , then is continuous and maps the compact set into itself. Therefore, has a fixed point . It is easy to see that the solution which passes through is an periodic solution of system (1.2). The proof is complete.
Next, we derive sufficient conditions which guarantee that the positive periodic solution of system (1.2) is globally stable. We first give the definition of global stability.
Definition 3.2. A positive periodic solution of system (1.2) is globally stable if each other solution of system (1.2) with positive initial value defined for all satisfies
Now, we give the main result in this section.
Theorem 3.3. In addition to (2.11), assume further that the following assumptions hold. Then the positive periodic solution of system (1.2) is globally stable.
Proof. Let be a positive periodic solution of system (1.2). We make the change of variables and , then system (1.2) is rewritten as
By the meanvalue theorem, it derives that
where the constants
Now, by (3.4) and (3.5), we choose the constant sufficiently small such that
In view of Propositions 2.2 and 2.3, there exists an such that , we have
Since , both and are between and . Meanwhile, both and are between and . From the first equation of system (3.7), it follows that
Similarly, it follows from (3.5) that
Denote then . Therefore, when ,
as a consequence, By using Definition 3.2, it follows that the positive periodic solution of system (1.2) is globally stable. This completes the proof.
Remark 3.4. Theorem 3.3 shows that is the global attractor of all positive solutions of system (1.2), then is the unique positive periodic solution of system (1.2).
4. Example and Numerical Simulation
In this paper, we have investigated the permanence and global stability of positive periodic solutions of a discrete competitive system. Each species is not isolated from its living environment, but competes with the other for the same resource. Sufficient conditions which guarantee the permanence, existence and global stability of positive periodic solutions are established, respectively. The theoretical results are confirmed by the following examples and their numerical results.
To verify the sufficient conditions for permanence of system (1.2), we assume that , , , , , , and the initial condition . Clearly, (2.11) in Theorem 2.4 are satisfied, and hence system (1.2) is permanent (see Figure 1).
(a)
(b)
Now, we further verify the sufficient conditions for the existence and global stability of positive periodic solutions of periodic system (1.2). Let us assume that all the coefficients of system (1.2) are periodic and listed in Table 1.

Besides, we choose the positive periodic solution with initial values , denoted by , and the positive solution with initial value denoted by . By Theorem 3.3, a simple calculation shows that the assumptions in (3.4) and (3.5) hold. So from Theorem 3.3 we know that periodic system (1.2) has a positive periodic solution which is globally stable. From Figures 2(a), 2(b), and 2(c), we see that with will tend to with . Similarly, from Figures 3(a), 3(b), and 3(c), we see that with will tend to with . Furthermore, Figures 4(a) and 4(b) show the phase portrait of periodic system (1.2) with for and for , respectively. From Figure 4(c), we can see that periodic system (1.2) has a positive periodic solution which is globally stable.
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
5. Acknowledgement
The work is supported by the Innovation Term of Educational Department of Hubei Province in China (T200804), the National Science Foundation of Hubei Province in China (2008CDB068), and the Innovation Project of Hubei Institute for Nationalities for postgraduate students. We would like to thank the Editor Professor A. Vecchio and the referee for careful reading of the original manuscript and valuable comments and suggestions that greatly improved the presentation of this work.
References
 H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, vol. 13 of Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, UK, 1995. View at: Zentralblatt MATH  MathSciNet
 G. F. Gause, The Struggle for Existence, Williams and Wilkins, Baltimare, Md, USA, 1934.
 B. Liu and L. Chen, “The periodic competing LotkaVolterra model with impulsive effect,” Mathematical Medicine and Biology, vol. 21, no. 2, pp. 129–145, 2004. View at: Google Scholar
 S. Ahmad, “On the nonautonomous VolterraLotka competition equations,” Proceedings of the American Mathematical Society, vol. 117, no. 1, pp. 199–204, 1993. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. Ahmad and F. Montes de Oca, “Average growth and extinction in a two dimensional LotkaVolterra system,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 9, no. 2, pp. 177–186, 2002. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 J. C. Eilbeck and J. LópezGómez, “On the periodic LotkaVolterra competition model,” Journal of Mathematical Analysis and Applications, vol. 210, no. 1, pp. 58–87, 1997. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 B. Lisena, “Competitive exclusion in a periodic LotkaVolterra system,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 761–768, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 F. Montes de Oca and M. Vivas, “Extinction in two dimensional LotkaVolterra system with infinite delay,” Nonlinear Analysis: Real World Applications, vol. 7, no. 5, pp. 1042–1047, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. Tineo, “Necessary and sufficient conditions for extinction of one species,” Advanced Nonlinear Studies, vol. 5, no. 1, pp. 57–71, 2005. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 Y. Song, M. Han, and Y. Peng, “Stability and Hopf bifurcations in a competitive LotkaVolterra system with two delays,” Chaos, Solitons & Fractals, vol. 22, no. 5, pp. 1139–1148, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 X. H. Tang and X. Zou, “Global attractivity of nonautonomous LotkaVolterra competition system without instantaneous negative feedback,” Journal of Differential Equations, vol. 192, no. 2, pp. 502–535, 2003. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Z. Y. Lu and Y. Takeuchi, “Permanence and global attractivity for competitive LotkaVolterra systems with delay,” Nonlinear Analysis: Theory, Methods & Applications, vol. 22, no. 7, pp. 847–856, 1994. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 Z. Teng, “On the nonautonomous LotkaVolterra $n$species competing systems,” Applied Mathematics and Computation, vol. 114, no. 23, pp. 175–185, 2000. View at: Publisher Site  Google Scholar  MathSciNet
 Y. Li and Y. Kuang, “Periodic solutions of periodic delay LotkaVolterra equations and systems,” Journal of Mathematical Analysis and Applications, vol. 255, no. 1, pp. 260–280, 2001. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Z. Liu, R. Tan, and Y. Chen, “Modeling and analysis of a delayed competitive system with impulsive perturbations,” The Rocky Mountain Journal of Mathematics, vol. 38, no. 5, pp. 1505–1523, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 M. Fan, K. Wang, and D. Jiang, “Existence and global attractivity of positive periodic solutions of periodic $n$species LotkaVolterra competition systems with several deviating arguments,” Mathematical Biosciences, vol. 160, no. 1, pp. 47–61, 1999. View at: Publisher Site  Google Scholar  MathSciNet
 R. M. May, Stability and Complexity in Model Ecosystems, Princeton Unversity Press, Princeton, NJ, USA, 1974.
 H. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980. View at: MathSciNet
 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
 J. D. Murray, Mathematical Biology, vol. 19 of Biomathematics, Springer, Berlin, Germany, 1989. View at: MathSciNet
 M. Fan and K. Wang, “Periodic solutions of a discrete time nonautonomous ratiodependent predatorprey system,” Mathematical and Computer Modelling, vol. 35, no. 910, pp. 951–961, 2002. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. View at: MathSciNet
Copyright
Copyright © 2009 Wenjie Qin 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.