Permanence and Global Stability of Positive Periodic Solutions of a Discrete Competitive System
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.
In biomathematics, one of the most challenging aspects of mathematical biology is competition modeling. Although the mathematical idea is simple , 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. Lotka-Volterra competitive model of two species communities is probably the best known model of mathematical ecology. In 1934, Gause  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 Lotka-Volterra competitive system are obtained (see [3–16]).
Although much progress has been seen for Lotka-Volterra 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 discrete-time 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. Discrete-time models can provide efficient computational models of continuous models for numerical simulations. So it is reasonable to study discrete-time competitive systems governed by difference equations.
In this paper, we will consider the dynamic behavior of a discrete-time competitive system. Let us first introduce its continuous time version which is motivated in  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 , 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 discrete-time 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.
In this section, we will establish sufficient conditions for the permanence of system (1.2).
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.
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.
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,
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.
Now, we give the main result in this section.
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 mean-value 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.
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).
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.
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.
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