Research Article | Open Access
Positive Periodic Solutions of a Periodic Discrete Competitive System Subject to Feedback Controls
Species living in a fluctuating medium and human exploitation activities might result in the duration of continuous changes. Such changes can be well-approximated as feedback controls. In this contribution a periodic discrete competitive system subject to feedback controls is proposed. By using the methods of discrete inequality, fixed point theorem, and analysis techniques, a good understanding of the existence and global asymptotic stability of positive periodic solutions is obtained. Some numerical investigations are provided to verify our analytical results.
Many mathematical models about bioecology play important roles for researchers to realize the interactions of ecological species. There has been increasing interest in studying the dynamical behaviors such as stability, permanence, and periodicity of competitive systems (see [1–5]). Recently, motivated by an autonomous competitive model of  in  we introduced a corresponding nonautonomous version where and can be interpreted as the density of two competing species at time , respectively. stand for the growth rates of species, and represent the effects of intraspecific competition, and are the rates of interspecific competition. Furthermore, we took the influence of almost periodic environment and impulsive perturbations into account and established sufficient conditions for the uniformly asymptotic stability of a unique positive almost periodic solution for the above system (for details see ).
Note that ecosystems in the real world are often disturbed by outside continuous forces which can lead to changes in biological coefficients such as survival rates. In the language of control, we call the disturbance functions as feedback control variables. Many good results on this direction are deliberated (see [8–12]) and some similar works on the topic have been done (see [13, 14]). This paper is concerned with a discrete model and is a continuation of the work in ; in this contribution we search for certain schemes (such as harvesting procedure) to ensure the system coexists under appropriate conditions. For this reason, we consider a discrete version which corresponds to differential version (i.e., system (1)), meanwhile, replacing abrupt external perturbations (impulses) in  by continuous external perturbations (feedback control variables), a nonautonomous discrete controlled system can be described as follows: Here , . are the first-order forward difference operators, stand for the densities of species at the th generation, represent the natural growth rates of species at the th generation, and stand for the intraspecific effects of the th generation of species on own population, and measure the interspecific effects of the th generation of species on species . The coefficients , , , , , , , and are -periodic sequences with . is the set of nonnegative integers.
It is well known that the discrete models governed by difference equations may be more appropriate than the continuous ones when populations have a short life expectancy or nonoverlapping generations. Also, discrete models can provide efficient computational methods of continuous models for numerical simulations. A very important ecological problem associated with the study of multispecies population interaction in a periodic environment is the positive periodic solution which plays the role played by the equilibrium of the autonomous model. In this paper, we will discuss the above discrete system (2) and focus on the existence and stability of positive periodic solution of system (2). To the best of our knowledge, there are few published papers concerning system (2).
The rest structure of this paper is as follows. In the next section, we establish sufficient conditions for the existence of positive periodic solutions. In Section 3, we further discuss the global asymptotic stability of positive periodic solutions. In Section 4, we carry out an example and its numerical simulations to substantiate our theoretical results. To simplify the reading of paper, we give the proofs of lemmas and theorems in appendices.
In this section, we establish sufficient conditions for the existence of positive periodic solutions of system (2). To do this, we first give two preliminary lemmas. For convenience, given a bounded sequence , let and be defined as , , and
Lemma 1. Every positive solution of system (2) satisfies and .
Lemma 2. Assuming that hold, then every positive solution of system (2) satisfies and .
3. Global Asymptotic Stability
In this section, we further investigate the stability of positive periodic solutions of system (2). Denote
4. An Example
In this section we give a numerical example and its corresponding simulations.
Example 1. Consider a discrete competitive system with feedback controls A computation shows that Furthermore, it follows from (3) that Obviously, the assumptions in (4) are satisfied, and moreover, one has so the assumptions of Theorem 4 are also satisfied. Thus, there exists a globally asymptotically stable positive periodic solution of system (6). Figure 1 shows that system (6) exists a positive periodic solution, and the two-dimensional phase portrait of periodic system (6) is displayed in Figure 2. From Figure 3, we can see that any positive solution tends to the periodic solution , and the two-dimensional phase portrait reflects the fact in Figure 4.
A. Proof of Lemma 1
We consider the following Cases I and II to show that Case I. Suppose that there is a such that , it follows from the first equation of system (2) that Here we use the fact that for and is the set of all positive real numbers. Thus .
One claims that for all . Otherwise, if there is a such that , then . Set namely, and ; then . From the above discussion one obtains that , which is a contradiction. Hence, for all ; then . Case I is complete.
Case II. Assume that for . especially exists, denoted by . One claims that . Otherwise, if , then it follows from the first equation of system (2) that which is a contradiction since This proves the claim. By and the fact that , one can show that Hence . We can prove that in the same way.
In the following, we prove that . For any , there exists an integer such that for all . We have from the third equation of system (2) that where . Since , we can choose a constant such that . By using the Stolz theorem, one has, as , So . Since is arbitrary, we obtain that .
Repeating a similar argument, we can prove that . The proof of Lemma 1 is complete.
B. Proof of Lemma 2
In view of Lemma 1, we obtain that there exist and such that, for , The first inequality of (4) implies that Similar to Lemma 1, we consider the following Cases I and II to show that Case I. Suppose that there is a nonnegative integer such that ; we have from the first equation of system (2) that and hence Note that (B.2) and is sufficiently small; one has from (B.5) that Furthermore, it follows from (B.4) and (B.6) that Hence , where
Now, we claim that for all . Otherwise, if there exists a such that , then . Set then . The above argument produces that , which is a contradiction. Thus for all . Since is sufficiently small, this reduces to Hence, . This verifies Case I.
Case II. We assume that for all large . In this case, exists, denoted by . One claims that ; otherwise, assume that . It follows from the first equation of system (2) that however, which is a contradiction. This fact implies that . Since can be sufficiently small, one yields , where is defined as (B.10).
Similarly, one can derive that .
In the following, we prove that . For any , there exists an integer such that for . We have from the third equation of system (2) that where . Since , we can choose a constant such that ; then by Stolz’s theorem one yields Then . Since can be sufficiently small, we have Analogously, The proof of Lemma 2 is complete.
C. Proof of Theorem 3
It follows from Lemmas 1 and 2 that is an invariant set of system (2). Define the continuous mapping on by for . Clearly, depends continuously on ; then is continuous and maps the compact set into itself. Hence, has a fixed point . It is easy to see that the solution which passes through is a -periodic solution of system (2). The proof of Theorem 3 is complete.
D. Proof of Theorem 4
Let be positive periodic solution and let be any positive solution of system (2), respectively. Let us make the change of variables as follows: By the mean-value theorem, we have from the first equation of system (2) that Similarly, we can arrive at the following desired results: where , and lie between and and , , , and lie between and .
By the inequalities in (5), we will now select a constant small enough such that From Lemmas 1 and 2, we know that there exists a such that, for , It is easy to see that , and are between and . Meanwhile, , and are between and . It follows from the equation of (D.2) that Analogously, Let ; then . Hence, when we have which implies So the positive periodic solution of system (2) is globally asymptotically stable. The proof of Theorem 4 is complete.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the referees for their valuable comments and suggestions that greatly improved the presentation of this work. The work is supported by the National Natural Science Foundation of China (no. 11261017), the Key Project of Chinese Ministry of Education (no. 212111), the Key Laboratory of Biological Resources Protection and Utilization of Hubei Province (PKLHB1329), and the Key Subject of Hubei Province (Forestry).
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