Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 1893181, 14 pages
https://doi.org/10.1155/2018/1893181

Research Article

Global Attractivity and Extinction of a Discrete Competitive System with Infinite Delays and Single Feedback Control

Yalong Xue,¹ Xiangdong Xie,¹ Qifa Lin,¹ and Fengde Chen²

¹College of Mathematics and Physics, Ningde Normal University, Ningde, Fujian 352100, China
²College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350108, China

Correspondence should be addressed to Xiangdong Xie; ndsyxxd@163.com

Received 25 March 2018; Accepted 20 May 2018; Published 25 June 2018

Academic Editor: Zhengqiu Zhang

Copyright © 2018 Yalong Xue 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.

Abstract.

A nonautonomous discrete two-species competition system with infinite delays and single feedback control is considered in this paper. Based on the discrete comparison theorem, a set of sufficient conditions which guarantee the permanence of the system is obtained. Then, by constructing some suitable discrete Lyapunov functionals, some sufficient conditions for the global attractivity and extinction of the system are obtained. It is shown that, by choosing some suitable feedback control variable, one of two species will be driven to extinction.

1. Introduction

Two or more species compete for the same limited food source or in some way inhibit each other’s growth. For example, competition may be for territory which is directly related to food resources. The importance of species competition in nature is obvious. Tradition two-species Lotka-Volterra competition system is as follows:

However, system (1) has a property which is considered as a disadvantage and that is the linearity of the above system. Ayala et al. [1] presented the following nonlinear competitive system with continuous time version:

Assume that each species needs some time to mature and the competition occurs after some time lag required for maturity of the species; Gopalsamy [2] discussed the following system with discrete delays:

Such systems are not well studied in the sense that most results are continuous time cases related (see [3, 4]). As we know, a discrete time system governed by difference equations is more approximate than the continuous ones when the populations have nonoverlapping generations or a short-life expectancy. Discrete time system can also provide efficient computation for numerical simulations (see [5–10]). Considering the biological parameters naturally being subject to almost periodic fluctuation in time, Tan and Liao [11] established the following nonautonomous discrete competition system:

In the real world, ecosystems are disturbed by unpredictable forces which can result in some changes of parameters. In order to accurately describe such a system, scholars introduced feedback control into ecosystems. Recently, the ecosystems with feedback controls have been extensively studied and obtained many interesting results (see [12–18]), noting that models in [10, 12–15] considered at least two feedback controls variables, which means that, for the different species, different control strategy is adopted, whereas, in the real world, the strategy adopted for one species may also affect the other species. For example, spraying pesticide not only can reduce the number of weeds but also have a negative impact on the growth of corps or beneficial animals [19]. Therefore, how to keep these negative effects caused by feedback controls to a minimum? One strategy is to reduce the number of feedback controls like [16–18]. Motivated by the above, in this paper, we study the following discrete competitive system with delays and single feedback control:

In system (5), is the density of species at the th generation and is the single feedback control variable.

Throughout this paper, we assume the following.

: for any nonnegative bounded sequence defined on , we use the notations and .

: , , , , , and are bounded nonnegative sequences of real numbers defined on such that

: , , , and are nonnegative bounded sequences such that

We consider the solution of system (5) with the following initial conditions:

where and . One can easily show that the solutions of (5) with initial condition (7) are defined and remain positive for all .

The remaining part of this paper is organized as follows. We introduce some useful lemmas in Section 2 and then state and prove the main results in Sections 3, 4, and 5, respectively. Two examples together with their numeric simulations are presented to show the feasibility of the main results in Section 6.

2. Preliminaries

This section is concerned with some lemmas which will be used for our main results. Consider the following difference equation:

where , are positive constants.

Lemma 1 (see [20]). Assume that , and, for any initial value , there exists a unique solution of (9), which can be expressed as follows:

where . Thus, for any solution of (10), we have

Lemma 2 (see [20]). Let and . For any fixed , is nondecreasing function with respect to , and, for , the following inequalities hold:

If , then for all .

Lemma 3 (see [5]). Assume that , satisfies , and

for , where is a positive constant. Then

Lemma 4 (see [5]). Assume that , satisfies , and

for , , and , where and are positive constants such that . Then

Similarly, according to the proof of lemma 2.3 in [5], we have the following lemma.

Lemma 5. Let be a nonnegative bounded sequences, and let be a nonnegative sequence such that . For any fixed , then

If holds, then

Proof. Let and . Given , let be an integer such that, for all ,

Therefore, for all ,

Then

Setting , we have

Let . If , the result is trivial. If , then, given , there exists an integer such that, for all ,

Therefore, for all ,

Then

Setting , we have

If , Given , there exists an integer such that, for all , , therefore, , and then . Setting , we have .
If and if , the result is trivial. If , given , there exists an integer such that, for all , , therefore, , and then . Setting , we have .
This ends the proof of Lemma 5.

3. Permanence

Concerned with the persistent property of system (5), we have the following result.

Theorem 6. Assume that

holds, and, then, for any positive solution of system (5), we have

where

Proof. From the first and second equations of system (5), we have

And so, from Lemma 3, for any solution of system (5), we can obtain

According to Lemma 5 and the above inequality, for , one has

For any , there exists a positive integer such that, for all ,

By the third equation of system (5) and (33), we have

Hence, by applying Lemma 1 and Lemma 2 to (34), we obtain

Setting , it follows that

According to Lemma 5 and the above inequality,

Condition (27) implies that, for enough small positive constant , the following inequalities hold:

It follows from (37) that there exists a positive integer such that, for all ,

Thus, for all , from (33), (39), and the first two equations of system (5), we have

where for , . Noting the fact that , for , we have

and then

Hence, according to Lemma 4,

Setting , it follows that

where for .
According to Lemma 5, from (44) we have that, for any small enough (without loss of generality, assume that ), there exists an , such that

For , from (45) and the last equation of system (5), we have

Hence, by applying Lemmas 1 and 2 to (44),

Setting , it follows that

This ends the proof Theorem 6.

4. Global Attractivity

Concerned with the stability property of system (5), we have the following result.

Theorem 7. Assume that there exist positive constants , , and , such that

hold; then, for any two positive solutions and of system (5), we have

where

Proof. By (49), we can choose enough small positive constants and such that

where

Let be any positive solution of system (5). For the above , from (31) and (36), there exists an enough large integer , such that

Now, let us define a Lyapunov functional

where , are positive constants and

Then, from the definition of , , one can easily see that for all . Also, for any fixed ,

Also, from the first equation of system (5) and using the Mean Value Theorem, for all ,

Similarly to the analysis of (58), we can obtain

where lies between and , .
From (58) and (59), we have

Summating both sides of the above inequalities from to ,

Hence,

Then, we have

Therefore

which means that

Consequently

This completes the proof of Theorem 7.

5. Extinction

Concerned with the extinction property of system (5), we have the following results.

Theorem 8. Assume that

hold, let be any positive solution of system (5), and then

where is defined in Theorem 6.

Theorem 9. Assume that

hold, let be any positive solution of system (5), and then

where is defined in Theorem 6.

Proof of Theorem 8. By condition (67), we can choose positive constants and such that

Thus, there exists a positive constant such that

There exists a constant such that Thus, for enough small positive constant , we have

Consider the following Lyapunov functional:

From (74), we obtain

From (73) and (75), we can obtain

Therefore,

From (31) and (36) we know that there exists an such that

and so

On the other hand, we also have

Combining inequalities (77), (79), and (80),

where

Hence we obtain that

This ends the proof of Theorem 8.

Proof of Theorem 9. Define the following Lyapunov functional:

Similarly to the analysis of the proof of Theorem 8, we have .
From Theorems 8 and 9 we know that, under some suitable assumption, one of the species in the system may be driven to extinction; in this case, one interesting problem is to investigate the stability property of the rest of the species.
Consider the following discrete equations:

Theorem 10. Assume that (67) holds and also

holds; then, for any positive solution of system (5) and any positive solution of system (85), we have

where is defined in Theorem 7.

Theorem 11. Assume that (69) holds and also

holds; then, for any positive solution of system (5) and any positive solution of system (86), we have

where is defined in Theorem 7.

Proof of Theorem 10. By condition (87), we can choose positive constants and such that

Thus, there exist enough small positive constants and such that

where is defined in (53).
From (81), we have

By applying the Direct Comparison Test to (81) and (93), we obtain and are absolute convergence.
Now, we define a Lyapunov functional:

and one could easily see that for all . Also, for any fixed , from (94) one could see that

It follows from system (5) and (85) and the Mean Value Theorem that

Summating both sides of the above inequality from to , we have

Hence

Then, from (95) we have

Therefore

which means that

Consequently

This completes the proof of Theorem 10.

Proof of Theorem 11. The proof of Theorem 11 is similar to that of Theorem 10, and we omit the details here.

6. Numerical Simulations

In this section, we give an example to check the feasibility of our result.

Example 12. Consider the following system:

One could easily see that conditions , , and are satisfied. Also, by calculating, one has

Now, let us take , , and , and then

Clearly, condition (49) is satisfied, and so from Theorem 7 we have and , where and are any two positive solutions of system (103).
Figure 1 shows the dynamic behaviors of system (103), which strongly supports the above assertions.

Figure 1: Dynamic behaviors of the solutions of system (103) with the initial conditions for , respectively.

Example 13. Consider the following system:

One could easily see that conditions , , and are satisfied. Also, by calculating, one has

Clearly, conditions (67) and (87) are satisfied, and so from Theorems 8 and 10 we know that will be driven to extinction, while species is globally attractive.

Figure 2 shows the dynamic behaviors of system (106), which strongly supports our results.

Figure 2: Dynamic behaviors of the solutions of system (106) with the initial conditions for , respectively.

7. Discussion

During the past decade, many scholars investigated the dynamic behaviors of the feedback control ecosystem. However, by using a feedback control variable to control all the species, it is much difficult. In this paper, we focused our attention on the nonlinear competition system with single feedback control, and the dynamic behavior of the system is investigated. The study shows that the feedback control variable plays a crucial role in both of global attractivity and partial extinction. i.e., under some suitable conditions the two species can survive well, but under other conditions one of two species will be driven to extinction.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The research was supported by the Special Project for Young Teachers in Ningde Normal University (2016Q35).

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