Global Attractivity and Extinction of a Discrete Competitive System with Infinite Delays and Single Feedback Control
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.
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.  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  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  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 . 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.
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 2 (see ). Let and . For any fixed , is nondecreasing function with respect to , and, for , the following inequalities hold: If , then for all .
Lemma 3 (see ). Assume that , satisfies , and for , where is a positive constant. Then
Lemma 4 (see ). Assume that , satisfies , and for , , and , where and are positive constants such that . Then
Similarly, according to the proof of lemma 2.3 in , 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.
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.
Concerned with the extinction property of system (5), we have the following results.
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