Dynamic Behaviors of a Discrete Lotka-Volterra Competition System with Infinite Delays and Single Feedback Control

and Applied Analysis 3 In the medical system, when doctor takes chemotherapeutic drugs as tools to cure the cancer patients, cancer cells will decrease rapidly, but at the same time, drugs also do harm to normal cells and body’s regulatory immune function. Yao et al. [8] studied the effect of chemotherapeutic drugs on cellular immunity in patients with lung cancer; they found that cell immunity is inhibited in patients with lung cancer; moreover, it is impaired considerably by chemotherapy. So how to keep the negative effect caused by the single strategy adopted for the weeds or cancer cells to a minimum? The above phenomenons motivated us to propose and study the discrete Lotka-Volterra competition system with infinite delays and single feedback control variable as follows:


Introduction
During the last decade, the study of the dynamic behaviors of discrete time models governed by difference equation has become one of the most important topics in mathematics biology; many interesting results concerned with permanence, extinction, and existence of positive periodic solution (almost periodic solution) and so forth have been extensively studied by many scholars; see  and the references cited therein.
As far as two-species discrete competition model is concerned, Chen and Zhou [1] proposed and studied the following discrete two-species Lotka-Volterra system: 2 ( + 1) where   (),  = 1, 2 represent the environmental carrying capacity of species  1 and  2 , respectively;   (),  = 1, 2 are the intrinsic growth rate of two species; and  1 (),  2 () represent the density of species  1 and  2 at the th generation, respectively.The authors obtained a set of sufficient conditions which ensure the persistence of system (1).Also, for the periodic case, they gave a set of sufficient conditions which guarantee the existence of a globally stable periodic solution of the system.Chen [2] argued that it is more realistic to incorporate delays into system (1), and he proposed and investigated the following model: hold; then system (2) is permanent.
However, the author did not investigate the stability and extinction property of the system (2), which are two of the most important topics on the study of population dynamics.
On the other hand, it is well known that, in the real world, ecosystems are continuously disturbed by unpredictable forces which can result in some changes of the biological parameters such as survival rates ( [3]).For having a more accurate description of such a system, scholars introduced feedback controls into ecosystems and studied a variety of systems with feedback controls.Based on the work of Chen and Zhou [1], X. X. Chen and F. D. Chen [4] proposed and studied the following nonautonomous two-species discrete competitive system with feedback controls: Some sufficient conditions for the persistence and global stability of system (3) were obtained.Xu et al. [5] further considered the following two-species nonautonomous Lotka-Volterra competitive system with delays and feedback controls: By using the comparison theorem of discrete differential equation and constructing a suitable discrete type Lyapunov functional, they obtained new sufficient conditions on the permanence of species and global attractivity for system (4).Their results show that feedback controls are harmless to the permanence of system (4); that is, feedback controls have no influence on the permanence of system (4).X. Chen and F. Chen [29] and Liao et al. [30] also proposed a discrete time periodic -species Lotka-Volterra competition system with feedback controls and deviating arguments; some sufficient conditions which ensure the existence of unique globally asymptotically stable periodic solution were obtained.Recently, Wu and Zhang [19] proposed a discrete autonomous Lotka-Volterra competition system with infinite delays and feedback controls; by using the iterative method, sufficient conditions which ensure the global attractivity of the system were obtained.
As we can see, those models considered in [4,5,19,29,30] contain two or more feedback control variables, which means that, for the different species, different control strategy is adopted.But, in the real world, the strategy adopted for one species may also affect the other species; in other words, such a strategy has influence on both species.For instance, in the agricultural system, spraying pesticide can reduce the number of weeds, but pesticide can also have a negative impact on the growth of crops or beneficial animals [6,7].In the medical system, when doctor takes chemotherapeutic drugs as tools to cure the cancer patients, cancer cells will decrease rapidly, but at the same time, drugs also do harm to normal cells and body's regulatory immune function.Yao et al. [8] studied the effect of chemotherapeutic drugs on cellular immunity in patients with lung cancer; they found that cell immunity is inhibited in patients with lung cancer; moreover, it is impaired considerably by chemotherapy.So how to keep the negative effect caused by the single strategy adopted for the weeds or cancer cells to a minimum?
The above phenomenons motivated us to propose and study the discrete Lotka-Volterra competition system with infinite delays and single feedback control variable as follows: (5) In system (5),   () ( = 1, 2) is the density of   species at the th generation and () is the single feedback control variable.
Throughout this paper, we assume the following.
We mention here that this is the first time such kind of model is proposed and studied, and, as far as system (5) is concerned, whether the single feedback control variable has influence on the persistent property of the system or not is an interesting problem.The aim of this paper is to investigate the dynamic behaviors of the system (5); in particular, we will find out the answer to the above problem.
The organization of this paper is as follows.We introduce some useful lemmas in the next section and then state and prove the main results in Sections 3, 4, and 5, respectively.Three examples together with their numeric simulations are presented to show the feasibility of the main results in Section 6.We end this paper by a brief discussion.

Lemmas
Now, let us consider the following difference equation: where ,  are positive constants.

Permanence
Concerned with the persistent property of the system (5), we have the following result.
Theorem 6. Assume that holds; then, for any positive solution ( 1 (),  2 (), ()) of the system (5), we have where Proof.From the first and second equations of system (5), we have And so, from Lemma 3, we can obtain lim sup According to Lemma 5, from the above inequality we have lim sup For any  > 0, there exists a positive integer  1 such that By the third equation of system (5), we have Hence, by applying Lemmas 1 and 2 to (25), we obtain lim sup Setting  → 0, it follows that lim sup Condition (18) implies that, for enough small positive constant  1 , the following inequalities hold: For the above  1 , it follows from ( 22) and ( 27) that there exists a positive integer Thus, for all  >  2 , from ( 28), (29), and the first two equations of system (5), we have where Noticing that then Hence, according to Lemma 4, we have lim inf Setting  1 → 0, it follows that lim inf where According to Lemma 5, from (34) we have that, for any  2 > 0 small enough (without loss of generality, assume that  < (1/2)min  {  }), there exists an For  ≥  3 , from (35) and the last equation of system (5), we have Hence, by applying Lemmas 1 and 2 to (36), we obtain lim sup Setting  2 → 0, it follows that lim sup This ends the proof Theorem 6.

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

Extinction
Concerned with the extinction property of the system (5), when the coefficients of the third equation are all constants, we could establish the following results. . (62) Thus, there exists a positive constant  such that There exists a constant  3 such that Consider the following discrete Lyapunov functional: From (65), we obtain From inequalities (63) and (64), we can obtain Therefore From ( 22) and ( 27) we know that there exists an  > 0 such that and so On the other hand, we also have Combining inequalities (68), (70), and (71), we have where Hence we obtain that lim This ends the proof of Theorem 8.
Proof of Theorem 9.By (60), we can choose positive constants  1 ,  2 , and  3 and constant  such that Define the following Lyapunov functional: From (76), we have Similarly to the analysis of ( 68)-( 73), we have lim This ends the proof of Theorem 9.
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.
Proof of Theorem 10.By conditions (81), we can choose positive constants  1 ,  2 such that Thus, there exist enough small positive constants  and  such that where   22 is defined in(43) Now, we define a Lyapunov functional where Then, from the definition of  1 (),  1 (), one could easily see that  1 () ≥ 0 for all  ∈  + .Also, for any fixed  * ∈  + , from (69) one could see that Also, from the third equation of system (5) we have From (70), we have Therefore, from (44), (88), and (90), for all  >  * , we have where  =  1   21 +  2  1 .Summating both sides of the above inequality from  * to , we have 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.
Remark 12.One of the purposes of this paper is to find out the influence of feedback control variable on the persistent property of the system.Obviously, the answer lies in the relations among conditions (A 1 ), (58), and (60).Now let us consider conditions (A 1 ) and (60); there may exist the following subcase: (101) and (100) (or (102) and (100)) show that conditions (A 1 ) and (60) could be satisfied together; that is, for the original permanent system (2), by choosing suitable feedback control variable, species  2 will be driven to extinction.
Remark 13.Similarly to the above analysis, if system (2) is permanent, by choosing suitable feedback control variable, the first species  1 will be driven to extinction.
Remark 14.From Theorem 7, we can find that feedback control variable has influence on the global attractivity of system (5); that is, only the feedback control variable is very low such that the third inequality of (39) holds; in other words, when inequality  = max{  1 ,   2 } <  3   /( 1 +  2 ) holds, the two species can be coexist.

Examples
In this section we will give three examples to illustrate the feasibility of main results.
Figure 1 shows the dynamic behaviors of system (103), which strongly supports the above assertions.
Example 2. Consider the following equations:  Clearly, conditions (58) and (81) are satisfied, and so from Theorems 8 and 10 we know that species  1 will be driven to extinction, while species  2 is globally attractive.
Figure 2 shows the dynamic behaviors of system (106), which strongly supports our results.
Example 3. Consider the following example: Clearly, condition (A 1 ) is satisfied; thus from Theorem A we know that system (108) is permanent.Figure 3 shows the dynamic behaviors of system (108).Now let us further incorporate the feedback control variable into system (108) and consider the following system: Thus, all the conditions of Theorem 9 are satisfied, and so the permanent  2 species in system (108) become extinct in system (110).Figure 4 shows the dynamic behaviors of system (110), which supports our assertion.

Discussion
(1) Li et al. [3] consider a continuous and autonomous Lotka-Volterra competitive system with infinite delays and feedback controls; if the Lotka-Volterra competitive system is globally stable, then they showed that the feedback controls only change the position of the unique positive equilibrium and retain the stable property.As a consequence of this result, feedback control variables have no influence on the persistent property of the system they considered; Xu et al. [5] proposed and studied the discrete nonautonomous Lotka-Volterra competitive system with delays and feedback controls (model ( 4)).Theorem 3.2 in [5] showed that feedback controls do not affect the persistent property of system (4).In this paper, we propose and study a discrete Lotka-Volterra competition system with single feedback control variable.Remarks 12 and 13 show that the feedback control variable plays an important role in the persistent property of the system (2).If the Lotka-Volterra competitive system is permanent, then we show that, by choosing suitable feedback control variable, one of the species will be driven to extinction; that is, feedback control variable, which represents the biological control or some harvesting procedure, is an unstable factor of the system.Such a finding overturns the previous recognition on feedback control variables ( [3,5]) and shows that it is better for us to take a single policy to protect the rare species through reducing the species' competitions than to take two different policies.
(2) In [2], Chen proposed system (2) and investigated the persistent property of the system (see Theorem A in introduction section); however, he did not investigate the stability property and extinction property of the system.As a direct corollary of Theorem 7, concerned with the stability property of system (2), we have the following.

Theorem
Remark 15.Wu and Zhang [19] obtained a set of sufficient conditions which ensure the global attractivity of a discrete autonomous Lotka-Volterra competition system with infinite delays and feedback controls; as a direct corollary of their main result, one could also obtain a set of sufficient conditions which ensure the global stability of the autonomous case of system (2).However, their result required the assumption   ≤ 1,  = 1, 2. While Theorem B  solves the case   > 1,  = 1, 2.
It is in this sense that our result Theorem B  supplements and complements the main result of Wu and Zhang [19].Now, concerned with the extinction property of the system (2), with some minor revision of the proof of Theorems 8 and 9, we could obtain the following results.
It is interesting to investigate the stability property of the rest of the species when one of the species in system (2) is driven to extinction.In this case, as a direct corollary of Theorems 10 and 11, by simple computation, we have the following.where  * 1 () is any positive solution of the system  1 ( + 1) =  1 () exp( 1 () −  11 () 1 ()).
Obviously, Theorems B-F complement and supplement the main results of Chen [2].