Permanence in a Discrete Mutualism Model with Infinite Deviating Arguments and Feedback Controls

It is well known that the long-term coexistence of species in mathematical ecology is an important and ubiquitous problem. Several mathematical concepts of coexistence of species are developed to deal with this aspect. Permanence is one important topic in these concepts. In recent years, permanence has received great attention and has been investigated in a number of notable studies. For example, Fan and Li [1] analyzed permanence of a delayed ratio-dependent predator-prey model with Holling type functional response. Mukherjee [2] addressed the permanence and global attractivity for facultative mutualism system with delay. Zhao and Jiang [3] focused on the permanence and extinction for nonautonomous Lotka-Volterra system. Chen [4] made a theoretical discussion on the permanence and global attractivity of Lotka-Volterra competition system with feedback control. Teng et al. [5] established the permanence criteria for a delayed discrete nonautonomous-species Kolmogorov systems. For more research on the permanence behavior of predator-prey models, one can see [6–19]. In 2007, Chen and You [20] investigated the permanence of the following two species integrodifferential model of mutualism:


Introduction
It is well known that the long-term coexistence of species in mathematical ecology is an important and ubiquitous problem.Several mathematical concepts of coexistence of species are developed to deal with this aspect.Permanence is one important topic in these concepts.In recent years, permanence has received great attention and has been investigated in a number of notable studies.For example, Fan and Li [1] analyzed permanence of a delayed ratio-dependent predator-prey model with Holling type functional response.Mukherjee [2] addressed the permanence and global attractivity for facultative mutualism system with delay.Zhao and Jiang [3] focused on the permanence and extinction for nonautonomous Lotka-Volterra system.Chen [4] made a theoretical discussion on the permanence and global attractivity of Lotka-Volterra competition system with feedback control.Teng et al. [5] established the permanence criteria for a delayed discrete nonautonomous-species Kolmogorov systems.For more research on the permanence behavior of predator-prey models, one can see [6][7][8][9][10][11][12][13][14][15][16][17][18][19].
Many authors [24][25][26][27][28][29][30][31][32][33] have argued that discrete time models governed by difference equations are more appropriate to describe the dynamics relationship among populations than continuous ones when the populations have nonoverlapping generations.Moreover, discrete time models can also provide efficient models of continuous ones for numerical simulations.Motivated by the above viewpoint, Li and Yang [34] considered the permanence of the following discrete model of mutualism with infinite deviating arguments: where   () ( = 1, 2) is the density of mutualism species  at the generation, {  ()}, {  ()}, {  ()}, {  ()} ( = 1, 2),  1 (), and  2 () are bounded nonnegative sequences.Applying the comparison theorem of difference equation and some lemmas, they derived some sufficient conditions which guarantee the permanence of system (2).It is well known that ecosystems in the real world are continuously distributed by unpredictable forces which can result in changes in the biological parameters such as survival rates [35].Of practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time.In the language of control variables, we call the disturbance functions as control variables.To the authors' knowledge, it is the first time to deal with system (2) with feedback control.
The main objective of this paper is to investigate the following discrete mutualism model with infinite deviating arguments and feedback controls: where   () ( = 1, 2) is the density of mutualism species  at the generation and and  2 () are bounded nonnegative sequences.Throughout this paper, we assume that Here, for any bounded sequence {()},   = sup ∈ {()} and   = inf ∈ {()}.
Let  = sup ∈ {  ()}, σ = inf ∈ {  ()},  = 1, 2. We consider (1) together with the following initial conditions: It is not difficult to see that solutions of (1) and ( 4) are well defined for all  ≥ 0 and satisfy The remainder of the paper is organized as follows.In Section 2, basic definitions and lemmas are given, some sufficient conditions for the permanence of system (1) are established.Brief conclusions are presented in Section 3.

Permanence
In order to obtain the main result of this paper, we will first state the definition of permanence and several lemmas which will be useful in the proving of the main result.
Lemma 2. Any solution of system (7) with initial condition (0) > 0 satisfies where Let us consider the first order difference equation: where  and  are positive constants.Following Theorem 6.2 of L. Wang and M. Q. Wang [37, page 125], we have the following Lemma 3.

Proposition 5. Assume that the condition (H) holds, then
where Proof.Let ( 1 (),  2 (),  1 (),  2 ()) be any positive solution of system (1) with the initial condition ( 1 (0),  2 (0),  1 (0),  2 (0)).It follows from the first equation of system (1) that Let  1 () = exp{ 1 ()} then ( 15) is equivalent to Summing both sides of ( 16) from  −  1 () to  − 1, we have which leads to Then Substituting (19) into the first equation of system (1), it follows that It follows from (20) and Lemma 2 that lim For any positive constant  > 0, it follows ( 21) that there exists a  1 > 0 such that for all  >  1 + From the second equation of system (1), we get Let  2 () = exp{ 2 ()} then ( 23) is equivalent to Summing both sides of (24) from  −  2 () to  − 1, we have which leads to Then Substituting (27) into the second equation of system (1), it follows that It follows from ( 28) and Lemma 2 that For any positive constant  > 0, it follows (29) that there exists a In view of the third and fourth equations of system (1), we can obtain Then Applying Lemmas 3 and 4, it immediately follows that lim Setting  → 0, it follows that lim This completes the proof of Proposition 5.
Proof.By applying Proposition 5, we can easily see that to end the proof of Theorem 6, it is enough to show that under the conditions of Theorem 6, lim In view of Proposition 5, for all  > 0, there exists a  3 > 0,  3 ∈ , for all  >  3 : It follows from the first equation of systems ( 1) and ( 36) that for all  >  3 + .

Conclusions
In the present paper, we have investigated the permanence of a discrete mutualism model with infinite deviating arguments and feedback controls.Sufficient conditions which ensure the permanence of the system are established.We have shown the effect of delay to the permanence of system and concluded that delay is an important factor to decide the permanence of the system.