Research Article | Open Access

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

**Academic Editor:**Zhan Zhou

#### Abstract

We propose and deal with a discrete mutualism model with infinite deviating arguments and feedback controls. Sufficient conditions which guarantee the permanence of the system are obtained by using the difference inequality theory. The paper ends with brief conclusions.

#### 1. 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 [6–19].

In 2007, Chen and You [20] investigated the permanence of the following two species integrodifferential model of mutualism: where , , , and are continuous functions bounded above and below by positive constants. ; and . Using the differential inequality theory, they obtained a set of sufficient conditions to ensure the permanence of system (1). For more background and biological adjustments of system (1), one could refer to [20–24] and the references cited therein.

Many authors [24–33] 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 is the density of mutualism species at the generation, , and 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 is the density of mutualism species at the generation and is the control variable. , and are bounded nonnegative sequences.

Throughout this paper, we assume that(H). Here, for any bounded sequence , and .

Let . We consider (3) together with the following initial conditions: It is not difficult to see that solutions of (3) and (4) are well defined for all 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 (3) are established. Brief conclusions are presented in Section 3.

#### 2. 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.

*Definition 1. *We say that system (3) is permanence if there are positive constants and such that for each positive solution of system (3) satisfies

Let us consider the following single species discrete model: where and are strictly positive sequences of real numbers defined for and . Similarly to the proofs Propositions 1 and 3 in [36], we can obtain the following Lemma 2.

Lemma 2. *Any solution of system (7) with initial condition 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.

Lemma 3 (see [37]). *Assume that , for any initial value , there exists a unique solution of (10) which can be expressed as follows:
**
where . Thus, for any solution of system (10), .*

Lemma 4 (see [37]). *Let . For any fixed , is a nondecreasing function with respect to , and for , the following inequalities hold:
**
If , then for all .*

Proposition 5. *Assume that the condition (H) holds, then
**
where
*

*Proof. *Let be any positive solution of system (3) with the initial condition . It follows from the first equation of system (3) that
Let then (15) is equivalent to
Summing both sides of (16) from to , we have
which leads to
Then
Substituting (19) into the first equation of system (3), it follows that
It follows from (20) and Lemma 2 that
For any positive constant , it follows (21) that there exists a such that for all
From the second equation of system (3), we get
Let then (23) is equivalent to
Summing both sides of (24) from to , we have
which leads to
Then
Substituting (27) into the second equation of system (3), it follows that
It follows from (28) and Lemma 2 that
For any positive constant , it follows (29) that there exists a such that for all
In view of the third and fourth equations of system (3), we can obtain
Then
Applying Lemmas 3 and 4, it immediately follows that
Setting , it follows that
This completes the proof of Proposition 5.

Theorem 6. *Assume that (H) holds, then system (3) is permanent.*

*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,
In view of Proposition 5, for all , there exists a , for all :
It follows from the first equation of systems (3) and (36) that
for all .

Let ; then (37) is equivalent to
Summing both sides of (38) from to leads to
Then
Thus
Substituting (36) and (41) into the first equation of (3), we have
for all .

By applying Lemmas 2 and 4, it immediately follows that
where
Setting in (43), then
where
By the second equation of systems (3) and (36), we can obtain
for all .

Let ; then (47) is equivalent to
Summing both sides of (48) from to leads to
Then
Thus
Substituting (36) and (51) into the second equation of (3), we have
for all .

By applying Lemmas 2 and 4, it immediately follows that
where
Setting in (53), then
where
Without loss of generality, we assume that . For any positive constant small enough, it follows from (45) and (55) that there exists enough large such that
for any .

From the third and fourth equations of systems (3) and (57), we can derive that
Hence
By applying Lemmas 2 and 3, it immediately follows that
Setting in the previous inequality leads to
This completes the proof of Theorem 6.

#### 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.

#### Acknowledgments

This work is supported by thr National Natural Science Foundation of China (no. 11261010 and no. 11101126), the Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), the Natural Science and Technology Foundation of Guizhou Province (J[2012]2100), the Governor Foundation of Guizhou Province ([2012]53), and the Doctoral Foundation of Guizhou University of Finance and Economics (2010).

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#### Copyright

Copyright © 2013 Changjin Xu and Yusen Wu. 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.