Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 397382, 7 pages

http://dx.doi.org/10.1155/2013/397382

Research Article

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

Changjin Xu¹ and Yusen Wu²

¹Guizhou Key Laboratory of Economics System Simulation, School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550004, China

²School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China

Received 22 March 2013; Accepted 4 September 2013

Academic Editor: Zhan Zhou

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.

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).

References

Y.-H. Fan and W.-T. Li, “Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response,” Journal of Mathematical Analysis and Applications, vol. 299, no. 2, pp. 357–374, 2004.

D. Mukherjee, “Permanence and global attractivity for facultative mutualism system with delay,” Mathematical Methods in the Applied Sciences, vol. 26, no. 1, pp. 1–9, 2003.

J. D. Zhao and J. F. Jiang, “Average conditions for permanence and extinction in nonautonomous Lotka-Volterra system,” Journal of Mathematical Analysis and Applications, vol. 299, no. 2, pp. 663–675, 2004.

F. D. Chen, “The permanence and global attractivity of Lotka-Volterra competition system with feedback controls,” Nonlinear Analysis. Real World Applications, vol. 7, no. 1, pp. 133–143, 2006.

Z. D. Teng, Y. Zhang, and S. J. Gao, “Permanence criteria for general delayed discrete nonautonomous n-Kolmogorov systems and its applications,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 812–828, 2010.

F. D. Chen, “Permanence of a discrete n-species food-chain system with time delays,” Applied Mathematics and Computation, vol. 185, no. 1, pp. 719–726, 2007.

J. Dhar and K. S. Jatav, “Mathematical analysis of a delayed stage-structured predator-prey model with impulsive diffsion between two predators territories,” Ecological Complexity, 2012.

S. Q. Liu and L. S. Chen, “Necessary-sufficient conditions for permanence and extinction in Lotka-Volterra system with distributed delays,” Applied Mathematics Letters, vol. 16, no. 6, pp. 911–917, 2003.

X. Y. Liao, S. F. Zhou, and Y. M. Chen, “Permanence and global stability in a discrete n-species competition system with feedback controls,” Nonlinear Analysis. Real World Applications, vol. 9, no. 4, pp. 1661–1671, 2008.

H. X. Hu, Z. D. Teng, and H. J. Jiang, “On the permanence in non-autonomous Lotka-Volterra competitive system with pure-delays and feedback controls,” Nonlinear Analysis. Real World Applications, vol. 10, no. 3, pp. 1803–1815, 2009.

Y. Muroya, “Permanence and global stability in a Lotka-Volterra predator-prey system with delays,” Applied Mathematics Letters, vol. 16, no. 8, pp. 1245–1250, 2003.

Z. Y. Hou, “On permanence of Lotka-Volterra systems with delays and variable intrinsic growth rates,” Nonlinear Analysis. Real World Applications, vol. 14, no. 2, pp. 960–975, 2013.

X. S. Xiong and Z. Q. Zhang, “Periodic solutions of a discrete two-species competitive model with stage structure,” Mathematical and Computer Modelling, vol. 48, no. 3-4, pp. 333–343, 2008.

R. Y. Zhang, Z. C. Wang, Y. M. Chen, and J. H. Wu, “Periodic solutions of a single species discrete population model with periodic harvest/stock,” Computers & Mathematics with Applications, vol. 39, no. 1-2, pp. 77–90, 2009.

Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems, World Scientific Publishing, River Edge, NJ, USA, 1996.

Y. G. Sun and S. H. Saker, “Positive periodic solutions of discrete three-level food-chain model of Holling type II,” Applied Mathematics and Computation, vol. 180, no. 1, pp. 353–365, 2006.

X. H. Ding and C. Lu, “Existence of positive periodic solution for ratio-dependent n-species difference system,” Applied Mathematical Modelling, vol. 33, no. 6, pp. 2748–2756, 2009.

Z. C. Li, Q. L. Zhao, and D. Ling, “Chaos in a discrete population model,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 482459, 14 pages, 2012.

H. Xiang, K.-M. Yan, and B.-Y. Wang, “Existence and global stability of periodic solution for delayed discrete high-order Hopfield-type neural networks,” Discrete Dynamics in Nature and Society, vol. 2005, no. 3, pp. 281–297, 2005.

F. D. Chen and M. S. You, “Permanence for an integrodifferential model of mutualism,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 30–34, 2007.

F. D. Chen, “Permanence in a discrete Lotka-Volterra competition model with deviating arguments,” Nonlinear Analysis. Real World Applications, vol. 9, no. 5, pp. 2150–2155, 2008.

Y. K. Li and G. T. Xu, “Positive periodic solutions for an integrodifferential model of mutualism,” Applied Mathematics Letters, vol. 14, no. 5, pp. 525–530, 2001.

P. Yang and R. Xu, “Global asymptotic stability of periodic solution in n-species cooperative system with time delays,” Journal of Biomathematics, vol. 13, no. 6, pp. 841–846, 1998.

Y. K. Li and H. T. Zhang, “Existence of periodic solutions for a periodic mutualism model on time scales,” Journal of Mathematical Analysis and Applications, vol. 343, no. 2, pp. 818–825, 2008.

R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000.

Y. K. Li and L. H. Lu, “Positive periodic solutions of discrete n-species food-chain systems,” Applied Mathematics and Computation, vol. 167, no. 1, pp. 324–344, 2005.

X. Chen and F. D. Chen, “Stable periodic solution of a discrete periodic Lotka-Volterra competition system with a feedback control,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1446–1454, 2006.

Y. Muroya, “Persistence and global stability in Lotka-Volterra delay differential systems,” Applied Mathematics Letters, vol. 17, no. 7, pp. 795–800, 2004.

Y. Muroya, “Partial survival and extinction of species in discrete nonautonomous Lotka-Volterra systems,” Tokyo Journal of Mathematics, vol. 28, no. 1, pp. 189–200, 2005.

X. T. Yang, “Uniform persistence and periodic solutions for a discrete predator-prey system with delays,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 161–177, 2006.

F. D. Chen, “Permanence for the discrete mutualism model with time delays,” Mathematical and Computer Modelling, vol. 47, no. 3-4, pp. 431–435, 2008.

W. P. Zhang, D. M. Zhu, and P. Bi, “Multiple positive periodic solutions of a delayed discrete predator-prey system with type IV functional responses,” Applied Mathematics Letters, vol. 20, no. 10, pp. 1031–1038, 2007.

Y. M. Chen and Z. Zhou, “Stable periodic solution of a discrete periodic Lotka-Volterra competition system,” Journal of Mathematical Analysis and Applications, vol. 277, no. 1, pp. 358–366, 2003.

X. P. Li and W. S. Yang, “Permanence of a discrete model of mutualism with infinite deviating arguments,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 931798, 7 pages, 2010.

X. P. Li and W. S. Yang, “Permanence of a discrete predator-prey systems with Beddington-DeAngelis functional response and feedback controls,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 149267, 8 pages, 2008.

F. D. Chen, “Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 3–12, 2006.

L. Wang and M. Q. Wang, Ordinary Difference Equation, Xinjiang University Press, 1991.