Discrete Dynamics in Nature and Society

Volume 2014, Article ID 682769, 9 pages

http://dx.doi.org/10.1155/2014/682769

## Note on the Persistence of a Nonautonomous Lotka-Volterra Competitive System with Infinite Delay and Feedback Controls

^{1}College of Zhicheng, Fuzhou University, Fuzhou, Fujian 350002, China^{2}Department of Science Research, Fuzhou Fujian Institute of Education, Fuzhou, Fujian 350001, China

Received 8 March 2014; Revised 15 May 2014; Accepted 20 May 2014; Published 24 July 2014

Academic Editor: Garyfalos Papashinopoulos

Copyright © 2014 Chunling Shi et al. 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 study a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls. We establish a series of criteria under which a part of -species of the systems is driven to extinction while the remaining part of the species is persistent. Particularly, as a special case, a series of new sufficient conditions on the persistence for all species of system are obtained. Several examples together with their numerical simulations show the feasibility of our main results.

#### 1. Introduction

In this paper, we consider the following nonautonomous -species Lotka-Volterra competitive system with infinite delay and feedback controls: where is the density of the th species at time and is the indirect control variable.

In particular, when the coefficients , , and for all and , the system (1) will degenerate into the following* pure delay type* system:

As is well known, systems such as (2) without feedback controls are very important mathematical models of multispecies populations dynamics. This is a generalization from Ahmad [1] about two-species system without delays to -species system of infinite delay. Systems without delays such as [1] have attracted the interest of many researchers (see, e.g., [2–5]), and systems with delays have been studied extensively in the past twenty years, and some good results on the permanence, extinction and persistence or uniform persistence, global stability, and almost periodic solution have been developed (see [6–18]). In [19], Montes de Oca and Pérez provided for us a very interesting work for system (2), who showed that if the coefficients are bounded and continuous and satisfy certain inequalities, then any solution with initial function of system (2) in an appropriate space will have of its components tenting to zero, while the remaining one will stabilize at a certain solution of a logistic differential equation. And for more works about single species dynamic behaviors of infinite delay, one could refer to [20, 21].

On the other hand, as was pointed out by Fan and Wang [22], feedback control is the basic mechanism by which systems, whether mechanical, electrical, or biological, maintain their equilibrium or homeostasis. Many scholars have done works on the ecosystem with feedback controls (see, e.g., [23–29] and the references cited therein). In [23], Shi et al. proposed the feedback control system (1). By using the method of multiple Lyapunov functionals and by developing a new analysis technique, Shi et al. established the sufficient conditions which guarantee part species of the -species driven to extinction. But in the paper [23], they did not discuss the survival problems for the remaining species. The main aim of this paper is to study the persistence of the remaining species of system (1). By the new method motivated by work [11, 27, 28], we will establish new sufficient conditions for which surplus species of system (1) remain persistent.

The organization of the paper is as follows. In the next section, some assumptions and lemmas are introduced. In Section 3, we state and prove our main results. Finally, several examples with their numerical simulations are presented to show the feasibility of the main results.

#### 2. Preliminaries

Throughout this paper, for system (1), we introduce the following hypotheses., , , , and are bounded and continuous, defined on . Furthermore, , , , and are nonnegative on , and . Here, we denote and ., , and , , , are piecewise continuous and satisfy There exists a positive constant such that for each There exist positive constants and such that for each

We will consider system (1) together with the initial conditions where , , and It is easy to verify that solutions of (1) satisfying the initial condition (6) are well defined for all and satisfy

We now introduce several lemmas which will be useful in the proofs of the main results.

We consider the following nonautonomous linear equation: where nonnegative functions and are bounded and continuous, defined on . We have the following results.

Lemma 1 (see [30]). *Suppose that there exist positive constants and such that
**
Then, there exist positive constants such that
**
for any positive solution of (9).*

Lemma 2 (see [23]). *Suppose that assumptions ()–() hold; then there exist constants and such that
**
for any positive solution , of system (1).*

*Remark 3. *If all parameters , , , , and of system (1) have the positive lower bound on , then, from Lemma 2.2 in [23], we can choose

Lemma 4 (see [6]). *Let be a nonnegative and bounded continuous function, and let be an integral function satisfying . Then
*

#### 3. Main Results

In this section, we discuss the persistence of part species of system (1), where integer . Let functions

Lemma 5. *Suppose that assumptions ()–() hold and there exists an integer such that for any there exists an integer such that
**
Then for each we have
**
for any positive solution , of system (1).*

The proof of the extinction of part species of system (1) could be found in [23] and we hence omit it here.

On the persistence of part species of system (1), we state and prove the following results.

Theorem 6. *Suppose that all assumptions of Lemma 5 hold and there exists a positive constant such that
**
Then, for each , there exist positive constants and , with , such that
**
for any positive solution of system (1).*

*Proof. *Let , be any positive solution of system (1). By Lemma 2, let ; for each , we have and . So, we only need to prove that there exists a positive constant such that and for all .

First of all, assumption (18) implies that there are positive constants and such that
for all and .

By Lemmas 2 and 5, we obtain that, for any constant , there is a such that, for all ,

Now, for any , we define the Lyapunov function as follows:
By assumptions () and () and Lemma 2, we have
So we see that has definition for all . From (23), we can obtain that for any there is a positive constant , and may be dependent on the positive solution of system (1) such that
Calculating the derivative of with respect to , we have
Let . From (21), for all , we have
Obviously, from inequality (20), we can find enough small positive constants and such that
for all . So for the above , when ,
Consider the auxiliary equation
then by (28), we obtain that
where is the solution of (29) with the initial condition . If for all , then is defined on . Integrating inequality (29) from to , we obtain
for all . Putting , , then, from (27) and (31), we have
Letting , we have , a contradiction. Hence, there is a such that . Now, we prove that
where , and the definition of implies . In fact, if (33) is not true, then there are and , , such that
Choosing the integer such that , then, by (27) and (29), it follows that
which is a contradiction.

From (24), (30), and (33), we can obtain that
Finally, we define the constants and ; then we have
Letting and , we have
for all .

Further, by Lemma 4 and (38), we can choose constants and such that for all and
Considering the second equation of system (1), from (39), for any , we obtain
We consider the following auxiliary equation:
Then by assumption () and applying Lemma 1 there exists a constant such that
for any positive solution of (41). Let be the solution of (41) with the initial condition ; then by the comparison theorem we have
Thus, we finally obtain
Let ; from (38) and (44), we obtain that and . This completes the proof of Theorem 6.

As consequences of Theorem 6 we have the following corollaries.

Corollary 7. *If, in system (1), for all , then system (1) will be reduced to the following -species competitive system with infinite delay:
**
Suppose that assumptions ()–() hold and there exists an integer such that for any there exists an integer such that
**
Furthermore, there exists a positive constant such that
**
Then, for each , there exist positive constants such that
**
and for each we have
**
for any positive solution of system (45).*

*Proof. *From the condition,
And the assumptions ()–() hold; from Corollary 7 in [23], for each we have
Further condition
holds, so we see, from Theorem 6, for each , that there exist positive constants such that

*Remark 8. *When , the conditions of Corollary 7 will reduce to the assumptions that ()–() hold and for any such that
We have that there exist positive constants such that
and for each we have
In comparison with the assumptions together with Proposition 2.2 given by Montes de Oca and Pérez [19], we can see that our assumptions in Corollary 7 are weaker.

*Remark 9. *When , from Corollary 7 we can easily obtain a criterion on the persistence of all species of system (45).

*Remark 10. *The conclusion of Corollary 7 improves that of Proposition 2.2 given by Montes de Oca and Pérez [19].

Corollary 11. *Suppose that ()–() hold and there exists a positive constant such that
**
Then, for each , there exist positive constants and , with , such that
**
for any positive solution , of system (1).*

*Remark 12. *From Corollary 11 we can easily obtain a criterion on the persistence of all species of system (1).

#### 4. Examples

In this section, we will give several examples to illustrate the conclusions of Corollary 7, Theorem 6, and Corollary 11. In the first part we will illustrate the conclusions of Corollary 7, in the second we will illustrate the conclusions of Theorem 6, and in the last we will illustrate the conclusions of Corollary 11.

*Example 1. *Consider the system
where
Obviously, we have that the period of system (59) is . By calculating, we obtain
We can choose , since all conditions of Corollary 7 hold; therefore, species and in system (59) are extinct, and only species is persistent (see Figure 1). However, conditions (1.5) of Proposition 2.2 given by Montes de Oca and Pérez [19] do not apply in this example.

*Example 2. *Consider the system
where
Obviously, we have that the period of system (62) is . By calculating, we obtain
From Remark 3, we can choose , , , , , , , and such that
All the conditions of Theorem 6 hold; therefore, species and coexist, and species in system (62) is extinct (see Figure 2).

*Example 3. *Consider the system
where
and the coefficients and the other kernels are as Example 2. In this case, we can choose , , , , , , and such that
All conditions of Corollary 11 hold, so all the species , , and are persistent (see Figure 3).

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work is supported by the Foundation of Fujian Education Bureau (nos. JA12369 and JA13361).

#### References

- S. Ahmad, “On the nonautonomous Volterra-Lotka competition equations,”
*Proceedings of the American Mathematical Society*, vol. 117, pp. 199–205, 1993. View at Publisher · View at Google Scholar - F. Montes de Oca and M. L. Zeeman, “Extinction in nonautonomous competitive Lotka-Volterra systems,”
*Proceedings of the American Mathematical Society*, vol. 124, no. 12, pp. 3677–3687, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Ahmad and A. C. Lazer, “Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system,”
*Nonlinear Analysis: Theory, Methods and Applications*, vol. 40, no. 1–8, pp. 37–49, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Ahmad and A. C. Lazer, “Average growth and extinction in a competitive Lotka-Volterra system,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 62, no. 3, pp. 545–557, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Montes de Oca and M. Vivas, “Extinction in two dimensional Lotka-Volterra system with infinite delay,”
*Nonlinear Analysis: Real World Applications*, vol. 7, no. 5, pp. 1042–1047, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. Z. He, “Almost periodic solutions of a competition system with dominated infinite delays,”
*The Tohoku Mathematical Journal*, vol. 50, no. 1, pp. 71–89, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - R. Xu, M. A. J. Chaplain, and L. Chen, “Global asymptotic stability in $n$-species nonautonomous Lotka-Volterra competitive systems with infinite delays,”
*Applied Mathematics and Computation*, vol. 130, no. 2-3, pp. 295–309, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S. Ahmad and M. R. M. Rao, “Asymptotically periodic solutions of $N$-competing species problem with time delays,”
*Journal of Mathematical Analysis and Applications*, vol. 186, no. 2, pp. 559–571, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Ahmad and M. R. Mohana Rao, “Stability criteria for $N$-competing species problem with time delays,”
*Nonlinear World*, vol. 1, no. 2, pp. 245–253, 1994. View at Google Scholar · View at MathSciNet - Z. Teng and Y. Yu, “Some new results of nonautonomous Lotka-Volterra competitive systems with delays,”
*Journal of Mathematical Analysis and Applications*, vol. 241, no. 2, pp. 254–275, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. D. Teng, “Permanence and stability of Lotka-Volterra type $N$-species competitive systems,”
*Acta Mathematica Sinica*, vol. 45, no. 5, pp. 905–918, 2002 (Chinese). View at Google Scholar · View at MathSciNet - Z. Hou, “On permanence of all subsystems of competitive Lotka-Volterra systems with delays,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 5, pp. 4285–4301, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. Hou, “Permanence and extinction in competitive Lotka-Volterra systems with delays,”
*Nonlinear Analysis. Real World Applications*, vol. 12, no. 4, pp. 2130–2141, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. Hu, K. Wang, and D. Wu, “Permanence and global stabil ity for nonautonomous
*N*-species Lotka-Volterra competitive system with impulses and infinite delays,”*Journal of Mathematical Analysis and Applications*, vol. 377, no. 1, pp. 145–160, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Q. Wang, M. Ding, Z. Wang, and H. Zhang, “Existence and attractivity of a periodic solution for an $N$-species Gilpin-Ayala impulsive competition system,”
*Nonlinear Analysis. Real World Applications*, vol. 11, no. 4, pp. 2675–2685, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Yan, A. Zhao, and J. J. Nieto, “Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems,”
*Mathematical and Computer Modelling*, vol. 40, no. 5-6, pp. 509–518, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - F. Chen, “Some new results on the permanence and extinction of nonautonomous Gilpin-Ayala type competition model with delays,”
*Nonlinear Analysis: Real World Applications*, vol. 7, no. 5, pp. 1205–1222, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Montes de Oca and L. Pérez, “Extinction in nonautonomous competitive Lotka-Volterra systems with infinite delay,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 75, no. 2, pp. 758–768, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. Yang, W. Wang, and J. Shen, “Permanence of a logistic type impulsive equation with infinite delay,”
*Applied Mathematics Letters*, vol. 24, no. 4, pp. 420–427, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Chen and C. Shi, “Dynamic behavior of a logistic type equation with infinite delay,”
*Acta Mathematicae Applicatae Sinica*, vol. 22, no. 2, pp. 313–324, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. Fan and L. Wang, “Global asymptotical stability of a logistic model with feedback control,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 4, pp. 2686–2697, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - C. Shi, Z. Li, and F. Chen, “Extinction in a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls,”
*Nonlinear Analysis: Real World Applications*, vol. 13, no. 5, pp. 2214–2226, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Chen, “Global asymptotic stability in $n$-species non-autonomous Lotka-Volterra competitive systems with infinite delays and feedback control,”
*Applied Mathematics and Computation*, vol. 170, no. 2, pp. 1452–1468, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Chen, Z. Li, and Y. Huang, “Note on the permanence of a competitive system with infinite delay and feedback controls,”
*Nonlinear Analysis. Real World Applications*, vol. 8, no. 2, pp. 680–687, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. Hu, Z. Teng, and S. Gao, “Extinction in nonautonomous Lotka-Volterra competitive system with pure-delays and feedback controls,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 4, pp. 2508–2520, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. Hu, Z. Teng, and H. 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - L. Nie, J. Peng, and Z. Teng, “Permanence and stability in multi-species non-autonomous Lotka-Volterra competitive systems with delays and feedback controls,”
*Mathematical and Computer Modelling*, vol. 49, no. 1-2, pp. 295–306, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - M. Fan, K. Wang, P. J. Y. Wong, and R. P. Agarwal, “Periodicity and stability in periodic $n$-species Lotka-Volterra competition system with feedback controls and deviating arguments,”
*Acta Mathematica Sinica (English Series)*, vol. 19, no. 4, pp. 801–822, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - H. Hu, Z. Teng, and H. Jiang, “Permanence of the nonautonomous competitive systems with infinite delay and feedback controls,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 4, pp. 2420–2433, 2009. View at Publisher · View at Google Scholar