Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 150841, 19 pages

doi:10.1155/2012/150841

Research Article

Partial Extinction, Permanence, and Global Attractivity in Nonautonomous n-Species Gilpin-Ayala Competitive Systems with Impulses

Juan Hou and Hanhui Liu

College of Applied Mathematics, Xinjiang University of Finance and Economics, Urumqi 830012, China

Received 3 May 2012; Accepted 21 June 2012

Academic Editor: Yong Zhou

Copyright © 2012 Juan Hou and Hanhui Liu. 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.

The qualitative properties of general nonautonomous n-species Gilpin-Ayala competitive systems with impulsive effects are studied. Some new criteria on the permanence, extinction, and global attractivity of partial species are established by using the methods of inequalities estimate and Liapunov functions.

1. Introduction

In [1], the general nonautonomous n-species Lotka-Volterra competitive systems with impulsive effects are investigated. By using the methods of inequalities estimate and constructing the suitable Liapunov functions, the sufficient conditions on the permanence of whole species and global attractivity of systems are established.

In [2], the authors studied the following general nonautonomous n-species Lotka-Volterra competitive systems with impulsive perturbations:

and got a series of criteria on the extinction of a part of n-species, the permanence of other part of n-species, and the global attractivity of the systems.

In [3], a periodic n-species Gilpin-Ayala competition system with impulses is studied and obtain some useful behaviors of the system.

In this paper, we investigate the general nonautonomous n-species Gilpin-Ayala competitive systems with impulsive effects.

where and are defined on and are bounded continuous functions, for all , and are constants for all and .

2. Preliminaries

Firstly, we introduce the following assumption.

Assumption H. There is a positive constant such that for each

and functions

are bounded on and .

For each , we consider the following logistic impulsive equation as the subsystem of system (1.2)

From the above assumption, we have the following results.

Lemma 2.1. Suppose that assumption H holds. Then we have the following:

(1)There exist positive constants and such that

for any positive solution of (2.3).(2) for any two positive solutions and of (2.3).

Proof. From assumption H, there are positive constants , , and such that for all we have

From the boundedness of function , there is a positive constant such that for any and

Firstly, we prove that there is a constant such that

for any positive solution of system (2.3). In fact, for any positive solution of system (2.3), we only need to consider the following three cases.

 Case I. There is a such that for all . Case II. There is a such that for all . Case III. is oscillatory about for all .

We first consider Case I. Since for all , then for , where is any positive integer, integrating system (2.3) from to , from (2.5) we have

Hence, as , which leads a contradiction.

Next, we consider Case III. From the oscillation of about , we can choose two sequences and satisfying and such that

For any , if for some integer , then we can choose integer and constant such that . Since

integrating this inequality from to , by (2.5) and (2.7) we obtain

where . If there is an integer such that , then we obviously have

Therefore, for Case III we always have

Lastly, if Case II holds, then we directly have

Choose constant , then we see that (2.8) holds.

Secondly, a similar argument as in the proof of (2.8) we can prove that there is a constant , such that

for any positive solution of system (2.3). Conclusion (1.1) is proved.

Now, we prove conclusion (1.2). Let and be any two positive solutions of system (2.3). From conclusion (1.1), it follows that there are positive constants and such that

Choose Liapunov function as follows:

For any , we have

Hence, is continuous for all and from the Mean-Value Theorem we can obtain

Calculating the upper right derivative of , then from (2.20) we obtain

where is the integer part of .

From this, we further have for any

From condition (2.5) we can obtain as . Hence, as . Further from (2.20) we finally obtain . Conclusion (1.2) is proved. This completes the proof of Lemma 2.1.

Applying Lemma 2.1 and the comparison theorem of impulsive differential equations, we easily prove the following result.

Lemma 2.2. Suppose that assumption H holds then there is a constant such that

for any positive solution of system (1.2).

3. Extinction

On the partial extinction of system (1.2), we have the following result.

Theorem 3.1. Suppose that assumption H holds. Let be a given integer and . If for any there is a such that for any

or

then species are extinction, that is, for any positive solution of system (1.2),

Proof. Firstly, from assumption H, that (2.7) still holds and there are constants and such that

for all and .

We first prove as . Without loss of generality, we assume that condition (3.2) holds. When condition (3.3) holds, a similar argument can be given. Since

where . Hence, we can choose positive constants , , and such that

for all and . Hence, from (3.5) we further obtain

for all and .

Consider the Liapunov function as follows:

Calculating the derivative, and from (3.1), we can obtain for any

for all and

for all . From (3.9), we further have

For any , there is an integer such that . Hence, by integrating (3.13) from to , we obtain

where

Since as , it follows that from (3.14)

Since

by the boundedness of on (see Lemma 2.2), we have

For any integer , assume that we have obtained as for all . Now, we prove that as . Suppose that condition (3.3) holds. When condition (3.2) holds, the argument is similar. Let , by (3.1), we have , then for , we have . Then we can choose positive constants , , and such that

for all , and , where .

Consider the Liapunov function as follows:

By calculating, we obtain for any

for all and

for all . From (3.3) and (3.19), we have

for all and . Hence, from (3.21), it follows that

Since as for all , by the boundedness of on , we obtain

Hence, for any small , there is a , such that

Combining (3.23), it follows that there is enough large such that for all ,

For any , we firstly choose an integer such that . Integrating (3.25) from to , then from (3.3) and (3.28), we have

where

Since as , we obtain from (3.29)

Since

by the boundedness of on , it follows that

Finally, by the induction principle, we obtain that as for all . This completes the proof of Theorem 3.1.

4. Permanence

In this section, we study the permanence of partial species of system (1.2). We state and prove the following result.

Theorem 4.1. Suppose that all the conditions of Theorem 3.1 hold. If for each

where is some fixed positive solution of (2.3), then species are permanent, that is, there are positive constants and such that for any positive solution of system (1.2)

Proof. From (4.1) and the boundedness of functions on , there are constants and such that for any and .

For any , from system (1.2), we have

we have

where is the solution of (2.3) with initial condition . From Lemma 2.1 and Theorem 3.1, for the above constant there is a such that for all

Let

where constant is given in (2.7). Obviously, and is independent of any positive solution of system (1.2).

Now, we prove that there is a such that

We only need to consider the following three cases for each .

 Case I. There is a such that for all . Case II. There is a such that for all . Case III. oscillates about for all .

For Case I, let , where is any integer. From (4.3)–(4.7) we obtain

Therefore, as which leads to a contradiction.

For Case III, we choose two sequences and satisfying and such that

For any , if for some integer , then we can choose an integer such that , where is a constant. Since for any from (4.6) and (4.7) we have

Integrating this inequality from to , then from (4.7) and (3.28)-(3.29) we have

If there exists an integer such that , then we obviously have

This shows that for Case III we always have

Finally, if Case II holds, then from for all , we can directly obtain that (4.9) holds.

Therefore, from Lemma 2.2 and (4.9), it follows that species are permanent. This proof of Theorem 4.1 is completed.

5. Global Attractivity

In this section, we further discuss the global attractivity of species . In order to obtain our results, we first consider the following subsystem which is composed of the species of system (1.2) and for convenience of statement we use the variable to denote the species of this subsystem,

We need the following lemma.

Lemma 5.1. Suppose that assumption H and condition (4.1) of Theorem 4.1 hold. Then subsystem (5.1) is permanent.

Lemma 5.1 can be proved by using the same method given in the proof of Theorem 4.1. We now state and prove the main result of this section.

Theorem 5.2. Suppose that all conditions of Theorem 3.1 and Theorem 4.1 hold. If there are positive constants , and and nonnegative integrable function defined on , satisfying for all , such that

for all , then for any positive solution of system (1.2) and any positive solution of subsystem (5.1)

Proof. Let be a positive solution of system (1.2) and be a positive solution of subsystem (5.1). By Theorem 3.1, we have as for all . From Theorem 4.1 and Lemma 5.1, there are positive constants and such that

for all . Choose the Liapunov function as follows:

Since

then is continuous for all . Calculating the upper right derivative of , we have

for all , where

By (5.2), we have

By (5.4), we further obtain

where . Applying the comparison theorem and the variation of constants formula of first-order linear differential equation, we have

for all . Since as , from the properties of function and (5.11), it is not hard to obtain as . That shows

This completes the proof of Theorem 5.2.

References

J. Hou, Z. Teng, and S. Gao, “Permanence and global stability for nonautonomous N-species Lotka-Valterra competitive system with impulses,” Nonlinear Analysis, vol. 11, no. 3, pp. 1882–1896, 2010.

J. Hou, Z. D. Teng, and S. J. Gao, “Partial extinction, permanence and global attractivity in nonautonomous n-species Lotka-Volterra competitive systems with impulses,” Applied Mathematics: A Journal of Chinese Universities, vol. 26, no. 2, pp. 161–176, 2011.

M. He, Z. Li, and F. Chen, “Permanence, extinction and global attractivity of the periodic Gilpin-Ayala competition system with impulses,” Nonlinear Analysis, vol. 11, no. 3, pp. 1537–1551, 2010.