Discrete Dynamics in Nature and SocietyVolume 2012 (2012), Article ID 125197, 12 pagesdoi:10.1155/2012/125197

Research Article

Permanence in Multispecies Nonautonomous Lotka-Volterra Competitive Systems with Delays and Impulses

Xiaomei Feng,1,2 Fengqin Zhang,² Kai Wang,³ and Xiaoxia Li²

¹College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China²Department of Mathematics, Yuncheng University, Yuncheng 044000, China³Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830011, China

Received 10 November 2011; Accepted 2 February 2012

Academic Editor: Zhen Jin

Copyright © 2012 Xiaomei Feng 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.

This paper studies multispecies nonautonomous Lotka-Volterra competitive systems with delays and fixed-time impulsive effects. The sufficient conditions of integrable form on the permanence of species are established.

1. Introduction

In this paper, we consider the nonautonomous -species Lotka-Volterra type competitive systems with delays and impulses where represents the population density of the th species at time , the functions , , , and are bounded and continuous functions defined on , , , for all , and impulsive coefficients for any and are positive constants.

In particular, when the delays for all and , then the system (1.1) degenerate into the following nondelayed non-autonomous -species Lotka-volterra system where and for and . For system (1.2), the author establish some new sufficient condition on the permanence of species and global attractivity in [1].

As we well know, systems like (1.1) and (1.2) without impulses are very important in the models of multispecies populations dynamics. Many important results on the permanence, extinction, global asymptotical stability for the two species or multi-species non-autonomous Lotka-Volterra systems and their special cases of periodic and almost periodic systems can be found in [2–14] and the references therein.

However, owing to many natural and man-made factors (e.g., fire, flooding, crop-dusting, deforestation, hunting, harvesting, etc.), the intrinsic discipline of biological species or ecological environment usually undergoes some discrete changes of relatively short duration at some fixed times. Such sudden changes can often be characterized mathematically in the form of impulses. In the last decade, much work has been done on the ecosystem with impulsive(see [1, 15–21] and the reference therein). Specially, the following system is considered in [22]: The author establish some new sufficient conditions on the permanence of species and global attractivity for system (1.3). However, the effect of discrete delays on the possibility of species survival has been an important subject in population biology. We find that infinite delays are considered in the system (1.3). In this paper, it is very meaningful that discrete delays are proposed in the impulsive system (1.1).

2. Preliminaries

Let . We define the Banach space of bounded continuous function with the supremum norm defined by: where , and Define , and for all and . Motivated by the biological background of system (1.1), we always assume that all solutions of system (1.1) satisfy the following initial condition: where .

It is obvious that the solution of system (1.1) with initial condition (2.2) is positive, that is, on the interval of the existence and piecewise continuous with points of discontinuity of the first kind at which it is left continuous, that is, the following relations are satisfied:

For system (1.1), we introduce the following assumptions:(H₁) functions and are bounded continuous on , and , and are nonnegative for all .(H₂) for each , there are positive constants such that and the functions are bounded for all and .

First, we consider the following impulsive logistic system where and are bounded and continuous functions defined on , for all , and impulsive coefficients for any are positive constants. We have the following results.

Lemma 2.1. Suppose that there is a positive constant such that and function is bounded on and . Then we have (a)there exist positive constants and such that for any positive solution of system (2.6);(b) for any two positive solutions and of system (2.6).

The proof of Lemma 2.1 can be found as Lemma  2.1 in [1] by Hou et al.

On the assumption (H₂), we firstly have the following result.

Lemma 2.2. If assumption holds, then there exist constants and such that for any

The proof of Lemma 2.2 is simple, we hence omit it here.

3. Main Results

Let be some fixed positive solution of the following impulsive logistic systems as the subsystems of system (1.1): On the permanence of all species for system (1.1), we have the following result.

Theorem 3.1. Suppose that assumptions hold. If there exist positive constants such that for each : and the functions are bounded for all and . Then the system (1.1) is permanent, that is, there are positive constants and such that for any positive solution of system (1.1).

Proof. Let be any positive solution of system (1.1). We first prove that the components of system (1.1) are bounded. From assumption (H₁) and the th equation of system (1.1), we have by the comparison theorem of impulsive differential equation, we have where is the solution of (3.1) with initial value . From the condition (3.2), we directly have Hence, from conclusion (a) of Lemma 2.1, we can obtain a constant , and there is a such that for all . Let and , we have Hence, we finally have

Next, we prove that there is a constant such that For any and directly from system (1.1), we have From condition (3.2), we can choose constants small enough and large enough such that for all and . Considering (3.5), by the comparison theorem of impulsive differential equation and the conclusion (b) of Lemma 2.1., we obtain for the above that there is a such that where is a globally uniformly attractive positive solution of system (3.1).

Claim 1. There is a constant such that for any positive solution of system (1.1). In fact, if Claim 1 is not true, then there is an integer and a positive solution of system (1.1) such that Hence, there is a constant such that On the other hand, by (3.13) there is a such that where and . By (3.11) and (3.16), we obtain for all . Thus, from (3.12) we finally obtain , which lead to a contradiction.

Claim 2. There is a constant such that for any positive solution of system (1.1).If Claim 2 is not true, then there is an integer and a sequence of initial function such that where constant is given in Claim 1. By Claim 1, for every m there are two time sequences and , satisfying: such that From the above proof, there is a constant such that for all . Further, there is an integer such that for all . From (3.11) and lemma 2.2., we can obtain where . Consequently, from (3.20) we have By (3.12), there is a large enough such that for all , and and , then, we obtain where . So, we choose such that for all , we have From (3.23), there is an integer such that for any and , we have where constant .So, when and , for any , from (3.11), (3.21), (3.25), and (3.26) we can obtain Consequently, from (3.20) and (3.25) it follows This leads to a contradiction. Therefore, Claim 2 is true. This completes the proof.When system (1.1) degenerates into the periodic case, then we can assume that there is a constant and an integer such that , , , and for all , and . From Remarks2.3 and  2.4 in [1], we can see the fixed positive solution of system (3.1) can be chosen to be the -periodic solution of system (3.1). Therefore, as a consequence of Theorem 3.1. we have the following result.

Corollary 3.2. Suppose that system (1.1) is -periodic and for each , Then, system (1.1) is permanent.

4. Numerical Example

In this section, we will give an example to demonstrate the effectiveness of our main results. We consider the following two species competitive system with delays and impulses: We take , , , , , , . Obviously, system (4.1) is periodic with period . For , we have , and for all . Consider the following impulsive logistic systems as the subsystems of system (4.1): According to the formula in [1], we can obtain that subsystem (4.2) has a unique globally asymptotically stable positive 2-periodic solution , which can be expressed in following form: where and ). Since we obtain that all conditions in Corollary 3.2 for system (1.1) holds. Therefore, from Theorem 3.1. we see that system (1.1) is permanent (see Figure 1).

Figure 1: Time series of and .

Acknowledgments

This paper was supported by the National Sciences Foundation of China (11071283), the Sciences Foundation of Shanxi (2009011005-3), the Young foundation of Shanxi province (no. 2011021001-1), research project supported by Shanxi Scholarship Council of China (2011-093), the Major Subject Foundation of Shanxi, and Doctoral Scientific Research fund of Xinjiang Medical University.

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