Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 656549 | 19 pages | https://doi.org/10.1155/2009/656549

Extinction in Nonautonomous Discrete Lotka-Volterra Competitive System with Pure Delays and Feedback Controls

Academic Editor: Elena Braverman
Received25 Jun 2009
Accepted12 Oct 2009
Published08 Nov 2009

Abstract

The paper discusses a nonautonomous discrete time Lotka-Volterra competitive system with pure delays and feedback controls. New sufficient conditions for which a part of the -species is driven to extinction are established by using the method of multiple discrete Lyapunov functionals.

1. Introduction

The coexistence and global stability of population models are the interesting subjects in mathematical biology. Many authors have argued that the discrete time models are governed by differential equations which are more appropriate than the continuous ones to describe the dynamics of population when the population has nonoverlapping generations, a lot has been done on discrete Lotka-Volterra systems.

May in [1] firstly considered the following autonomous discrete two-species Lotka-Volterra competitive system:

and studied the stable points, stable cycles, and the chaos behaivor. Further, Lu and Wang [2] studied the permanence and global attractivity of this system.

Chen and Zhou in [3] considered the following periodic discrete two-species Lotka-Volterra competitive system:

and studied the permanence and existence of a periodic solution, and further, sufficient conditions are established on the global stability of the periodic solution.

Zhang and Zhou in [4] investigated the following nonautonomous discrete two-species Lotka-Volterra competitive system:

Some sufficient conditions were obtained for the permanence of the system.

Wang et al. in [5] studied the following general nonautonomous discrete -species Lotka-Volterra systems:

By applying the linear method and constructing the appropriate Lyapunov functionals, the author established the sufficient conditions which guarantee that any positive solution of this system is stable and attracts others, and obtained some applications of main results.

Muroya in [6, 7] considered the following general nonautonomous discrete -species Lotka-Volterra systems:

and related pure delays models, that is, for all . The author obtained the permanence and the global asymptotically stable by applying mean-value conditions and the method of constructing discrete Lyapunov functionals.

Liao et al. in [8] discussed the following general discrete nonautonomous -species competitive system with feedback controls:

Some sufficient conditions are established on the permanence and the global stability of the system.

Recently, we see that in [9, 10] the authors studied the following nonautonomous continuous Lotka-Volterra competitive system with pure delays and feedback controls:

The sufficient conditions for which a part of the -species is driven to extinction and the surplus part of the -species remains permanence are established.

However, we see that for general discrete -species population systems the results for which a part of the -species is driven to extinction and the surplus part of the -species remains the permanence, up to now, are still not obtained. Therefore, motivated by the above works, in this paper we study the following discrete nonautonomous Lotka-Volterra competitive system with pure delays and feedback controls

The main purpose is to establish a criterion for which guarantee the part species in system (1.8) is driven to extinction. The method used in this paper is to constructing the multiple discrete Lyapunov functions. On the permanence of the surplus species , owing to the length of this paper, we will give the discussion in another paper.

This paper is organized as follows. In next section, as preliminaries, some assumptions and useful lemmas are introduced. In Section 3, the main results of this paper on the extinction of a part of the -species of system (1.8) are established. In Section 4, an example is presented to illustrate the feasibility of our results.

2. Preliminaries

Let denote the set of all nonnegative integers. For any bounded sequence , we denote , . Throughout this paper, we introduce the following assumptions.

is a bounded sequence defined on ; , , and are nonnegative bounded sequences defined on ; , , and are nonnegative integers.Sequences satisfy for all .There exist positive integers and such that for each There exists positive integer such that for each

Let and . We denote by the interior of . For any nonnegative constants and with , we denote by the set of all integers in the interval . For some integer , we denote by the space of all nonnegative discrete time function with norm .

Let . Motivated by the biological background of system (1.8), in this paper we only consider all solutions of system (1.8) that satisfy the following initial conditions:

where and . For any , let , by the fundamental theory of difference equations, system (1.8) has a unique solution satisfying the initial condition (2.3), where and . It is obvious that solution is positive, that is, and for all .

We first consider the following nonautonomous difference inequality system:

where and are bounded sequences defined on and for all . We have the following result.

Lemma 2.1 (see [11]). Assume that there exists an integer such that then there exists a constant such that for any nonnegative solution of system (2.4) with initial value , where is some integer,

Next, we consider the following nonautonomous linear difference equation:

where and are nonnegative bounded sequences defined on . We have the following results.

Lemma 2.2 (see [11]). Assume that there exists an integer such that then there exists a constant such that for any nonnegative solution of system (2.7) with initial value , where is some integer,

Lemma 2.3 (see [11]). Assume that the conditions of Lemma 2.2 hold, then for any constants and there exist positive constants and such that for any and , when for all , one has where is the solution of (2.7) with initial value .

Lemma 2.4. Assume that assumptions hold, then there exists a constant such that for any positive solution of system (1.8).

Proof. Let be any positive solution of system (1.8). For each , we have then, for any integer and with , summing inequality (2.12) from to , we obtain Therefore, for any integer , from (2.13) and the first equation of system (1.8) we obtain Since for any and where , we have from (2.14), for any , where .
We consider the following auxiliary equation:
where , then by assumption and applying Lemma 2.1 there exists a constant such that for any positive solution of (2.17). Therefore, from the comparison theorem of difference equation, we finally obtain Further form inequality (2.19), there exists a positive constant such that Thus, from the second equation of system (1.8), we obtain for all . We consider the following auxiliary equation: then by assumption and applying Lemma 2.2, there exists a constant such that for any positive solution of (2.22). Let be the solution of (2.22) with initial value , then from the comparison theorem of difference equation, we have for all . Thus, we finally obtain Let , then from (2.19) and (2.24) we finally see that the conclusions of Lemma 2.4 hold.

3. Main Results

In this section, we discuss the extinction of the part of species of system (1.8). Define functions as follows:

where .

Theorem 3.1. Assume that assumptions hold and there exists an integer such that for any there exists an integer such that then for each one has for any positive solution of system (1.8).

Proof. From assumption , there exist constant and integer such that We first prove that . Let and . From conditions (3.2) we can find positive constants and integer such that for all and . Consequently, for all and .
Let be any positive solution of system (1.8). Constructing the following discrete Lyapunov functional
By calculating, we obtain From inequalities (3.7)–(3.11), we can obtain For any , we choose an integer such that , then from (3.6) and (3.14) we further have where On the other hand, from assumptions , we have for all , where constant is given in Lemma 2.4. Hence, there exist a positive constant such that From (3.15) and (3.18), we obtain for all , where From (3.19), we finally obtain
Next, we consider the second equation of system (1.8), applying Lemma 2.3 we can easily obtain as .
Now, we suppose that for any , we have obtained
for all . We further will prove From conditions (3.2), we can choose positive constants , , , , and integer such that for all and , where . Consequently, for all and . Constructing the following discrete Lyapunov functional Calculating , similarly to , we can obtain for all . From inequalities (3.26)–(3.30), we further obtain for all , where From (3.19) we have as . Hence, Thus, from (3.22) we can obtain that there exists an integer such that for all . By calculating, from (3.33) we obtain From this, a similar argument as in the proof of (3.15)–(3.19), we further can obtain where and are two positive constants. From (3.38), we finally obtain
Next, we consider the second equation of system (1.8), applying Lemma 2.3 we can easily obtain as .
Finally, according to the induction principle, we have
This completes the proof of Theorem 3.1.

As consequences of Theorem 3.1, we consider the following two special cases of system (1.8).

Case 1. Nondelayed nonautonomous discrete -species Lotka-Volterra competitive systems with feedback controls For system (3.41), assumptions and become into the following form is a bounded sequence defined on ; , and are nonnegative bounded sequences defined on .There exist positive integers and such that for each

Directly from Theorem 3.1, we have the following corollary.

Corollary 3.2. Assume that assumptions , , and hold and there exists a integer such that for any there exists an integer such that then for each one has for any positive solution of system (3.41).

Case 2. Pure delayed nonautonomous discrete -species Lotka-Volterra competitive systems without feedback controls For system (3.41), assumption becomes into the following form. is a bounded sequence defined on ; and are nonnegative bounded sequences defined on ; and are nonnegative integers.

Directly from Theorem 3.1, we have the following corollary.

Corollary 3.3. Assume that assumptions and hold and there exists an integer such that for any there exists an integer such that then for each one has for any positive solution of system (3.45).

Remark 3.4. By comparison, we easily see that the results obtained in this paper are a very good extension of the corresponding results obtained in [9] on the extinction of species for nonautonomous continuous Lotka-Volterra competitive system with pure-delays and feedback controls to discrete ones.

4. An Example

We consider the following periodic discrete three-species Lotka-Volterra competitive system with pure-delays and feedback controls:

where