Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 710353 | 15 pages | https://doi.org/10.1155/2009/710353

Global Dynamics of Discrete Competitive Models with Large Intrinsic Growth Rates

Academic Editor: Yong Zhou
Received15 May 2009
Accepted27 Jun 2009
Published06 Aug 2009

Abstract

The global dynamics of discrete competitive model of Lotka-Volterra type with two species is considered. Earlier works have shown that the unique positive equilibrium is globally attractive under the assumption that the intrinsic growth rates of the two competitive species are less than 1+ln 2, and further the unique positive equilibrium is globally asymptotically stable under the stronger condition that the intrinsic growth rates of the two competitive species are less than or equal to 1. We prove that the system can also be globally asymptotically stable when the intrinsic growth rates of the two competitive species are greater than 1 and globally attractive when the intrinsic growth rates of the two competitive species are greater than 1 + ln 2.

1. Introduction

In this paper, we further consider the global dynamics of discrete Lotka-Volterra model

with positive initial conditions . Here is the density of population at th generation, is the intrinsic growth rate of population . represents the intensity of intraspecific competition or interspecific action of the two species. It is assumed that and are positive constants throughout this paper.

The discrete Lotka-Volterra models have wide applications in applied mathematics. They were first established in biomathematical background and then have proved to be a rich source in analysis for the dynamical systems in different research fields such as physics, chemistry, and economy [1].

Model (1.1) was first introduced in May [2] then was investigated by many authors [314]. The difference system (1.1) is autonomous, some of the works mentioned above are the nonautonomous case of (1.1). Many results about the global dynamics of (1.1) such as permanence, global attractivity, global asymptotical stability have been obtained. For example, it is shown in [10] that (1.1) can be globally asymptotically stable when . And from [3] we know that (1.1) can be globally attractive under the assumption that .

It is well known that for the single-species Logistic model the positive equilibrium is globally asymptotically stable if and only if and there exists periodic cycles when . When , (1.2) exhibits chaotic behavior (e.g., see [15]). That is, the global dynamics of (1.2) is very complex when the intrinsic growth rate is large. It is clear that (1.1) is a coupling of two equations described by (1.2). And it is proved in [16] that (1.1) also exhibits chaotic behavior when . Therefore, questions can be proposed naturally: how to investigate the global dynamics of (1.1) when ? Can model (1.1) be also globally asymptotically stable when ? Can model (1.1) be globally attractive when ?

Our aim of this paper is to obtain some global dynamics of (1.1) when the intrinsic growth rate is large () and give answers to the above questions. First we obtain permanent result of (1.1), then global attractivity of (1.1) is obtained through geometrical properties of (1.1). Last, we obtain the global asymptotical stability of (1.1) by applying a theorem in [10]. After these theoretical results for (1.1) obtained, we give numerical examples to confirm these theoretical results and show that our theoretical results imply that (1.1) can be globally attractive when and (1.1) can also be globally asymptotically stable when .

The paper is organized as follows. We give some preliminaries in Section 2. In Section 3, we discuss permanence, global attractivity, and global asymptotical stability of (1.1) theoretically. Numerical examples are given in Section 4 to show the feasibility of the assumptions of the main results and on the other hand, to show that our main results can be applied to larger intrinsic growth rates than earlier works. Brief conclusion is given in Section 5.

2. Preliminaries

A pair of sequences of real positive numbers that satisfies (1.1) is a positive solution of (1.1). It is clear that the solutions of system (1.1) with initial values are positive ones, which is accordant with the biological background of (1.1). That is, we only need to investigate the dynamics of system (1.1) in the plane domain

If a solution of (1.1) is a pair of real constants , then it is an equilibrium of (1.1).

Lemma 2.1. Assume that then system (1.1) has four equilibria.

Proof. Solving the following scalar equation system: We obtain that the four equilibria of system (1.1) are respectively. Here and the following, we denote

The equilibria and are the so-called “boundary equilibrium.” If we further assume that which implies that , then is the unique positive equilibrium of (1.1).

Lemma 2.2. Denote , , then the maximum of in the domain is (1)(2)Denote , , then the maximum of in domain is(1)(2)

Proof. For any fixed , let we get . Note that therefore, the maximum of in domain exists. Direct computation gives , we omit the details. Similarly, exists and its value can be obtained directly.

Lemma 2.3. (1) If :   is monotonously increasing, then for each positive sequence , If : is monotonously decreasing, then for each positive sequence ,
(2) For any positive sequences one has

Proof. One can refer to [17] for the proof of this lemma.

Next we give some definitions that will be used in this paper.

Definition 2.4. System (1.1) is permanent if there exist positive constants and such that

Definition 2.5. System (1.1) is strongly persistent if each positive solution of (1.1) satisfies

Definition 2.6. The solution of system (1.1) with initial values , is said to be stable if for any , there is a such that if , we have for all positive integers , where is the solution of (1.1) with initial values .

Definition 2.7. Suppose that is the positive equilibrium solution of (1.1). If for each positive solution of system (1.1), we have as , we say (1.1) is globally attractive or the equilibrium of (1.1) is globally  attractive.

Definition 2.8. The positive equilibrium solution of (1.1) or system (1.1) is said to be globally asymptotically stable if this equilibrium is stable and globally attractive.
The following lemma can be found in [10].

Lemma 2.9. Consider the following difference system: Assume that (i)there exist positive constant and positive constants such that for all large ;(ii)system (2.14) is strongly persistent;(iii)for any positive solution of system (2.14), for all large.
Then system (2.14) is globally asymptotically stable.

3. Main Results

In this section, we will obtain the permanence, global attractivity, and global asymptotical stability of system (1.1) when .

Lemma 3.1. For every positive solution of system (1.1) with initial values , one has where

Proof. Note that for all , therefore Here we used for . Then The proof of is similar.

Lemma 3.2. Assume that is the solution of (1.1) with initial values   and then where and are the same as in Lemma 3.1.

Proof. The proof of this lemma is similar to that of [3, Proposition 2].

Note that , therefore, system (1.1) is permanent from Lemmas 3.1 and 3.2 under the assumption (3.8).

Theorem 3.3. Assume that (3.8) is satisfied then system (1.1) with initial values   is permanent.

Theorem 3.4. Assume that (2.6), and (3.8) hold. The coefficients of (1.1) satisfy   and (1)(2)Further, assume that where   and   are defined in Lemma 2.2. Then the unique positive equilibrium of (1.1) is globally attractive.

Proof. If we denote for any positive solution of system (1.1) with initial conditions , we have from Theorem 3.3 and Definition 2.4. Moreover, from (2) of Lemma 2.3.
Note (3.13), the inequalities (3.14)–(3.17) can be written as follows:
From (3.18)–(3.21), it is clear that lies in the domain while lies in the domain    (see (2.7). Therefore, from (3.11) and Lemma 2.2, the maximum of in domain is , the maximum of in domain is . Then But in domain , only the point satisfies these two inequalities, then
At this point, we claim that Note (3.11), we must consider the following four cases to prove claim (3.25):Case (i):Case (ii):Case (iii):Case (iv):
It is easy to verify that the function is monotonously increasing when and monotonously decreasing when . With this fact and Lemma 2.3, the proof of the claim is given as below.
Case (i)
We rearrange the two equations of (1.1) as Note that we have for sufficiently large. Then That is The inequalities (3.24), (3.29) together with (3.30) imply that From (3.13) and (3.24), we get
Case (ii)
Similarly, we have From (3.13), (3.24), and (3.33), we get . And from (3.13), (3.24), and (3.34), follows.
The proof of Case (iii) is similar to that of Case (ii).
Case (iv)
We have Therefore, are consequent from (3.13), (3.24), and (3.35).
The proof of claim (3.25) is completed. Note (3.24) and (3.25), for any positive solution of system (1.1). That is, (1.1) is globally attractive according to Definition 2.7.

Theorem 3.5. Assume that the assumptions of Theorem 3.4 are satisfied, moreover, then the unique positive equilibrium of system (1.1) is globally asymptotically stable.

Proof. From Theorem 3.3, system (1.1) is strongly persistent. That is, condition (ii) of Lemma 2.9 is satisfied.
implies that Set , it is clear that . Thus, condition (i) of Lemma 2.9 is satisfied.
Let be any positive solution of system (1.1). We show below that for all large . By Theorem 3.4, we know that is globally attractive. That is From (3.38) we first select , such that Further from (3.40), we know that there exists and , such that respectively. Then denote , we get for from (3.41). That is, (3.39) is true for all sufficiently large . Therefore, condition (iii) of Lemma 2.9 is satisfied. The proof is completed by applying Lemma 2.9.

Theorem 3.6. Assume that (2.6), and (3.8) hold, the coefficients of (1.1) satisfy and then the positive equilibrium of system (1.1) is globally asymptotically stable.

Proof. From the proof of Theorem 3.4, we know that lies in domain . Therefore, we obtain from Lemma 2.2. That is, condition (iii) of Lemma 2.9 is satisfied. Conditions (i) and (ii) of Lemma 2.9 are also satisfied. Then the positive equilibrium of system (1.1) is globally asymptotically stable by applying Lemma 2.9.

4. Numerical Examples

In this section, we give two numerical examples to show the feasibility of the assumptions of the results. The first example also shows that system (1.1) can be globally attractive when the intrinsic growth rates of the two species are greater than and this result can be obtained by Theorem 3.4.

Example 4.1. Consider the following case of system (1.1): then
We see that the conditions of Theorem 3.4 are satisfied. Therefore, the positive equilibrium of system (1.1) is globally attractive (see Figure 1). But this result cannot be obtained by [3, Theorem 3] when consider the autonomous case of this theorem(the model studied in [3] is nonautonomous). In fact, the condition of [3, Theorem 3] must satisfy when , that is, . In Example 4.1, .
The following example shows that system (1.1) can be globally asymptotically stable when the intrinsic growth rates of the two species are greater than 1, and this result can be obtained by Theorem 3.6.

Example 4.2. Consider the following case of system (1.1): then
It is clear that the conditions of Theorem 3.6 are satisfied. Thus by Theorem 3.6 the positive equilibrium of system (1.1) is globally asymptotically stable (see Figure 2).
Example 4.2 shows that our results improve [12, Theorem 3] by providing estimates for the smallness of  . The work in [10, Theorem 2] states that if , then the positive equilibrium is globally asymptotically stable. Thus the global asymptotical stability of system (1.1) in the case of Example 4.2 cannot be obtained by [10, Theorem 2] because of .

5. Conclusion

In this paper, we further discuss the global dynamics of a discrete autonomous competitive model of Lotka-Volterra type. Sufficient conditions are obtained to guarantee the permanence, global attractivity, and global asymptotical stability of the system. These conditions are expressed by the coefficients of the model and can be easily verified. Numerical examples are also given to show the feasibility of the conditions.

Earlier works have shown that the system of this type can be globally attractive when the intrinsic growth rates of the two species are less than ([3], for single-species system see [18]). It is shown in [10] that the system can be globally asymptotically stable when the intrinsic growth rates of the two species are less than 1. In [16], it is shown that the system can exhibit chaotic behavior when the intrinsic growth rates of the two species are equal and greater than 3.13. But the global dynamics of the system is not clear enough when the intrinsic growth rates of the two species are greater than 1 and less than 3.13. We obtain that the system can also be globally asymptotically stable when the intrinsic growth rates of the two competitive species are greater than 1 and globally attractive when the intrinsic growth rates of the two competitive species are greater than .

For the global stability of the system, the following condition in Theorem 3.5: can be reduced to the following by direct computation: And the above inequalities imply that can be greater than 1 while the system is globally asymptotically stable.

Acknowledgments

The authors would like to thank the Editor Professor Yong Zhou and the referees for their valuable comments and suggestions. This work was supported by the Foundation of Jiangsu Polytechnic University (ZMF09020020).

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Copyright © 2009 Chunqing Wu and Jing-an Cui. 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.


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