Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 205481 | 9 pages | https://doi.org/10.1155/2009/205481

Permanence of a Discrete Periodic Volterra Model with Mutual Interference

Academic Editor: Antonia Vecchio
Received05 Dec 2008
Accepted12 Feb 2009
Published01 Apr 2009

Abstract

This paper discusses a discrete periodic Volterra model with mutual interference and Holling II type functional response. Firstly, sufficient conditions are obtained for the permanence of the system. After that, we give an example to show the feasibility of our main results.

1. Introduction

In 1971, Hassell introduced the concept of mutual interference between the predators and preys. Hassell [1] established a Volterra model with mutual interference as follows: where denote mutual interference constant and

Motivated by the works of Hassell [1], Wang and Zhu [2] considered the following Volterra model with mutual interference and Holling II type functional response: Sufficient conditions which guarantee the existence, uniqueness, and global attractivity of positive periodic solution are obtained by employing Mawhin's continuation theorem and constructing suitable Lyapunov function.

On the other hand, it has been found that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations (see [315]). However, to the best of the author's knowledge, until today, there are still no scholars propose and study a discrete-time analogue of system (1.2). Therefore, the main purpose of this paper is to study the following discrete periodic Volterra model with mutual interference and Holling II type functional response: where is the density of prey species at th generation and is the density of predator species at th generation. Also, denote the intrinsic growth rate and density-dependent coefficient of the prey, respectively, denote the death rate and density-dependent coefficient of the predator, respectively, denote the capturing rate of the predator and represent the transformation from preys to predators. Further, is mutual interference constant and is a positive constant. In this paper, we always assume that , , , are positive -periodic sequences and . Here, for convenience, we denote , , and , where

This paper is organized as follows. In Section 2, we will introduce a definition and establish several useful lemmas. The permanence of system (1.3) is then studied in Section 3. In Section 4, we give an example to show the feasibility of our main results.

From the view point of biology, we only need to focus our discussion on the positive solution of system (1.3). So it is assumed that the initial conditions of (1.3) are of the formOne can easily show that the solution of (1.3) with the initial condition (1.4) are defined and remain positive for all where

2. Preliminaries

In this section, we will introduce the definition of permanence and several useful lemmas.

Definition 2.1. System (1.3) is said to be permanent if there exist positive constants which are independent of the solution of the system, such that for any positive solution of system (1.3) satisfies

Lemma 2.2. Assume that satisfieswhere and are positive sequences, and Then, one haswhere

Lemma 2.3. Assume that satisfieswhere and are positive sequences, and Also, and Then, one haswhere

Proof. The proofs of Lemmas 2.2 and 2.3 are very similar to that of [8, Lemmas 1 and 2], respectively. So, we omit the detail here.

The following Lemma 2.4 is Lemma 2.2 of Fan and Li [12].

Lemma 2.4. The problemwith has at least one periodic positive solution if both and are -periodic sequences with Moreover, if is a constant and then for sufficiently large, where is any solution of (2.6).

The following comparison theorem for the difference equation is Theorem 2.1 of L. Wang and M. Q. Wang [15, page 241].

Lemma 2.5. Suppose that and with () for and Assume that is nondecreasing with respect to the argument If and are solutions ofrespectively, and ( thenfor all

3. Permanence

In this section, we establish a permanent result for system (1.3).

Proposition 3.1. If holds, then for any positive solution of system (1.3), there exist positive constants and , which are independent of the solution of the system, such that

Proof. Let be any positive solution of system (1.3), from the first equation of (1.3), it follows thatBy applying Lemma 2.2, we obtainwhere
Denote . Then, from the second equation of (1.3), it follows thatwhich leads toConsider the following auxiliary equation:By Lemma 2.4, (3.7) has at least one positive -periodic solution and we denote one of them as Now and Lemma 2.4 imply for sufficiently large, where is any solution of (3.7). Consider the following function:It is not difficult to see that is nondecreasing with respect to the argument Then, applying Lemma 2.5 to (3.6) and (3.7), we easily obtain that So which together with that transformation producesThus, we complete the proof of Proposition 3.1.

Proposition 3.2. Assume thatholds, then for any positive solution of system (1.3), there exist positive constants and , which are independent of the solution of the system, such thatwhere can be seen in Proposition 3.1.

Proof. Let be any positive solution of system (1.3). From , there exists a small enough positive constant such thatAlso, according to Proposition 3.1, for above , there exists such that for Then, from the first equation of (1.3), for we haveLet so the above inequality follows thatBecause and we haveHere, we use the fact From (3.12) and (3.15), by Lemma 2.3, we haveSetting in the above inequality leads towhereFor above , there exists such that for So from (3.5), we obtain thatConsider the following auxiliary equation:By Lemma 2.4, (3.21) has at least one positive -periodic solution and we denote one of them as
Let
Then,
SetThen,In the following we distinguish three cases.
Case 1. is eventually positive. Then, from (3.25), we see that for any sufficiently large . Hence, which implies that
Case 2. is eventually negative. Then, from (3.24), we can also obtain (3.26).
Case 3. oscillates about zero. In this case, we let be the positive semicycle of where denotes the first element of the th positive semicycle of From (3.25), we know that if Hence, From (3.25), and we can obtain
From (3.22) and (3.24), we easily obtainSetting in the above inequality leads towhich together with that transformation we haveThus, we complete the proof of Proposition 3.2.

Theorem 3.3. Assume that and hold, then system (1.3) is permanent.

It should be noticed that, from the proofs of Propositions 3.1 and 3.2, one knows that under the conditions of Theorem 3.3, the set is an invariant set of system (1.3).

4. Example

In this section, we give an example to show the feasibility of our main result.

Example 4.1. Consider the following system where , , , , , ,,

By simple computation, we have . Thus, one could easily see thatClearly, conditions and are satisfied, then system (1.3) is permanent.

Figure 1 shows the dynamics behavior of system (1.3).

Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions which greatly improve the presentation of the paper. This work was supported by the Program for New Century Excellent Talents in Fujian Province University (NCETFJ) and the Foundation of Fujian Education Bureau (JB08028).

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Copyright © 2009 Lijuan Chen and Liujuan Chen. 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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