Abstract

This paper discuss a discrete periodic Volterra model with mutual interference and Beddington-DeAngelis functional response. By using the comparison theorem of difference equation, sufficient conditions are obtained for the permanence of the system. After that,we give an example to show the feasibility of our main result.

1. Introduction

In 1971, Hassell introduced the concept of mutual interference and established a Volterra model with mutual interference as follows: (see [1]) Recently, Wang and Zhu [2] proposed following system: Motivated by the works of Wang and Zhu [2], Lin and Chen [3] considered an almost periodic Volterra model with mutual interference and Beddington-DeAngelis function response as follows, which is the generalization of the model (1.2): Sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained by applying the comparison theorem of the differential equation and constructing a suitable Lyapunov functional.

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 [4, 5]). This motivated us to propose and study the discrete analogous of predator-prey system (1.4): where is the density of prey species at th generation, 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-dependant coefficient of the predator, respectively, is the capturing rate of the predator, is the rate of conversion of nutrients into the reproduction of the predator. Further, is mutual interference constant. In this paper, we assume that all the coefficients , are all positive -periodic sequences and . Here, for convenience, we denote , and where .

The remaining part of this paper is organized as follows: in Section 2 we will introduce some definitions and establish several useful lemmas. The permanence of system (1.4) is then studied in Section 3. In Section 4, we give an example to show the feasibility of our main result.

By the biological meaning, we will focus our discussion on the positive solution of system (1.4). So it is assumed that the initial conditions of (1.4) are of the form One can easily show that the solution of (1.4) with the initial condition (1.5) 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 (1.4) is said to be permanent if there exist positive constants , which are independent of the solution of system (1.4), such that for any position solution of system (1.4) satisfies

Lemma 2.2 ([6]). Assume that satisfies and for , where and are nonnegative sequences bounded above and below by positive constants. Then

Lemma 2.3 ([6]). Assume that satisfies and , where and are nonnegative sequences bounded above and below by positive constants and . Then

Lemma 2.4 ([7]). The problem with 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).

Lemma 2.5 ([8]). Suppose that and with for and . Assume that is nondecreasing with respect to the argument . If and are solutions of respectively, and then for all .

3. Permanence

In this section, we establish a permanent result for system (1.4).

Proposition 3.1. If holds, then for any positive solution of system (1.4), there exist positive constants and , which are independent of the solution of the system, such that

Proof. Let be any positive solution of system (1.4), from the first equation of (1.4), it follows that By applying Lemma 2.2, we obtain where Denote . Then from the second equation of (1.4), it follows that which leads to Consider 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 , produces This ends the proof of Proposition 3.1.

Proposition 3.2. Assume that hold, then for any positive solution of system (1.4), there exist positive constants and , which are independent of the solution of the system, such that where can be seen in Proposition 3.1.

Proof. Let be any positive solution of system (1.4). From , there exists a small enough positive constant such that
Also, according to Proposition 3.1, for above , there exists such that for , Then from the first equation of (1.4), for , we have Let , so the above inequality follows that From (3.12) and (3.15), by Lemma 2.3, we have Setting in the above inequality leads to where From above , there exists such that , . So from (3.5), we obtain that Consider the following auxiliary equation: By Lemma 2.4, (3.20) has at least one positive -periodic solution and we denote one of them as .
Let Then, Set Then In the following we distinguish three cases.
Case 1. is eventually positive. Then, from (3.24), we see that for any sufficient large . Hence, which implies that Case 2. is eventually negative. Then, from (3.23), we can also obtain (3.25).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 (3.24), we know that if . Hence, . From (3.24), and , we can obtain From (3.21) and (3.23), we easily obtain Setting in the above inequality leads to which together with that transformation , we have Thus, we complete the proof of Proposition 3.2.

Theorem 3.3. Assume that and hold, then system (1.4) is permanent.

It should be noticed that, from the proof 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 (1.4).

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 that
Clearly, conditions and are satisfied,then system (4.1) is permanent.
Figure 1 shows the dynamics behavior of system (4.1).

(a)

(b)

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(b)

Figure 1 

Dynamics behavior of system (4.1) with initial condition .

Example 4.2. Consider the following system: where .
By simple computation, we have . Thus, one could easily see that
Clearly, condition is satisfied and condition is not satisfied, but the system (4.3) is permanent. It shows that conditions and are sufficient for the system (1.4) but not necessary.
Figure 2 shows the dynamics behavior of system (4.3).