#### Abstract

We propose a discrete multispecies cooperation and competition predator-prey systems. For general nonautonomous case, sufficient conditions which ensure the permanence and the global stability of the system are obtained; for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system are obtained.

#### 1. Introduction

In this paper, we consider the dynamic behavior of the following non-autonomous discrete -species cooperation and competition predator-prey systems where is the density of prey species at th generation. is the density of predator species at th generation.

Dynamic behaviors of population models governed by difference equations had been studied by a number of papers, see [1–4] and the references cited therein. It has been found that the autonomous discrete systems can demonstrate quite rich and complicated dynamics, see [5, 6]. Recently, more and more scholars paid attention to the non-autonomous discrete population models, since such kind of model could be more appropriate.

May [7] suggested the following set of equations to describe a pair of mutualists: where are the densities of the species at time , respectively. are positive constants. He showed that system (1.2) has a globally asymptotically stable equilibrium point in the region .

Bai et al. [8] argued that the discrete case of cooperative system is more appropriate, and they proposed the following system:

Chen [9] investigated the dynamic behavior of the following discrete -species Lotka-Volterra competition predator-prey systems he investigated the dynamic behavior of the system (1.4).

The aim of this paper is, by further developing the analysis technique of Huo and Li [10] and Chen [9], to obtain a set of sufficient condition which ensure the permanence and the global stability of the system (1.1); for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system (1.1) are obtained.

We say that system (1.1) is permanent if there are positive constants and such that for each positive solution of system (1.1) satisfies for all .

Throughout this paper, we assume that , , , , , , , are all bounded nonnegative sequence, and use the following notations for any bounded sequence

For biological reasons, we only consider solution with

Then system (1.1) has a positive solution passing through .

The organization of this paper is as follows. In Section 2, we obtain sufficient conditions which guarantee the permanence of the system (1.1). In Section 3, we obtain sufficient conditions which guarantee the global stability of the positive solution of system (1.1). As a consequence, for periodic case, we obtain sufficient conditions which ensure the existence of a globally stable positive solution of system (1.1).

#### 2. Permanence

In this section, we establish permanence results for system (1.1).

Lemma 2.1 (see [11]). *Let . For any fixed is a non-decreasing function with respect to , and for , the following inequalities hold:
**
If , then for all .*

Now let one consider the following single species discrete model: where and are strictly positive sequences of real numbers defined for and .

Lemma 2.2 (see [12]). *Any solution of system (2.5) with initial condition satisfies
**
where
*

Proposition 2.3. *Assume that
**
holds, then
**
where
*

*Proof. *Let be any positive solution of system (1.1), from the th equation of (1.1) we have
By applying Lemmas 2.1 and 2.2, it immediately follows that
For any positive constant small enough, it follows from (2.9) that there exists enough large such that

From the th equation of the system (1.1) and (2.10), we can obtain
Condition (2.5) shows that Lemmas 2.1 and 2.2 could be applied to (2.11), and so, by applying Lemmas 2.1 and 2.2, it immediately follows that
Setting in the above inequality leads to
This completes the proof of Proposition 2.3.

Now we are in the position of stating the permanence of the system (1.1).

Theorem 2.4. *In addition to (2.5), assume further that
**
then system (1.1) is permanent, where
*

*Proof. *By applying Proposition 2.3, we see that to end the proof of Theorem 2.4, it is enough to show that under the conditions of Theorem 2.4,
From Proposition 2.3, , there exists a ,
From the th equation of system (1.1) and (2.17), we have
for all .

By applying Lemmas 2.1 and 2.2 to (2.18), it immediately follows that
Setting in (2.19) leads to

Then, for any positive constant small enough, from (2.20) we know that there exists an enough large such that
Equations (2.17), (2.21) combining with the th equation of the system (1.1) leads to
under the condition (2.14), by applying Lemmas 2.1 and 2.2 to (2.22), it immediately follows that
Setting in (2.23) leads to
This ends the proof of Theorem 2.4.

It should be noticed that, under the assumption of Theorem 2.4, the set is an invariant set of system (1.1).

#### 3. Global Stability

Now we study the stability property of the positive solution of system (1.1).

Theorem 3.1. *Assume that
**
Then for any two positive solution and of system (1.1), one has
*

*Proof. *Let
Then system (1.1) is equivalent to
So,
where , to complete the proof, it suffices to show that
In view of (3.1), we can choose small enough such that
For the above , according to Theorem 2.4 in Section 2, there exists a such that
for all .

Noticing that implies that lies between and lies between and . From (3.5), we get
Let , then . In view of (3.9), for , we get
This implies
From (3.11), we have
This ends the proof of Theorem 3.1.

#### 4. Existence and Stability of Periodic Solution

In this section, we further assume that the coefficients of the system (1.1) satisfies (4.1).

There exists a positive integer such that for ,

Our first result concerned with the existence of a positive periodic solution of system (1.1).

Theorem 4.1. *Assume that (2.5) and (2.14) hold, then system (1.1) admits at least one positive -periodic solution which ones denotes by .*

*Proof. *As noted at the end of Section 2,
is an invariant set of system (1.1). Thus, we can define a mapping on by
for . Obviously, depends continuously on . Thus, is continuous and maps the compact set into itself. Therefore, has a fixed point. It is easy to see that the solution passing through this fixed point is an -periodic solution of the system (1.1). This completes the proof of Theorem 4.1.

Theorem 4.2. *Assume that (2.5), (2.14), and (3.1) hold, then system (1.1) has a global stable positive -periodic solution.*

*Proof. *Under the assumption of Theorem 4.2, it follows from Theorem 4.1 that system (1.1) admits at least one positive -periodic solution. Also, Theorem 3.1 ensures the positive solution to be globally stable. This ends the proof of Theorem 4.2.