Global Attractivity and Periodic Solution of a Discrete Multispecies Cooperation and Competition Predator-Prey System
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.
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  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.  argued that the discrete case of cooperative system is more appropriate, and they proposed the following system:
The aim of this paper is, by further developing the analysis technique of Huo and Li  and Chen , 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.
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).
In this section, we establish permanence results for system (1.1).
Lemma 2.1 (see ). 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 .
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).
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.
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
Then system (1.1) is equivalent to
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
There exists a positive integer such that for ,
Our first result concerned with the existence of a positive periodic solution of system (1.1).
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.
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.
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