Research Article | Open Access

# Global Attractivity of a Periodic Delayed -Species Model of Facultative Mutualism

**Academic Editor:**Beatrice Paternoster

#### Abstract

Two classes of periodic -species Lotka-Volterra facultative mutualism systems with distributed delays are discussed. Based on the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin and the Lyapunov function method, some new sufficient conditions on the existence and global attractivity of positive periodic solutions are established.

#### 1. Introduction

Mutualism is the interaction of two species of organisms that benefits both [1]. In general, mutualism may be either obligate or facultative. Obligate mutualist may survive only by association, and facultative mutualist, while benefiting from the presence of each other, may also survive in the absence of any of them [2]. As it is well known, in recent years the nonautonomous and periodic population dynamical systems are extensively studied. The basic and important studied questions for these systems are the persistence, permanence, and extinction of species, global stability of systems and the existence of positive periodic solutions, positive almost periodic solutions and strictly positive solutions, and so forth. Many important and influential results have been established and can be found in many articles and books. Particularly, the existence of positive periodic solutions for various type population dynamical systems has been extensively studied in [1–16] and the references cited therein.

In [7], the authors studied the following delayed two-species model of facultative mutualism: By using the technique of coincidence degree and the Lyapunov functionals method, the sufficient conditions for the existence and globally asymptotic stability of positive periodic solutions are obtained for system (1). In [2], the authors considered the following periodic delayed two-species model of facultative mutualism: By means of the methods of coincidence degree and the Lyapunov functional, the sufficient conditions for the existence and globally asymptotic stability of positive periodic solutions are established for system (2). In [12], the following -species periodic Lotka-Volterra type competitive systems with feedback controls and finite and infinite distributed delays are discussed: where . By using the technique of coincidence degree and the Lyapunov functionals method, the sufficient conditions for the existence and global stability of positive periodic solutions are obtained for system (3).

Motivated by the above works, in this paper, we investigate the following two classes of species periodic model of facultative mutualism with finite distributed delays:

By using the technique of coincidence degree developed by Gaines and Mawhin in [17] and the Lyapunov functional method, we will establish some new sufficient conditions which guarantee that the system has at least one positive periodic solution and is globally attractive.

The organization of this paper is as follows. In the next section we will present some basic assumptions and main definitions and lemmas. In Section 3, conditions for the existence and global attractivity of positive periodic solution. In Section 4, two examples are given to illustrate that our main results are applicable. In the final section, we will discuss what we study in this paper and what we had in this paper.

#### 2. Preliminaries

In systems (4) and (5), we have that represent the density of species at time , respectively; represent the intrinsic growth rate of species at time , respectively; represent the intrapatch restriction density of species at time , respectively; represent the mutualism coefficients between species at time , respectively, while is a constant and may be . System (4) involves positive feedback terms
which can be regarded as the passive effect on the growth rate of a species, and system (5) involves negative feedback terms
which are due to gestation. In this paper, we always assume the following:(**H1**)?? are -periodic continuous functions with ; and are positive -periodic continuous functions; are nonnegative integrable functions on satisfying .

From the viewpoint of mathematical biology, in this paper for systems (4) and (5) we only consider the solution with the following initial condition: where are nonnegative continuous functions defined on satisfying .

In this paper, for any -periodic continuous function we denote the following:

In order to obtain the existence of positive -periodic solutions of systems (4) and (5), we will use the continuation theorem developed by Gaines and Mawhin in [17]. For the reader’s convenience, we will introduce the continuation theorem in the following.

Let and be two normed vetor spaces. Let ??be a linear operator and ??be a continuous operator. The operator is called a Fredholm operator of index zero, if?? and is a closed set in . If is a Fredholm operator of index zero, then there exist continuous projectors and such that and . It follows that is invertible and its inverse is denoted by and denote by an isomorphism of onto Ker?. Let be a bounded open subset of , we say that the operator is -compact on , where denotes the closure of in , if is bounded and is compact.

Lemma 1 (see [17]). *Let be a Fredholm operator of index zero and let be -compact on . If*(a)*for each and , ;*(b)*for each , ;*(c)*,**then the operator equation has at least one solution lying in .*

#### 3. Main Results

Now, for the convenience of statements, we denote the function

The following theorem is about the existence and global attractivity of positive periodic solutions of system (4).

Theorem 2. *Suppose that assumption holds and there exists a constant such that
**
and the algebraic equation
**
has a unique positive solution. Then, system (4) has a positive -periodic solution which is globally attractive.*

*Proof. *We firstly consider the existence of positive periodic solutions of system (4). For system (4), we introduce new variables such that
Then, system (4) is rewritten in the following form:
In order to apply Lemma 1 to system (14), we introduce the normed vector spaces and as follows. Let denote the space of all continuous function . We take
with norm
It is obvious that and are the Banach spaces.

We define a linear operator and a continuous operator as follows:
where
Further, we define continuous projectors and as follows:
We easily see that?? and . It is obvious that is closed in and . Since for any there are unique and with
such that , we have co?dim?. Therefore, is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to ) is given in the following form:
For convenience, we denote as follows:
Thus, we have
From formulas (23), we easily see that and are continuous operators. Furthermore, it can be verified that is compact for any open bounded set by using Arzela-Ascoli theorem and is bounded. Therefore, is -compact on for any open bounded subset .

Now, we reach the position to search for an appropriate open bounded subset for the application of the continuation theorem (Lemma 1) to system (4).

Corresponding to the operator equation with parameter , we have
where is given in (22).

Assume that is a solution of system (24) for some parameter . By integrating system (24) with the interval , we obtain the following:
Consequently,
From the continuity of , there exist constants such that
From (26) and (27), we obtain
Therefore, we further have
For each and , we have
Hence, from (26) we further obtain
Consequently,
From the assumptions of Theorem 2, we can obtain
Hence,
Consequently,
From (35), we further obtain
On the other hand, directly from system (14) we have
From (36) and (37), we have, for any ,

Further, from (29) and (37), we have, for any ,

Therefore, from (38) and (39) we have
It can be seen that the constants are independent of parameter .

For any , from (18) we can obtain
where
We consider the following algebraic equation:
From the assumption of Theorem 2, the equation has a unique positive solution . Hence, the equation has a unique solution .

Choosing constant large enough such that and , we define a bounded open set as follows:
It is clear that satisfies conditions (a) and (b) of Lemma 1. On the other hand, by direct calculating we can obtain
where

From the assumption of Theorem 2, we have

From this, we finally have
This shows that satisfies condition of Lemma 1. Therefore, system (14) has a -periodic solution . Further, from (13), system (4) has a positive -periodic solution .

Next, we will consider the global attractivity of positive periodic solutions of system (4). Choose positive constants , such that
From the assumption of Theorem 2, there exists constant such that for all we have
Let be any solution of system (4), we define Lyapunov function as follows:
Calculating the upper right derivation of along system (4) for , we have
Further, we define a Lyapunov function as follows:
Calculating the upper right derivation of , from (52) we finally can obtain, for all ,

Integrating from to on both sides of (54) and by (50) produces
then
By the definition of and (53), we have
Therefore, for we have
which, together with (49), leads to
and, hence, . From the boundedness of and (58), it follows that are bounded for . It is obvious that both and satisfy the equations of system (4), then by system (4) and the boundedness of and , we know that the derivatives and are bounded. Furthermore, we can obtain that and their derivatives remain bounded on . Therefore is uniformly continuous on . Thus, from (56), we have
Therefor,
This completes the proof of Theorem 2.

From the proof of Theorem 2, on the existence and global attractivity of positive periodic solutions of system (5), we have the following result.

Corollary 3. *Suppose that assumption holds and there exists a constant such that
**
and the algebraic equation
**
has a unique positive solution. Then, system (5) has a positive -periodic solution which is globally attractive. *

#### 4. Two Examples

*Example 4. *First, we consider the following delayed system:
Corresponding to system (4), ,??,??, by direct calculation, we can get
and the following equations have unique positive solutions:
where
It is clear that all the conditions of Theorem 2 hold. Hence, system (64) has a positive periodic solution which is globally attractive.

*Example 5. *Next, we consider the following delayed system:
Corresponding to system (4), ,??,??, by direct calculation we can get
and the following equations have a unique positive solutions:
where
Clearly, the conditions of Theorem 2 do not hold.

From Figure 1 we can see that system (68) has no globally attractive positive periodic solution.

*Remark 6. *From these two examples, we can see that if the conditions of Theorem 2 hold, then the system has a globally attractive positive periodic solution. If the conditions of Theorem 2 do not hold, then the system has no globally attractive positive periodic solution.

#### 5. Conclusions

Mawhin’s continuation theorem is a powerful tool for studying the existence of periodic solutions of periodic high-dimensional time-delayed problems. When dealing with time-delayed problem, it is very convenient and the result is relatively simple. The most critical thing in the using of the theorem is the calculation of topological degree, that is, the condition (c) of the theorem.

In this paper, motivated by [2, 7] of Liu et al. we propose two classes of periodic -species Lotka-Volterra facultative mutualism systems with distributed delays. By applying the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin and the Lyapunov function method, we easily obtain sufficient conditions for the existence and global attractivity of positive periodic solutions of the system. From Theorem 2 and Corollary 3, we can see that the distributed time delays have effect on the existence and global attractivity of positive periodic solutions, and conditions (11) and (62) are very crucial to find the criteria for globally attractive positive periodic solutions. Further, the conditions (11) and (62) indicate that the undelayed intraspecific competition dominates the delayed intraspecific reproduction, and the intraspecific competition is more significant than the interspecific cooperation.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grants nos. 11271312, 11261056, and 11261058), the China Postdoctoral Science Foundation (Grant no. 20110491750), and the Natural Science Foundation of Xinjiang (Grant no. 2011211B08).

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#### Copyright

Copyright © 2013 Ahmadjan Muhammadhaji and Zhidong Teng. 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.