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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 940287, 19 pages
http://dx.doi.org/10.1155/2012/940287
Research Article

Application of Mawhin's Coincidence Degree and Matrix Spectral Theory to a Delayed System

1Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
2Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, 2000 Maribor, Slovenia
3School of Electrical and Electronic Engineering, Nanyang Technological University, 639798, Singapore
4School of Basic Sciences, Indian Institute of Technology Mandi, Mandi 175001, India

Received 11 August 2012; Accepted 27 August 2012

Academic Editor: Jifeng Chu

Copyright © 2012 Yong-Hui Xia et al. 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.

Abstract

This paper gives an application of Mawhin’s coincidence degree and matrix spectral theory to a predator-prey model with M-predators and N-preys. The method is different from that used in the previous work. Some new sufficient conditions are obtained for the existence and global asymptotic stability of the periodic solution. The existence and stability conditions are given in terms of spectral radius of explicit matrices which are much different from the conditions given by the algebraic inequalities. Finally, an example is given to show the feasibility of our results.

1. Introduction and Motivation

1.1. History and Motivations

Mawhin's coincidence degree theory has been applied extensively to study the existence of periodic solutions for nonlinear differential systems (e.g. see [116] and references therein). The most important step of applying Mawhin’s degree theory to nonlinear differential equations is to obtain the priori bounds of unknown solutions to the operator equation . However, different estimation techniques for the priori bounds of unknown solutions to the equation may lead to different results. Most of papers obtained the priori bounds by employing the inequalities: These inequalities lead to a relatively strong condition given in terms of algebraic inequality or classic norms (see e.g., [316]). Different from standard consideration, in this paper, we employ matrix spectral theory to obtain the priori bounds, not the above inequalities. So in this paper, the existence and stability of periodic solution for a multispecies predator-prey model is studied by jointly employing Mawhin's coincidence degree and matrix spectral theory.

1.2. Model Formulation

One of classical Lotka-Vlterra system is predator-prey models which have been investigated extensively by mathematicians and ecologist. Many good results have been obtained for stability, bifurcations, chaos, uniform persistence, periodic solution, almost periodic solutions. It has been observed that most of works focus on either two or three species model. There are few paper considering the multispecies model. To model the dynamic behavior of multispecie predator-prey system, Yang and Rui [17] proposed a predator-prey model with M-predators and N-preys of the form: where denotes the density of prey species at time , denotes the density of predator species at time . The coefficients , and are nonnegative continuous periodic functions defined on . The coefficient is the intrinsic growth rate of prey species , is the death rate of the predator species , measures the amount of competition between the prey species and , measures the amount of competition between the predator species and , and the constant denotes the coefficient in conversing prey species into new individual of predator species . By using the differential inequality, Zhao and Chen [18] improved the results of Yang and Rui [17]. Recently, Xia et al. [19] obtained some sufficient conditions for the existence and global attractivity of a unique almost periodic solution of the system (1.2).

It is more natural to consider the delay model because most of the species start interacting after reaching a maturity period. Hence many scholars think that the delayed models are more realistic and appropriate to be studied than ordinary model. Delayed system is important also because sometimes time delays may lead to oscillation, bifurcation, chaos, instability which may be harmful to a system. Inspired by the above argument, Wen [20] considered a periodic delayed multispecies predator-prey system as follows: where , , , , , , are assumed to be continuous -periodic functions and the delays , , , are assumed to be positive constants. The system (1.3) is supplemented with the initial condition: where It is easy to see that for such given initial conditions, the corresponding solution of the system (1.3) remains positive for all . The purpose of this paper is to obtain some new and interesting criteria for the existence and global asymptotic stability of periodic solution of the system (1.3).

1.3. Comparison with Previous Work

To obtain the periodic solutions of the system (1.3), the method used in [20] is based on employing the differential inequality and Brower fixed point theorem. Different from consideration taken by [20], our method is based on combining matrix spectral theory with Mawhin's degree theory. In our method, we study the global asymptotic stability by combining matrix's spectral theory with Lyapunov functional method. The existence and stability conditions are given in terms of spectral radius of explicit matrices. These conditions are much different from the sufficient conditions obtained in [20].

1.4. Outline of This Work

The structure of this paper is as follows. In Section 2, some new and interesting sufficient conditions for the existence of periodic solution of system (1.3) are obtained. Section 3 is devoted to examining the stability of the periodic solution obtained in the previous section. In Section 4, some corollaries are presented to show the effectiveness of our results. Finally, an example is given to show the feasibility of our results.

2. Existence of Periodic Solutions

In this section, we will obtain some sufficient conditions for the existence of periodic solution of the system (1.3).

2.1. Preliminaries on the Matrix Theory and Degree Theory

For convenience, we introduce some notations, definitions, and lemmas. Throughout this paper, we use the following notations.(i)We always use , unless otherwise stated.(ii)If is a continuous -periodic function defined on , then we denote We use to denote a column vector, is an matrix, denotes the transpose of , and is the identity matrix of size . A matrix or vector (resp., ) means that all entries of are positive (resp., nonnegative). For matrices or vectors and , (resp., ) means that (resp., ). We denote the spectral radius of the matrix by .

If , then we have a choice of vector norms in , for instance , , and are the commonly used norms, where We recall the following norms of matrices induced by respective vector norms. For instance if , the norm of the matrix induced by a vector norm is defined by In particular one can show that (column norm), (row norm).

Definition 2.1 (see [1, 21]). Let be normed real Banach spaces, let be a linear mapping, and be a continuous mapping. The mapping is called a Fredholm mapping of index zero, if dimKer  and Im  is closed in . If is a Fredholm mapping of index zero and there exist continuous projectors and such that Im , , then is invertible. We denote the inverse of that map by . If is an open bounded subset of , the mapping will be called -compact on if is bounded and is compact. Since is isomorphic to , there exists an isomorphism .

Definition 2.2 (see [1, 22]). Let be open and bounded, and , that is, is a regular value of . Here, , the critical set of and is the Jacobian of at . Then the degree deg is defined by with the agreement that . For more details about Degree Theory, the reader may consult Deimling [22].

Lemma 2.3 (Continuation Theorem [1]). Let be an open and bounded set and be a Fredholm mapping of index zero and be -compact on (i.e., is bounded and is compact). Assume(i) for each , , ;(ii) for each , and . Then has at least one solution in .

Definition 2.4 (see [23, 24]). A real matrix is said to be an -matrix if , , , and .

Lemma 2.5 (see [23, 24]). Let be an matrix and , then , where denotes the identity matrix of size .

Now we introduce some function spaces and their norms, which will be valid throughout this paper. Denote The norms are given by Obviously, and , respectively, endowed with the norms and are Banach spaces.

2.2. Result on the Existence of Periodic Solutions

Theorem 2.6. Assume that the following conditions hold:: the system of algebraic equations: has finite solution with ; :, where ,
Then system (1.3) has at least one positive -periodic solution.

Proof. Note that every solution of the system (1.3) with the initial condition is positive. By using the following changes of variables: the system (1.3) can be rewritten as Obviously, system (1.3) has at least one -periodic solution which is equivalent to the system (2.11) having at least one -periodic solution. To prove Theorem 2.6, our main tasks are to construct the operators (i.e., , , , and ) appearing in Lemma 2.3 and to find an appropriate open set satisfying conditions (i), (ii) in Lemma 2.3.
For any , in view of the periodicity, it is easy to check that Now, we define the operators as follows: Define, respectively, the projectors and by It is obvious that the domain of in is actually the whole space, and Moreover, are continuous operators such that It follows that is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to ) exists, which is given by Then and are defined by Clearly, and are continuous. By using the generalized Arzela-Ascoli theorem, it is not difficult to prove that is relatively compact in the space . The proof of this step is complete.
Then, in order to apply condition (i) of Lemma 2.3, we need to search for an appropriate open bounded subset , denoted by Specifically, our aim is to find an appropriate . Corresponding to the operator equation for each , we have Since , each , as components of , is continuously differentiable and -periodic. In view of continuity and periodicity, there exists such that , , and there also exists such that , . Accordingly, , , and we get That is, Note that and , which implies It follows that Let Using (2.25), the inequalities (2.24) become or which implies where Set . It follows from (2.28) and () that In view of and Lemma 2.5, we get . Let Using (2.30) and (2.31), we get or Then which implies On the other hand, it follows from (2.31) that that is Estimating (2.20), by using (2.25), (2.33), and (2.37), we have The above relations imply Further, it follows form the definition of norm that Let us set the following: where is any positive constant.
Then for any solution of , we have for all . Obviously, are independent of and the choice of . Consequently, by taking this , the open subset satisfies that , that is, the open subset satisfies the assumption (i) of Lemma 2.3.
Now in the last step of the proof, we need to verify that for the given open bounded set obtained in Step 2, the assumption (ii) of Lemma 2.3 also holds. That is, for each , and .
Take . Then, in view of , is a constant vector in , denoted by and with the property By operating by gives It is easy to obtain that and , where is the Brouwer degree and is the identity mapping since . We have shown that the open subset satisfies all the assumptions of Lemma 2.3. Hence, by Lemma 2.3, the system (2.11) has at least one positive -periodic solution in . By (2.10), the system (1.3) has at least one positive -periodic solution. This completes the proof of Theorem 2.6.

3. Globally Asymptotic Stability

Under the assumption of Theorem 2.6, we know that system (1.3) has at least one positive -periodic solution, denoted by . The aim of this section is to derive a set of sufficient conditions which guarantee the existence and global asymptotic stability of the positive -periodic solution .

Before the formal analysis, we recall some facts which will be used in the proof.

Lemma 3.1 (see [25]). Let be a nonnegative function defined on such that is integrable on and is uniformly continuous on . Then .

Lemma 3.2 (see [23, 24]). Let be a matrix with nonpositive off-diagonal elements. is an -matrix if and only if there exists a positive diagonal matrix such that

Theorem 3.3. Assume that all the assumptions in Theorem 2.6 hold. Then system (1.3) has a unique positive -periodic solution which is globally asymptotically stable.

Proof. Let be any positive solution of system (1.3). It is easy to see that . Thus, in view of Lemma 2.5 and Definition 2.4, is an -matrix, where denotes an identity matrix of size . Therefore, by Lemma 3.2, there exists a diagonal matrix with positive diagonal elements such that the product is strictly diagonally dominant with positive diagonal entries, namely, Now, we define a Lyapunov function as follows: Calculating the upper right derivative of and using (3.2), we get where It follows from (3.4) that . Obviously, the zero solution of (1.3) is Lyapunov stable. On the other hand, integrating (3.4) over leads to or Noting that , it follows that Therefore, by Lemma 3.1, it is not difficult to conclude that Theorem 3.3 follows.

4. Corollaries and Remarks

In order to illustrate some features of our main results, we will present some corollaries and remarks in this section. From the proofs of Theorems 2.6 and 3.3, one can easily deduce the following corollary.

Corollary 4.1. In addition to , further suppose that or is an -matrix. Then system (1.3) has a unique positive -periodic solution which is globally asymptotically stable.

Now recall that for a given matrix , its spectral radius is equal to the minimum of all matrix norms of , that is, for any matrix norm . Therefore, we have the following corollary.

Corollary 4.2. In addition to , if one further supposes that there exist positive constants such that one of the following inequalities holds. (1) < , or equivalently, for all , (2), where has been defined in Theorem 2.6.(3) < , or equivalently, for all , Then system (1.3) has a unique positive -periodic solution which is globally asymptotically stable.

Proof. For any matrix norm and any nonsingular matrix , also defines a matrix norm. Let us denote , then the conditions (1.2) and (1.3) correspond to the column norms and Frobenius norm of matrix , respectively. Condition (2.10) corresponds to the row norms of and note that . Now Corollary 4.2 follows immediately.

5. Example

In this section, an example and its simulations are presented to illustrate the feasibility and effectiveness of our results.

Example 5.1. Consider the following periodic predator-prey model with -predators and -prey: Simple computation leads to By using mathematica, we see that . Thus, the system (5.1) has a periodic solution which is globally asymptotically stable. Figure 1 shows the asymptotic behavior of the periodic solution.

940287.fig.001
Figure 1: Asymptotic behavior of system (5.1) with initial values , respectively, .

Remark 5.2. In this example, one can observe that though the spectral , the matrix norms of the matrix are all bigger than . For instance, the column norm: is

Acknowledgments

This work was supported by NNSFC (no. 11271333 and no. 11171090). The first author also acknowledges the supports by the foundation of Slovene human resources development and scholarship fund. He thanks Professor Valery G. Romanovski and Professor Marko Robnik for warm hospitality during his stay at CAMTP in University of Maribor as a Research Fellow.

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