Abstract and Applied Analysis

Volume 2014 (2014), Article ID 156948, 12 pages

http://dx.doi.org/10.1155/2014/156948

Research Article

Existence and Stability of Periodic Solution to Delayed Nonlinear Differential Equations

Xiang Gu,¹ Huicheng Wang,¹ P. J. Y. Wong,² and Yonghui Xia¹

¹Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

²School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, 639798

Received 23 February 2014; Accepted 17 March 2014; Published 14 April 2014

Academic Editor: Yongli Song

Copyright © 2014 Xiang Gu 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.

The main purpose of this paper is to study the periodicity and global asymptotic stability of a generalized Lotka-Volterra’s competition system with delays. Some sufficient conditions are established for the existence and stability of periodic solution of such nonlinear differential equations. The approaches are based on Mawhin’s coincidence degree theory, matrix spectral theory, and Lyapunov functional.

1. Introduction and Motivation

In the past few decades, differential equations have been used in the study of population dynamics, ecology and epidemiology, malaria transmission, and so forth (see, e.g., [1–10]). One of the rudimentary population systems is the nonautonomous -species competitive model:

Based on Mawhin’s coincidence degree theory, spectral theory, and novel estimation techniques for the priori bounds of unknown solutions to the equation , Xia and Han [8] studied the existence and stability of periodic solution for (1). But model (1) is doubted by Gilpin and Ayala [11], they thought that the model is not reasonable enough. In order to fit data in the experiments conducted in Ayala et al. [12] and to yield significantly more accurate results on the competitive model, Chen [13] proposed a more complicated model as follows:

where provides a nonlinear measure of intraspecific interference and provides a measure of interspecific interference. Chen studied the permanence of (2) by average method. For the sake of convenience, in what follows, the new factor introduced by Gilpin and Ayala is called Gilpin-Ayala effect. On the other hand, many scholars think that the delayed models are more realistic. Because time delays may lead to oscillation, bifurcation, chaos, and instability which may be harmful to a system. In fact, May [14] has shown that if a time delay is incorporated into the resource limitation of the logistic equation, then it has destabilizing effect on the stability of the system (also see Cooke and Grossman [15]). But sometimes, the delays may be harmless under some restriction and this is more important in some sense (e.g., see [16]). A very basic and important ecological problem in the study of multispecies population dynamics concerns the global existence and global asymptotic stability of positive periodic solutions. It is doubted whether the existence and stability of periodic solutions can be affected by the delays or Gilpin-Ayala effect. For this reason, in the present paper, we consider the Gilpin-Ayala type delayed system as follows:

where is the population density of the th species; is the intrinsic exponential growth rate of the th species; , measure the amount of competition between the th species and the th species (); and , provide a nonlinear measure of intraspecific interference. For the point of biological view, the coefficients are assumed to be continuous -periodic functions; we always assume that , , , , , are nonnegative and , are strictly positive. And system (3) is supplemented with the initial condition

where , , and is the set of all bounded continuous functions from into . It is easy to see that for such given initial value condition, the corresponding solution of (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 system (3).

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 (3) are obtained. Section 3 is devoted to examining the stability of this periodic solution. In Section 4, some corollaries and discussion are presented. Finally, some examples and their simulations are given to show the effectiveness and 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 system (3).

2.1. Preliminaries on the Matrix Theory and Degree Theory

For convenience, we introduce some notations, definitions, and lemmas. If is a continuous -periodic function defined on , 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 also 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) and , (row norm).

Definition 1 (see [17, 18]). Let , be normed real Banach spaces, let be a linear mapping, and let be a continuous mapping. The mapping is called a Fredholm mapping of index zero if dim Ker  and is closed in . If is a Fredholm mapping of index zero and there exist continuous projectors and such that Im  and , it follows that 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 (see [17, 18]). 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 is referred to [18].

Lemma 3 (continuation theorem [17]). Let be an open and bounded set. Let be a Fredholm mapping of index zero and let be -compact on (i.e., is bounded and is compact). Assume,

(i)for each , and ;(ii)for each , and .

Then has at least one solution in .

Definition 4 (see [19, 20]). A real matrix is said to be an -matrix if , , , and .

Lemma 5 (see [19, 20]). Let be an matrix and ; then , where denotes the identity matrix of size .

In what follows, we will introduce some function spaces and their norms, which are valid throughout this paper. Denote

And, 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 6. Assume that the following conditions hold:

the system of algebraic equations

 has finite solutions with and ;, , , , ;, where and

Then system (3) has at least one positive -periodic solution.

Proof. Note that every solution , of system (3) with the initial value condition is positive. Make the change of variables

Then system (3) is the same as

Obviously, system (3) that has at least one -periodic solution is equivalent to system (14) that has at least one -periodic solution. To prove Theorem 6, our main tasks are to construct the operators (i.e., , , , and ) appearing in Lemma 3 and to find an appropriate open set satisfying conditions (i) and (ii) in Lemma 3. To this end, we proceed with three steps.

Step  1. In this step, we intend to construct the operators appearing in Lemma 3 and verify that they satisfy the conditions of Lemma 3. For any , in view of the periodicity, it is easy to check that

And define the operators and as follows:

where

Define, respectively, the projectors and by

It can be found 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

where

Clearly, and are continuous. Now we turn to show the fact that for any open bounded set , denoted by

the mapping is -compact on . Here, the constants are independent of the choice of . In view of Definition 1, to show the above fact, it suffices to show that is bounded and is compact. We first arrive at

which implies that is bounded in the space . Secondly, we will show that is relatively compact in the space . In fact, it follows from (22) that

where and

This, combining with (22), gives

which implies that is bound in the space .

On the other hand, we prove that is equicontinuous. In view of uniform continuity of , , and , for any , there exists such that, for any , provided that , we have

Since any is equicontinuous, for the same , there exists such that, for any , provided that , we have

It follows from (29) and (30) that

Thus, it follows from (26) that

On the other hand, the mean value theorem together with (26) gives

where lies between and . Taking , it follows from (32) and (33) that implies

which implies that is equicontinuous.

Therefore, by the generalized Arzela-Ascoli theorem, we have that is relatively compact in the space . The proof of this step is complete.

Step  2. In this step, we are in a position to search for appropriate open bounded subsets satisfying condition (i) of Lemma 3. Specifically, our aim is to search for an appropriate defined by in Step  1 such that satisfies condition (i) of Lemma 3. To this end, assume that is a solution of the equation for each ; that is,

Since , each , , as components of , is continuously differentiable and -periodic. In view of continuity and periodicity, there exists such that , . Accordingly, and we arrive at

That is,

Noticing that implies

It follows from () and (37) that

Here we used (). Letting , it follows from (39) that

or

which implies

Set . It follows from (42) that

In view of and Lemma 5, . Let

Then it follows from (43) and (44) that

which implies

On the other hand, it follows from (44) that

Estimating (2), by using (45) and (47), we have

We can choose a large enough real number () such that

Set . Then for any solution of , we have

for all . Obviously, are independent of and the choice of . Consequently, taking , the open subset satisfies that for each , ; that is, the open subset satisfies assumption (i) of Lemma 3.

Step  3. In what follows, we verify that for the given open bounded set , assumption (ii) of Lemma 3 also holds. That is, for each , and .

Take . Then, in view of , is a constant vector in , denoted by and by the property

Operate by , and we obtain that, for ,

We claim that , for . If this is not valid, suppose that there exists a certain such that ; that is,

or

That is,

Letting , we have

In view of (51), we get

Note that . It follows from (2), (57), and (47) that

which is a contradiction. Therefore, for any , for all . That is, , for . Furthermore, in view of () and Definition 2, it is easy to see that

where is the Brouwer degree and is the identity mapping since .

So far, we have shown that the open subset satisfies all the assumptions of Lemma 3. Hence, by Lemma 3, system (14) has at least one positive -periodic solution in . By (13), system (3) has at least one positive -periodic solution, denoted by . This completes the proof of Theorem 6.

3. Globally Asymptotic Stability

Under the assumption of Theorem 6, we know that system (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 global asymptotic stability of the positive -periodic solution . As pointed out in Section 1, because has been changed to in (3), the previous method in Xia and Han [8] cannot be applied to study the stability of system (3) directly. Before the formal analysis, we recall some facts which will be used in the proof.

Lemma 7 (see [21]). Let be a nonnegative function defined on such that is integrable on and is uniformly continuous on . Then .

Lemma 8 (see [19, 20]). Let be a matrix with nonpositive off-diagonal elements. is an -matrix if and only if there exists a positive diagonal matrix such that

We need a lemma which can follow immediately from Theorem 2.1 in Xia et al. [22]. We consider the following logistic equation:

Since is nonnegative and are strictly positive, it follows immediately from Lemma 2.1 in Xia et al. [22] that system (61) has a unique positive solution, denoted by , with , which is globally asymptotically stable. Then, as a special case of Theorem 2.1 in [22], we have the following.

Lemma 9. Suppose that

(); then system (3) is bounded above and below.

Lemma 10 (see [23]). If , and ,  , then

Theorem 11. Assume that ; if (H₁)–(H₄) hold, then system (3) has a unique positive -periodic solution which is globally asymptotically stable.

Proof. By Lemma 9, system (3) is bounded below. Thus there exist positive constants such that . We proceed the proof of this theorem with two steps.

Step  1. Choose positive constants , , such that , where and

We claim that the positive constants can be definitely chosen. By similar arguments in [23], one can prove this fact.

Step  2. Prove the global asymptotic stability of system (3). Changing variables , by noticing that , then system (3) changes to

In view of , system (3) has at least a positive periodic solution. By the linear transformation , we know that system (64) also has at least a positive periodic solution, denoted by . In order to show the global asymptotic stability of system (3), it suffices to show that of system (64) is globally asymptotically stable. For this purpose, let be any other positive solution of system (64). And we define a Lyapunov functional as follows:

Calculating the upper right derivative of along (64), it follows from (65) that

Note that and . By Lemma 10, we have

It follows from (66) and (67) that

From Step  1, we know . Thus, in view of Lemma 5 and Definition 4, is an -matrix, where denotes an identity matrix of size . Therefore, by Lemma 8, there exists a diagonal matrix with positive diagonal elements such that the product is strictly diagonally dominant with positive diagonal entries; namely,

It follows from (68) and (69) that . Obviously, the zero solution of (64) is Lyapunov stable. On the other hand, integrating (69) over leads to

or

Noting that , it follows that

Therefore, by Lemma 7, it is not difficult to conclude that

Which implies the global asymptotical stability of system (64). By the linear transformation ,  , the positive periodic solution of (3) is also globally asymptotically stable. This completes the proof of Theorem 11.

4. Remark and Discussion

To illustrate the generality of our results, we will give a corollary in this section. 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, a corollary of Theorem 11 is stated as follows.

Corollary 12. In addition to (H₁), (H₂), and (H₄), 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 6,(3), or equivalently, for all , then system (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 . Then conditions (1)-(2) correspond to the column norms and Frobenius norm of matrix , respectively. Condition (3) corresponds to the row norms of and note that . Corollary 12 follows immediately.

Remark 13. The corollary implies that the conditions given in terms of the spectral radius are much better than the classic norms.

Now we consider a special case of system (3). Take , , and ; then system (3) reduces to the classical LV competition system which has been well studied in Xia and Han [8]:

Remark 14. In this case, Theorem 11 and Corollary 12 reduce to the main results in Xia and Han [8].

5. Discussion

As we know, dynamic systems are often classified into two categories of either continuous-time or discrete-time systems. However, many real-world phenomena are neither purely continuous-time nor purely discrete-time. This leads to the development of dynamic systems with impulses, which display a combination of characteristics of both the continuous-time and discrete-time systems and hence provide a more natural framework for mathematical modeling of many real-world phenomena. Whether the new method proposed in this paper can be applied to study the existence and global asymptotic stability of the LV systems with impulses remains open.

6. Examples

In this section, some examples and their simulations are presented to illustrate the feasibility and effectiveness of our results.

Example 1. Consider the two-species competitive system

Corresponding to system (3), we have , , , , . Simple computation shows

And . Thus, by Theorem 11, system (75) has a unique positive equilibrium which is globally asymptotically stable.

Remark 15. In this example, one can observe that, though the spectral , the matrix norms (including the row norm, the column norm, and the Frobenius norm) of matrix may be bigger than 1. For instance, the column norm

Example 2. Consider the three-species competitive system

Corresponding to system (3), we have , , , , , , , and . Then, simple computation leads to

Hence, by using MATLAB, we get

Thus, by Theorem 11, system (78) has a unique positive equilibrium which is globally asymptotically stable.

Remark 16. In this example, one can observe that though the spectral , the matrix norms (including the row norm, the column norm, and the Frobenius norm) of matrix may be bigger than 1. For instance, the column norm

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant no. 11271333 and ZJNSFC.

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