Existence and Stability of Periodic Solution to Delayed Nonlinear Differential Equations

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.


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][2][3][4][5][6][7][8][9][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 , , , , , = 1, 2, . . . , , are nonnegative and , are strictly positive. And system (3) is supplemented with the initial condition and BC is the set of all bounded continuous functions from [− , 0] into R + . It is easy to see that for such given initial value condition, the corresponding solution of (3) remains positive for all ≥ 0. 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.

Existence of Periodic Solutions
In this section, we will obtain some sufficient conditions for the existence of periodic solution of system (3).

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 R, denote = min We use = ( 1 , . . . , ) ∈ R to denote a column vector, D = ( ) × is an × matrix, D denotes the transpose of D, and is the identity matrix of size . A matrix or vector D > 0 (resp., D ≥ 0) means that all entries of D are positive (resp., nonnegative). For matrices or vectors D and , D > (resp., D ≥ ) means that D − > 0 (resp., D − ≥ 0). We also denote the spectral radius of the matrix D by (D). If V = (V 1 , V 2 , . . . , V ) ∈ R , then we have a choice of vector norms in R ; for instance, ‖V‖ 1 , ‖V‖ 2 , and ‖V‖ ∞ are the commonly used norms, where We recall the following norms of matrices induced by respective vector norms. For instance, if A = ( ) × , the norm of the matrix ‖A‖ induced by a vector norm ‖ ⋅ ‖ is defined by ‖AV‖ . (7) In particular one can show that Definition 1 (see [17,18]). Let , be normed real Banach spaces, let : Dom ⊂ → be a linear mapping, and let : → be a continuous mapping. The mapping is called a Fredholm mapping of index zero if dim Ker = codim Im < +∞ and Im is closed in . If is a Fredholm mapping of index zero and there exist continuous projectors : → and : → such that Im = Ker and Ker = Im = Im( − ), it follows that | dom ∩ Ker : ( − ) → Im is invertible. We denote the inverse of that map by . If Ω is an open bounded subset of , the mapping will be called -compact on Ω if (Ω) is bounded and ( − ) : Ω → is compact. Since Im is isomorphic to Ker , there exists an isomorphism : Im → Ker .

Then
= has at least one solution in Ω ∩ Dom .
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 ‖ ⋅ ‖ 1 and ‖ ⋅ ‖ 0 are Banach spaces.
Proof. Note that every solution ( ) = ( 1 ( ), 2 ( ), . . . , ( )) of system (3) with the initial value condition is positive. Make the change of variables Then system (3) is the same aṡ 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 Ω 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 : Dom ⊂ → and : → as follows: where 4 Abstract and Applied Analysis 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 ) : Im → Dom ∩ Ker exists, which is given by Then where Clearly, and ( − ) are continuous. Now we turn to show the fact that for any open bounded set Ω ⊂ , denoted by the mapping is -compact on Ω. Here, the constants ℎ are independent of the choice of ( ). In view of Definition 1, to show the above fact, it suffices to show that (Ω) is bounded and ( − ) : Ω → is compact. We first arrive at which implies that (Ω) is bounded in the space ( , ‖ ⋅ ‖ 0 ). Secondly, we will show that ( ( − ) ) (Ω) is relatively compact in the space ( , ‖ ⋅ ‖ 1 ). In fact, it follows from (22) that where = / and This, combining with (22), gives which implies that On the other hand, we prove that ( ( − ) ) (Ω) is equicontinuous. In view of uniform continuity of , , and , for any > 0, there exists 1 > 0 such that, for any , ∈ R, provided that | − | < 1 , we have Since any ( ) = ( 1 ( ), 2 ( ), . . . , ( )) ∈ Ω is equicontinuous, for the same , there exists 0 < 2 ≤ 1 such that, for any , ∈ R, provided that | − | < 2 , we have It follows from (29) and (30) that Abstract and Applied Analysis 5 Thus, it follows from (26) that On the other hand, the mean value theorem together with (26) gives where lies between and . Taking = min{ /2 , 2 }, it follows from (32) and (33) that | − | < implies which implies that ( ( − ) )(Ω) is equicontinuous. Therefore, by the generalized Arzela-Ascoli theorem, we have that ( ( − ) )(Ω) is relatively compact in the space ( , ‖ ⋅ ‖ 1 ). The proof of this step is complete.

Globally Asymptotic Stability
Under the assumption of Theorem 6, we know that system (3) has at least one positive -periodic solution, denoted bỹ ( ) = (̃1( ), . . . ,̃( )) . 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.
Proof. By Lemma 9, system (3) is bounded below. Thus there exist positive constants > 0 such that ( ) ≥ . We proceed the proof of this theorem with two steps. (63) We claim that the positive constants can be definitely chosen. By similar arguments in [23], one can prove this fact.
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 = 1, = 0, and ≡ 0; 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].

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 continuoustime and discrete-time systems and hence provide a more natural framework for mathematical modeling of many realworld 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.

Examples
In this section, some examples and their simulations are presented to illustrate the feasibility and effectiveness of our results.
Remark 16. In this example, one can observe that, though the spectral (K) < 1, the matrix norms (including the row Hence, by using MATLAB, we get (K) = max . eigenvalues [K] = 0.7667 < 1.
Thus, by Theorem 11, system (78) has a unique positive equilibrium which is globally asymptotically stable.
Remark 18. In this example, one can observe that though the spectral (K) < 1, the matrix norms (including the row norm, the column norm, and the Frobenius norm) of matrix K may be bigger than 1. For instance, the column norm