Existence and Stability of Periodic Solution to Delayed Nonlinear Differential Equations

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 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 1 2 Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 Correspondence should be addressed to Yonghui Xia; 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. 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 𝑛-species competitive model: 𝑛 𝑦𝑖̇ (𝑡) = 𝑦𝑖 (𝑡) [𝑏𝑖 (𝑡) − ∑ 𝑎𝑖𝑗 (𝑡) 𝑦𝑗 (𝑡)] , 𝑗=1 [ ] 𝑖 = 1, 2, . . . , 𝑛. (1) Based on Mawhin’s coincidence degree theory, spectral theory, and novel estimation techniques for the priori bounds of unknown solutions to the equation 𝐿𝑥 = 𝜆𝑁𝑥, 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: 𝑛 𝛼 𝑦𝑖̇ (𝑡) = 𝑦𝑖 (𝑡) [𝑟𝑖 (𝑡) − ∑𝑎𝑖𝑗 (𝑡) 𝑦𝑗 𝑖𝑗 (𝑡)] , 𝑗=1 [ ] 𝑖 = 1, 2, . . . , 𝑛, (2) where 𝛼𝑖 provides a nonlinear measure of intraspecific interference and 𝛼𝑖𝑗 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 GilpinAyala 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 2 Abstract and Applied Analysis 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: 𝑛 𝛼 𝑦𝑖̇ (𝑡) = 𝑦𝑖 (𝑡) [𝑟𝑖 (𝑡) − ∑𝑎𝑖𝑗 (𝑡) 𝑦𝑗 𝑖𝑗 (𝑡) 𝑗=1 [ 𝑛 𝛽 − ∑𝑏𝑖𝑗 (𝑡) 𝑦𝑗 𝑖𝑗 (𝑡 − 𝜏𝑖𝑗 )] , 𝑗=1 ] 𝑛 𝑖 = 1, 2, . . . , 𝑛, 𝑠 ∈ [−𝜏, 0] , 𝜙𝑖 (0) > 0, 𝑖 = 1, 2, . . . , 𝑛, 󵄨 󵄨 ‖V‖1 = ∑ 󵄨󵄨󵄨V𝑖 󵄨󵄨󵄨 , 𝑗=1 (3) where 𝑦𝑖 is the population density of the 𝑖th species; 𝑟𝑖 is the intrinsic exponential growth rate of the 𝑖th species; 𝑎𝑖𝑗 , 𝑏𝑖𝑗 measure the amount of competition between the 𝑖th species and the 𝑗th species (𝑖 ≠ 𝑗); and 𝛼𝑖𝑗 , 𝛽𝑖𝑗 provide a nonlinear measure of intraspecific interference. For the point of biological view, the coefficients are assumed to be continuous 𝜔-periodic functions; we always assume that 𝑟𝑖 , 𝑎𝑖𝑗 , 𝑏𝑖𝑗 , 𝜏𝑖𝑗 , 𝑖, 𝑗 = 1, 2, . . . , 𝑛, are nonnegative and 𝑎𝑖𝑖 , 𝑏𝑖𝑖 are strictly positive. And system (3) is supplemented with the initial condition 𝑦𝑖 (𝑠) = 𝜙𝑖 (𝑠) , D, and 𝐸𝑛 is the identity matrix of size 𝑛. 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 𝐸, D > 𝐸 (resp., D ≥ 𝐸) means that D − 𝐸 > 0 (resp., D − 𝐸 ≥ 0). We also denote the spectral radius of the matrix D by 𝜌(D). If V = (V1 , V2 , . . . , V𝑛 )𝑇 ∈ R𝑛 , then we have a choice of vector norms in R𝑛 ; for instance, ‖V‖1 , ‖V‖2 , and ‖V‖∞ are the commonly used norms, where (4) where 𝜏 = max1≤𝑖≤𝑛 {𝜏𝑖𝑗 }, 𝜙 = (𝜙1 , . . . , 𝜙𝑛 ) ∈ BC([−𝜏, 0], R+𝑛 ), and BC is the set of all bounded continuous functions from [−𝜏, 0] into R+𝑛 . It is easy to see that for such given initial value condition, the corresponding solution of (3) remains positive for all 𝑡 ≥ 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. 2. Existence of Periodic Solutions In this section, we will obtain some sufficient conditions for the existence of periodic solution of system (3). 1/2 { 𝑛 󵄨 󵄨2 } ‖V‖2 = {∑󵄨󵄨󵄨V𝑖 󵄨󵄨󵄨 } } {𝑗=1 󵄨󵄨 󵄨󵄨 ‖V‖∞ = max 󵄨󵄨V𝑖 󵄨󵄨 . 1≤𝑖≤𝑛 (6) , We recall the following norms of matrices induced by respective vector norms. For instance, if A = (𝑎𝑖𝑗 )𝑛×𝑛 , the norm of the matrix ‖A‖ induced by a vector norm ‖ ⋅ ‖ is defined by ‖A‖𝑝 = sup V∈R𝑛 ,V ≠ 0 ‖AV‖𝑝 ‖V‖𝑝 = sup ‖AV‖𝑝 = sup ‖AV‖𝑝 . (7) ‖V‖𝑝 =1 ‖V‖𝑝 ≤1 In particular one can show that ‖A‖1 = max1≤𝑗≤𝑛 ∑𝑛𝑖=1 |𝑎𝑖𝑗 | (column norm) and ‖A‖2 = [𝜆 max (A𝑇 A)]1/2 = [max . eigenvalue of (A𝑇 A)]1/2 , ‖A‖∞ = max1≤𝑖≤𝑛 ∑𝑛𝑗=1 |𝑎𝑖𝑗 | (row norm). Definition 1 (see [17, 18]). Let 𝑋, 𝑍 be normed real Banach spaces, let 𝐿 : Dom 𝐿 ⊂ 𝑋 → 𝑍 be a linear mapping, and let 𝑁 : 𝑋 → 𝑍 be a continuous mapping. The mapping 𝐿 is called a Fredholm mapping of index zero if dim Ker 𝐿 = codim Im 𝐿 < +∞ and Im 𝐿 is closed in 𝑍. If 𝐿 is a Fredholm mapping of index zero and there exist continuous projectors 𝑃 : 𝑋 → 𝑋 and 𝑄 : 𝑍 → 𝑍 such that Im 𝑃 = Ker 𝐿 and Ker 𝑄 = Im 𝐿 = Im(𝐼 − 𝑄), it follows that 𝐿 | dom 𝐿 ∩ Ker 𝑃 : (𝐼 − 𝑃)𝑋 → Im 𝐿 is invertible. We denote the inverse of that map by 𝐾𝑃 . If Ω is an open bo (...truncated)