Positive Periodic Solutions of Cooperative Systems with Delays and Feedback Controls

This paper studies a class of periodic n species cooperative Lotka-Volterra systems with continuous time delays and feedback controls. Based on the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin, some new sufficient conditions on the existence of positive periodic solutions are established. 1. Introduction Mathematical ecological system has become one of the most important topics in the study of modern applied mathematics. Its dynamical behavior includes persistence, permanence and extinction of species, global stability of systems, the existence of positive periodic solutions, positive almost periodic solutions, and strictly positive solutions. The existence of positive periodic solutions has already become one of the most interesting subjects for scholars. In the recent years, the application of fixed-point theorems to the existence of positive periodic solutions in mathematical ecology has been studied extensively, for example, Brouwer’s fixed point theorem [1–4], Schauder’s fixed-point theorem [5–8], Krasnoselskii’s fixed-point theorem [9–14], Horn’s fixed-point theorem [15, 16], and Mawhins continuation theorem [17–36], and so forth. In particular, Mawhins continuation theorem is a powerful tool for studying the existence of periodic solutions of periodic high-dimensional time-delayed problems. When dealing with a time-delayed problem, it is very convenient and the result is relatively simple [30]. Recently, a considerable number of mathematical models with delays have been proposed in the study of population dynamics. One of the most celebrated models for population is the Lotka-Volterra system. Subsequently, a lot of the literature related to the study of the existence of positive periodic solutions for various Lotka-Volterra-type population dynamical systems with delays by using the method of continuation theorem was published and extensive research results were obtained [17–21, 24–34]. On the other hand, in some situations, people may wish to change the position of the existing periodic solution but to keep its stability. This is of significance in the control of ecology balance. One of the methods for the realization of it is to alter the system structurally by introducing some feedback control variables so as to get a population stabilizing at another periodic solution. The realization of the feedback control mechanism might be implemented by means of some biological control scheme or by harvesting procedure [21]. In fact, during the last decade, the existence of positive periodic solutions for the