Existence of Positive Periodic Solutions for Periodic Neutral Lotka-Volterra System with Distributed Delays and Impulses

By using a fixed-point theorem of strict-set-contraction, we investigate the existence of positive periodic solutions for a class of the following impulsive neutral Lotka-Volterra system with distributed delays: Some verifiable criteria are established easily. 1. Introduction It is well known that natural environments are physically highly variable, and in response, birth rates, death rates, and other vital rates of populations vary greatly in time. Theoretical evidence suggests that many population and community patterns represent intricate interactions between biology and variation in the physical environment (see [1–4]). Thus, the focus in theoretical models of population and community dynamics must be not only on how populations depend on their own population densities or the population densities of other organisms, but also on how populations change in response to the physical environment. It is reasonable to study the models of population with periodic coefficients. In addition to the theoretical and practical significance, the Lotka-Volterra model is one of the famous models for dynamics of population; therefore it has been studied extensively [5–9]. In view of the above effects, by applying a fixed-point theorem of strict-set-contraction, Li [10] established criteria to guarantee the existence of positive periodic solutions of the following neutral Lotka-Volterra system with distributed delays: where are -periodic functions and and satisfying , , . On the other side, birth of many species is an annual birth pulse or harvesting. To have a more accurate description of many mathematical ecology systems, we need to consider the use of impulsive differential equations [11–13]. Some qualitative properties such as oscillation, periodicity, asymptotic behavior, and stability properties have been investigated extensively by many authors over the past few years [14–18]. However, to our knowledge, there are few published papers discussing the existence of periodic solutions for neutral Lotka-Volterra system with distributed delays and impulses. In this paper, we are concerned with the following neutral Lotka-Volterra system with distributed delays and impulses: where are -periodic functions and and satisfying , . Moreover, ？ (here represents the right limit of at the point ), ; that is, changes decreasingly suddenly at times ;？？ is a constant, , . We assume that there exists an integer such that , where . The main purpose of this paper is by using a fixed-point theorem of strict-set-contraction [19, 20] to establish new criteria to guarantee the existence