Global Positive Periodic Solutions of Generalized -Species Gilpin-Ayala Delayed Competition Systems with Impulses

We consider the following generalized -species Lotka-Volterra type and Gilpin-Ayala type competition systems with multiple delays and impulses: , , ; , , By applying the Krasnoselskii fixed-point theorem in a cone of Banach space, we derive some verifiable necessary and sufficient conditions for the existence of positive periodic solutions of the previously mentioned. As applications, some special cases of the previous system are examined and some earlier results are extended and improved. 1. Introduction In the recent decades, the traditional Lotka-Volterra competition systems have been studied extensively. One of the models is the following competition system: Many results concerned with the permanence, global asymptotic stability, and the existence of positive periodic solutions of system (1) are obtained; we refer to [1–10] and the reference therein. However, the Lotka-Volterra type models have often been severely criticized. One of the criticisms is that, in such a model, the per capita rate of change of the density of each species is a linear function of densities of the interacting species. In 1973, Ayala et al. [11] conducted experiments on fruit fly dynamics to test the validity of ten models of competitions. One of the models accounting best for the experimental results is given by In order to fit data in their experiments and to yield significantly more accurate results, Gilpin and Ayala [12] claimed that a slightly more complicated model was needed and proposed the following competition model: where is the population density of the ith species, is the intrinsic exponential growth rate of the ith species, is the environmental carrying capacity of species in the absence of competition, provides a nonlinear measure of interspecific interference, and provides a measure of interspecific interference. [13–15] obtained sufficient conditions which guarantee the global asymptotic stability of system (3). Chen [16] investigated the following -species nonautonomous Gilpin-Ayala competitive model: For each , they established a series of criteria under which of the species of system (4) were permanent while the remaining species were driven to extinction. In [17], Fan and Wang further studied the following delay Gilpin-Ayala type competition model: They obtained a set of easily verifiable sufficient conditions for the existence of at least one positive periodic solution of the system (5) by applying the coincidence degree theory. Recently, in [18], Chen investigated the following -species Gilpin-Ayala type competition systems: He established a series of