Existence of Periodic Solution for a Non-Autonomous Stage-Structured Predator-Prey System with Impulsive Effects

In recent years, non-autonomous predator-prey systems have been widely studied [1-6]. There has been a growing interest in the study of mathematical models of populations dispersing among patches in the nature world [3,7-9]. In the classical predator-prey models it is usually assumed that each individual predator admits the same ability to feed on prey. However, it is different for some species whose individuals have a life history that takes them through two stages, immature and mature, where immature predators are raised by their parents, so many models with time delays and stage structure for both prey and predator were investigated and rich dynamics have been observed [4,6,10-12]. In this paper, we are considered the effects of prey diffusion in two patches and maturation delay for predator on the dynamics of an impulsive predator-prey model. We discuss the differential equation: (See 1.1) Where we suppose that the system is composed of two patches connected by diffusion.   1 x t and   2 x t represent the densities of prey species in patch I and II at time t,   1 y t and   2 y t represent the densities of the immature and mature predator at time t in patch II, respectively.   1 , x t   2 x t can diffuse between patch I and II while the predator species is confined to patch II.  represents a constant time to maturity.    1,2 i a t i  is the intrinsic                                                      