In the natural world, there are many species whose individuals have a life history that takes them through two stages, juvenile stage and adult stage . Individuals in each stage are identical in biological characteristics, and some vital rates (rates of survival, development, and reproduction) of individuals in a population almost always depend on stage structure. Furthermore, there is a strong interaction relationship between the mature population and the immature population , which is to some extent relevant to the persistence and extinction of the related population. Consequently, it is constructive to investigate the dynamics of such ecosystem without ignorance of stage structure for population.Before 1990, the stage structure has attracted some attention from several authors [6,20,21,23,24,31,48,53,29]. The well-known biological system of single species with stage structure is established by O. Aiello and H. Freedman [1], which represents a milestone in this research field. In the following decades, much research efforts are put into the investigation of biological dynamical systems with stage structure for population. Some related research results can be found in [1,2,3,10,14,26,8,39,34,35,36,37,38,40,44,41,42,43,45,47,51,52,54,25,16,9] and the references therein.Recently, the dynamics of a class of stage-structured prey-predator models with discrete time delay have been studied by several authors [4,55,18,58,5] and the references therein.Arino et al. [4] suggested that the time delay to adulthood should be state dependent and careful formulation of such state dependent time delays can lead to models that produce periodic solutions. Xu et al. [55] studied the persistence and stability of a delayed prey-predator model with stage structure for predator. Gourley et al. [18] formulated a general and robust prey predator model with stage-structure with constant maturation time delay and performed a systematic mathematical and computational Q. Zhang et al.: Complex
