Periodic Solutions of a Delayed Predator-prey Model with Stage Structure for Predator

The traditional Lotka-Volterra-type predator-prey model has received great attention from both theoretical and mathematical biologists, and has been well studied (see, e.g., [5, 8, 9]). It is assumed in the classical Lotka-Volterra predator-prey model that each individual predator admits the same ability to attack prey. This assumption seems not to be realistic for many animals. In the natural world, there are many species whose individuals have a life history that takes them through two stages, immature and mature. Stagestructured models have received much attention in recent years. In [1], a stage-structured model of single species growth consisting of immature and mature individuals was proposed and discussed. In [2], it was further assumed that the time from immaturity to maturity is itself state-dependent. An equilibrium analysis and eventual lower and upper bounds of positive solutions for the model were addressed. Recently, Wang and Chen [11] proposed a predator-prey model with stage structure for the predator to analyze the influence of a stage structure for the predator on the dynamics of predator-prey models. In [11], the authors classify individuals of the predator as belonging to either the immature or the mature and suppose that the immature predator does not have ability to attack prey. This seems reasonable for a number of mammals, where the immature predators are raised by their parents; the rate they attack prey can be ignored. Sufficient conditions are derived in [11] for the uniform persistence and global stability of a positive equilibrium of the proposed model. We note that any biological or environmental parameters are naturally subject to fluctuation in time. As Cushing [3] pointed out, it is necessary and important to consider models with periodic ecological parameters or perturbations which might be quite