Persistence and periodicity of a delayed ratio-dependent predator-prey model with stage structure and prey dispersal

A delayed periodic ratio-dependent predator–prey model with prey dispersal and stage structure for predator is investigated. It is assumed that immature individuals and mature individuals of the predator species are divided by a fixed age, and that immature predators don't have the ability to attack prey, and that predator species is confined to one of the patches while the prey species can disperse between two patches. We first discuss the uniform persistence and impermanence of the model. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, a set of easily verifiable sufficient conditions are derived for the existence of positive periodic solutions of the proposed model. Numerical simulations are presented to illustrate the feasibility of our main result