This paper aims to study the prey refuge impact on the dynamic behaviour of a stage structure predator-prey model. The model consists of four ecological species: prey in the protected and unprotected area and immature and mature predators. It assumes the grown predator can feeds only on the prey in an unreserved area. The conditions that guarantee the existence of the possible fixed points are found. Further, the local stability around all of the equilibria is considered. Then, using the Lyapunov direct method, the essential conditions for the global stability of the equilibria are adopted. Numerical simulations are illustrated to confirm our results. It concluded that the protected area positively affects the system's co-existence.

We propose a stage structure predator-prey model with a partially dependent predator and prey conservation. It is taken that the environment has been divided into two disjoint regions, namely, unreserved and reserved areas, where a predator is not allowed to enter the latter. The first model describes four species: prey refuge (prey in the reserved zone), prey in the unreserved zone, mature and immature predators. The predator is partially dependent on the prey in the unprotected area. The existence of ecological equilibria and their local and global stability is investigated. By using the Lyapunov theorem, sufficient conditions on the global stability of the equilibriums are obtained. Some numerical simulations show the viability of our results. The results show that the reserved area has a stabilizing impact on the stage structure predator-prey model.