Abstract: In nature, population growth often has a process of growing and development. At different age stages, both predators and prey will show different growth characteristics. In addition, the delay has a great influence on the topological structure of differential equation solutions. In many cases, the change of the delay will destroy the stability of the positive equilibrium point and produce Hopf bifurcation. Therefore, this paper takes the growth time from young predator to adult predator as the delay, constructs a time-delayed predator-prey system with stage structure for both predator and prey. Using the persistence theory for infinite-dimensional systems and Hurwitz criterion, the permanent persistence condition of this system and the local stability condition of the system's coexistence equilibrium are given. Choosing the delay as a bifurcation parameter, we derive the existence of the Hopf bifurcation in this system, and then using normal form theory and center manifold arguments, we discuss the direction of the Hopf bifurcation and the stability of period solutions bifurcating from the Hopf bifurcations. Finally, the critical value $\tau_{0n}$ that causes Hopf bifurcation is obtained by choosing the qualified parameters satisfying the theorem conditions, and numerical results are presented to verify the theoretical conclusion.

Key words: predator-prey system, time delay, stage structure, stability, Hopf bifurcation