Global Hopf bifurcation of a delayed diffusive Gause-type predator-prey system with the fear effect and Holling type III functional response

In this paper, a delayed diffusive predator-prey system with the fear effect and Holling type III functional response is considered, and Neumann boundary condition is imposed on this system. First, we explore the stability of the unique positive constant steady state and the existence of local Hopf bifurcation. Then the global attraction domain G∗ of system (4) is obtained by the comparison principle and the iterative method. Through constructing the Lyapunov function, we investigate uniform boundedness of periodic solutions’periods. Finally, we prove the global continuation of periodic solutions by  the global Hopf bifurcation theorem of Wu. Moreover, some numerical simulations that support the analysis results are given.