Hopf bifurcation analysis of a multiple delays stage-structure predator-prey model with refuge and cooperation

In this paper, a multiple delays stage-structure predator-prey model with refuge and cooperation is established. First, the local asymptotic stability of the trivial equilibrium and the predator extinction equilibrium are discussed by analyzing the characteristic equations of the system. Second, taking time delays as the bifurcation parameters, the existence of Hopf bifurcation at the positive equilibrium is given. Next, the direction of Hopf bifurcation and the stability of the periodic solutions are analyzed based on the center manifold theorem and normal form theory. Moreover, the optimal harvesting policy of the system is showed by using Pontryagin's maximum principle. Finally, we give the global sensitivity analysis of some parameters by calculating the partial rank correlation coefficients, and some numerical simulations are performed to verify the correctness and feasibility of the theoretical results by using the MATLAB software.