Bifurcation analysis of a special delayed predator-prey model with herd behavior and prey harvesting

In this paper, we propose a predator-prey system with square root functional response, two delays and prey harvesting, in which an algebraic equation stands for the economic interest of the yield of the harvest effort. Firstly, the existence of the positive equilibrium is discussed. Then, by taking two delays as bifurcation parameters, the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained. Next, some explicit formulas determining the properties of Hopf bifurcation are analyzed based on the normal form method and center manifold theory. Furthermore, the control of Hopf bifurcation is discussed in theory. What's more, the optimal tax policy of system is given. Finally, simulations are given to check the theoretical results.