Hopf bifurcation in a spatial predator-prey model with the square root functional response for the predator

‎In this paper‎, ‎we consider a diffusive predator-prey model‎, ‎in which the prey population lives in groups and has a social behavior‎. ‎We show that Hopf bifurcation and the existence of a center manifold may occur‎. ‎The linear stability analysis shows that a Hopf bifurcation occurs in the corresponding homogeneous system‎. ‎Next‎, ‎we study the effect of diffusion parameters on homogeneous dynamics‎. ‎By choosing a proper bifurcation parameter‎, ‎we prove that a Hopf bifurcation occurs in the nonhomogeneous system‎. ‎We compute the normal form of this bifurcation up to the third order and obtain the direction of the Hopf bifurcation‎. ‎Finally‎, ‎we provide numerical simulations to illustrate our analytical findings‎.