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Hopf bifurcation in a delayed predator-prey system with general group defense for prey
Chuanying Zhang,Ranchao Wu,Mengxin Chen
Keywords:Stability, Hopf bifurcation, Time delay, Group defense
Abstract:
      In this paper, a delayed predator-prey system with group defence for prey is investigated. Firstly, in the absence of spatial diffusion and time delay, the stability of positive equilibrium and existence of the Hopf bifurcation are investigated, as well as the direction of the Hopf bifurcation, which is determined by applying the first Lyapunov number. Then, the occurrence of the Hopf bifurcation in the diffusion-driven delayed system is further explored. By using the center manifold reduction and the normal form theorem, the conditions ensuring the occurrence of Hopf bifurcation, its direction and its stability are formulated in terms of different parameters. Finally, some numerical simulations are carried out to verify the theoretical results and the existence of the homogeneous periodic solution is exhibited by setting different values of parameters. Moreover, stable temporal periodic solutions and spatially inhomogeneous periodic solutions are identified from the numerical simulations. The obtained results are also explained and discussed from the practical point of view.