Supercritical Neimark‐Sacker bifurcation of a discrete‐time Nicholson‐Bailey model

Abstract

We study the local dynamics and supercritical Neimark‐Sacker bifurcation of a discrete‐time Nicholson‐Bailey host‐parasitoid model in the interior of . It is proved that if α>1, then the model has a unique positive equilibrium point , which is locally asymptotically focus, unstable focus and nonhyperbolic under certain parametric condition. Furthermore, it is proved that the model undergoes a supercritical Neimark‐Sacker bifurcation in a small neighborhood of the unique positive equilibrium point , and meanwhile, the stable closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the period or quasiperiodic oscillations between host and parasitoid populations. Some numerical simulations are presented to verify theoretical results.