Holling II predator–prey impulsive semi-dynamic model with complex Poincaré map

Planar impulsive semi-dynamic systems arising biological applications including integrated pest management have been paid great attention recently. However, most of works only focus on very special cases of proposed models, and the complete dynamics are far from being resolved due to complexity. Therefore, a planar impulsive Holling II prey–predator semi-dynamic model has been employed with aims to develop analytical techniques and provide a comprehensive qualitative analysis of global dynamics for whole parameter space. To do this, we initially assume that the proposed ODE model does not exist positive steady state. We determine the Poincaré map for impulsive point series defined in the phase set and analyze its properties including monotonicity, continuity, discontinuity and convexity. We address the existence, local and global stability of an order-1 limit cycle and obtain sharp sufficient conditions for the global stability of the boundary order-1 limit cycle. Moreover, the existence of an order-3 limit cycle indicates that the proposed model exists any order limit cycles. If the proposed ODE model exists an unstable focus, then the results show that a finite or an infinite countable discontinuity points for the Poincaré map imply the model exists a finite or an infinite number of order-1 limit cycles. The bifurcation analyses show that the model undergoes a transition to chaos via a cascade of period-adding bifurcation and also multiple attractors can coexist.