Poincaré Map Approach to Global Dynamics of the Integrated Pest Management Prey-Predator Model

An integrated pest management prey-predator model with ratio-dependent and impulsive feedback control is investigated in this paper. Firstly, we determine the Poincaré map which is defined on the phase set and discuss its main properties including monotonicity, continuity, and discontinuity. Secondly, the existence and stability of the boundary order-one periodic solution are proved by the method of Poincaré map. According to the Poincaré map and related differential equation theory, the conditions of the existence and global stability of the order-one periodic solution are obtained when , and we prove the sufficient and necessary conditions for the global asymptotic stability of the order-one periodic solution when . Furthermore, we prove the existence of the order-k periodic solution under certain conditions. Finally, we verify the main results by numerical simulation