Antipredator behavior of a nonsmooth ecological model with a state threshold control strategy

A nonsmooth ecological model was proposed and analyzed, focusing on IPM, state-dependent feedback control strategies, and anti-predator behavior. The main objective was to investigate the impact of anti-predator behavior on successful pest control, pest outbreaks, and the dynamical properties of the proposed model. First, the qualitative behaviors of the corresponding ODE model were presented, along with an accurate definition of the Poincaré map in the absence of internal equilibrium. Second, we investigated the existence and stability of order-k (where k = 1, 2, 3) periodic solutions through the monotonicity and continuity properties of the Poincaré map. Third, we conducted numerical simulations to investigate the complexity of the dynamical behaviors. Finally, we provided a precise definition of the Poincaré map in situations where an internal equilibrium existed within the model. The results indicated that when the mortality rate of the insecticide was low or high, the boundary order-1 periodic solution of the model was stable. However, when the mortality rate of the insecticide was maintained at a moderate level, the boundary order-1 periodic solution of the model became unstable; in this case, pests and natural enemies could coexist.