Dynamic of a delayed predator–prey model with birth pulse and impulsive harvesting in a polluted environment

•A delayed predator–prey model was studied.•Birth pulse, impulsive harvesting, pollution are considered, which are more reasonable.•A threshold is obtained for the permanence of the system.•Numerical simulations support our theoretical analysis.

In this paper, we propose a delayed predator–prey model with birth pulse and impulsive harvesting in a polluted environment. Existence conditions of the predator-extinction periodic solution are derived by developing the discrete dynamical system, which is determined by the stroboscopic map. Further, we discuss the global attractivity of predator-extinction periodic solution and permanence of the system, and obtain the threshold conditions. The results provide a dependable theoretical strategies to protect population from extinction in a polluted environment. Finally, the numerical simulations are presented for verifying the theoretical conclusions.