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.