Mathematical study of stage-structured pests control through impulsively released natural enemies with discrete and distributed delays

Many insect pest species have two major stages in their life-history: an immature larval stage and a mature adult stage. The immature larval stage may have a significant duration, during which the insect may not be the same natural predators as in the adult stage. Keeping this biological background in mind, we propose a stage-structured natural enemy–pest model with distributed and discrete delays for natural enemy and pest, respectively. It is also assumed that the natural enemies are released in an impulsive manner. The global attractivity of the pest-extinction periodic solution of the system is obtained using comparison techniques on impulsive delay differential equations. Furthermore, the permanence conditions for the coexistence of the species are explored, which indicates that the maturation time delay and the amount of natural enemies released in a pulse can affect the dynamics of the system. The threshold values for maturation delay of pest and the releasing amount natural enemy are obtained for extinction and permanence of the species. Finally, numerical simulation is given in support of validation of the theoretical results to provide a strategic basis for the biological pest management.