Abstract
We established a mathematical model based on the sense of biological survey in the field of agriculture and introduced various control methods on how to prevent the crops from destructive pests. Basically, there are two main stages in the life cycle of natural enemies like insects: mature and immature. Here, we construct a food chain model of plant pest natural enemy. In natural enemies, there are two stages of construction. Also, we consider three classes of diseases in the pest population, namely, susceptible, exposed, and infectious in this proposed work. In order to categorize the considered models into the class of Impulsive Differential Equations (IDEs), in our study, we specifically consider two ecosystems, which define the impact of control mechanisms on the impulsive releasing of virus particle natural enemies and infectious pests at particular time. Additionally, the importance of spraying virus particles in pest control is discussed; then, we obtain two types of periodic solutions for the system, namely, plant pest extinction and pest extinction. By utilizing the small amplitude perturbation techniques and Floquet theory of the impulsive equation, we obtain the local stability of both periodic solutions. Moreover, the comparison technique of IDE shows the sufficient conditions for the global attractivity of a pest extinction periodic solution. With the assistance of the comparison results, we draw a numerical calculation for the addressed models. Finally, we extend the study of the two models for pest management models: with and without the existence of virus particle.