Study of a prey–predator model with preventing crop pest using natural enemies and control

In this article, we propose a mathematical model of an ecological model that is presented of prey, pest, and natural enemy of pest. Different equilibrium points are obtained, their existence and stability of the steady states are investigated. Further, a mathematical model is obtained in the presence of the control variable for designing optimal pest control problems and to study the effect on crop pest. An optimal control technique is one that minimizes the loss due to crop damage and causes minimum harm to the environment. Several chemicals, biological, physical control, and mechanisms are used. We apply Pontryagin’s maximum principle to find the existence, characterization, and necessary conditions of the optimal control. Numerical simulations are further verified to support analytical results and to illustrate a different system that can be observed in the model.