Dynamics of a crop, pest and predator model in an agricultural system

Abstract

A three-species model of the interactions of crop, pest, and predator has been constructed in this work. In the absence of pests and harvesting of crops, the rate of crop growth is considered as logistic. It is considered that pests consume crops whereas predators consume pests. Harvesting of the crops has been considered here. Holling type II functional response has been considered for crop and pest consumption. It is also assumed that the predators may consume pests via Holling type II functional response. Positivity and boundedness of the solution of the model have been investigated. The various equilibrium points of the model are evaluated, and the local and global stability are then investigated. Studies on the Hopf bifurcation existence criterion in connection to a crucial model parameter have been conducted. The optimal crop harvesting has then been determined using an objective function that has been defined. The Pontryagin’s maximum theory is used to calculate the ideal crop harvesting rate. The consumption of pests by predators has been proven to help keep the system stable up to a point, but a higher rate of pest consumption by predators can potentially cause instability. According to observation, the model may become unstable as the mortality rate of predators of pests rises. It is found that the increase in optimal harvesting of crops may lead the model towards stability. It has been noted that increased rate of crops consumption by pests may result in lower agricultural yields.