Predator-Prey Model in the Growth Phase of Rice Plants with Pest Control Using Pesticides

Abstract

In Indonesia, Rice is the primary food crop in Indonesia, with rice being the staple food for more than half of the population. However, pest infestations at each stage of rice growth pose a serious threat that can reduce crop yields. This study developed a predator-prey mathematical model to describe the interactions between rice plants at different growth stages, such as the vegetative, reproductive, and maturation phases, and three major pests: rice stem borers, brown planthoppers, and rats. The model also incorporates Pontryagin's maximum principle for optimal control through pesticide application. The method yields state and costate equations, which are solved using the fourth-order Runge-Kutta forward-backward sweep method with the assistance of Scilab 2024 software. The objective of this research is to examine the model, analyze its stability and controllability, and conduct numerical simulations. The results indicate that the model has a stable equilibrium point, suggesting that the system is controllable. The simulation shows that pesticide application is effective in controlling pests during all stages of rice growth. The populations of rice stem borers, brown planthoppers, and rats decrease significantly, with each pest population reduced to fewer than 10 individuals. These results demonstrate that pesticide use helps maintain ecological balance, and reduces pest damage to rice plants.