Bistability and Double‐Peak Hormetic Effects of a Discrete‐Time Pest Model

ABSTRACT In formulating an integrated pest management (IPM) strategy, the judicious application of insecticides is crucial for ensuring continuous crop production and maintaining ecological balance. However, devising an effective strategy for insecticide use presents a significant challenge, particularly due to the potential for improper dosage or timing, which could trigger so‐called hormetic effects—where low doses of insecticides may stimulate pest growth, while high doses may suppress it. This paper introduces a novel mathematical model based on the method of piecewise constant arguments, aimed at optimizing control strategies to prevent large‐scale pest outbreaks. The model specifically considers the strategy of applying insecticides during the time interval between consecutive pest generations. Firstly, we discuss the existence and stability of the fixed points when insecticides are sprayed at the end of generations, analyzing the system's bifurcation behavior based on intrinsic growth rate and predation density parameters. Next, we consider spraying insecticides within each generation, using two‐parameter and one‐parameter bifurcation diagrams to analyze the existence and stability of the fixed points, as well as the system's intricate dynamical behaviours. Ultimately, the main theoretical results are validated through numerical simulations. Experimental results reveal that when insecticide application timing occurs within a critical range, low‐dose applications can promote pest population growth, whereas high‐dose applications may trigger a secondary outbreak following initial suppression, resulting in the double‐peak hormetic effects. Additionally, the paper comprehensively discusses the impact of pest growth rates, the timing of insecticide application, and the intensity of insecticide on the system dynamics. The findings provide valuable insights for selecting the appropriate timing for pest prevention, contributing to the optimization of pest management strategies and the sustainable development of agricultural ecosystems.