Sliding mode dynamics of a non-smooth Filippov predator-prey system for integrated pest management

In recent years, the problem of pests seriously affects the yield and quality of crop, posing a major challenge to the safe production of crop, which have seriously hindered the development of China's agriculture. How to quickly and accurately monitor pests, timely grasp the occurrence dynamics of pests, and prevent and control pests is of great significance for reducing crop yield losses. Considering the discontinuity of spraying pesticides and releasing natural enemies in the process of pest control, and the Filippov system's ability to accurately depict switching states and human intervention measures, a non-smooth Filippov predator-prey system with threshold strategies is investigated incorporating several different functional responses, such as Holling II functional response and ratio functional response etc, which should be selectively applied dependent on the population of the prey. The aim of this study is to investigate the complex dynamics including bistabilities of the ecosystem when the relative populations of the prey and predator is substantially different, by modelling the non-smooth Filippov system with multiple switchable functional responses for the very first time, which is believed to be more realistic for modeling the dynamics of real ecosystem, thus the solution of the present work may be more suitable for real world applications such as for the integrated pest management. The validity of the proposed system is assessed by simulation, and bifurcation set of equilibria and the global stability of equilibria has been numerically obtained through an arbitrary set of parameters. Moreover, the dynamic behaviors of proposed system, such as the existence of various equilibria and their global stabilities; the existence of various domains such as the sliding domain, escaping domain and crossing domain, have been analyzed in great details in the present work. It is shown that the sliding region and escaping region cannot coexist when the density of the prey and predators is substantially different. It is further demonstrated that the real equilibrium and pseudo-equilibrium points can coexist when the population of the prey is less than that of the predator; and only the virtual equilibrium and pseudo-equilibrium can coexist in the case of when the population of the prey is more than that of the predator.In particular, it is noted that all trajectories of the prey and predators population are eventually converging into certain equilibrium points as it is demonstrated in the numerical simulation. This implies that there exists global asymptotic stability of equilibrium points under the proposed system, in which the population of preys eventually reaches a steady state of density at the real equilibrium and pseudo-equilibrium points. This work also highlights the significant role of the threshold in the process of pest controls: it is seen from this work that different types of equilibrium points can occur dependent on the choice of the economic threshold (ET). The conclusions obtained will be applied to Unmanned Aerial Vehicle (UAV) to spray pesticides and release natural enemies in a timely and quantitative manner, thereby achieving efficient and rapid monitoring and control of large-scale crop. This can more effectively ensure stable and high crop yields, provide theoretical guidance for scientific prevention and control, and is of great significance for reducing the burden on farmers, promoting agricultural development, and realizing agricultural modernization.