Non-smooth plant disease models with economic thresholds

In order to control plant diseases and eventually maintain the number of infected plants below an economic threshold, a specific management strategy called the threshold policy is proposed, resulting in Filippov systems. These are a class of piecewise smooth systems of differential equations with a discontinuous right-hand side. The aim of this work is to investigate the global dynamic behavior including sliding dynamics of one Filippov plant disease model with cultural control strategy. We examine a Lotka–Volterra Filippov plant disease model with proportional planting rate, which is globally studied in terms of five types of equilibria. For one type of equilibrium, the global structure is discussed by the iterative equations for initial numbers of plants. For the other four types of equilibria, the bounded global attractor of each type is obtained by constructing appropriate Lyapunov functions. The ideas of constructing Lyapunov functions for Filippov systems, the methods of analyzing such systems and the main results presented here provide scientific support for completing control regimens on plant diseases in integrated disease management.

► This is the first mathematical model of plant diseases using Filipov systems. ► We determine economic thresholds for proportional planting to control plant diseases. ► We find five types of equilibria and determine the global stability properties of each. ► Infected plants are maintained below the threshold; susceptible plants also survive.