Sliding motion and global dynamics of a Filippov fire-blight model with economic thresholds

Abstract

Cutting off infected branches has always been an effective method for removing fire-blight infection in an orchard. We introduce a Filippov fire-blight model with a threshold policy: cutting off infected branches and replanting susceptible trees. The dynamics of the proposed piecewise smooth model are described by differential equations with discontinuous right-hand sides. For each susceptible threshold value ST, we investigate the global dynamical behaviour of the Filippov system, including the existence of all the possible equilibria, their stability and sliding-mode dynamics, as we vary the infected threshold level IT. Our results show that model solutions ultimately approach the equilibrium that lies in the region above IT or below IT or on I=IT, or the equilibrium ET=(ST,IT) on the surface of discontinuity. Furthermore, control strategies should be taken when the solution of this system approaches the equilibrium that lies in the region above IT. The findings indicate that proper choice of susceptible and infected threshold levels can either preclude an outbreak of fire blight or lead the number of infected trees to a desired level.