Rich Sliding Motion and Dynamics in a Filippov Plant-Disease System

Abstract

In order to reduce the spread of plant diseases and maintain the number of infected trees below an economic threshold, we choose the number of infected trees and the number of susceptible plants as the control indexes on whether to implement control strategies. Then a Filippov plant-disease model incorporating cutting off infected branches and replanting susceptible trees is proposed. Based on the theory of Filippov system, the sliding mode dynamics and conditions for the existence of all the possible equilibria and Lotka–Volterra cycles are presented. We find that model solutions ultimately approach the positive equilibrium that lies in the region above the infected threshold value [Formula: see text], or the periodic trajectories that lie in the region below [Formula: see text], or the pseudo-attractor [Formula: see text], as we vary the susceptible and infected threshold values. It indicates that the plant-disease transmission is tolerable if the trajectories approach [Formula: see text] or the periodic trajectories lie in the region below [Formula: see text]. Hence an acceptable level of the number of infected trees can be achieved when the susceptible and infected threshold values are chosen appropriately.