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.

Regularization of the Boundary Equilibrium Bifurcation in Filippov System with Rich Discontinuity Boundaries

“…In order to understand the disease transmission dynamics and provide economic and environmentally acceptable control strategies, many mathematical models have been constructed [46]. More recently, a Filippov plant disease model that incorporates cutting off infected branches and replanting susceptible trees has been built [15]. The model [15] is constructed based on the following regulations:…”

“…There are also some analyses of the Filippov system with two discontinuity boundaries separating R 2 into four regions; see [11][12][13][14]. Recently, some work on a Filippov system with two discontinuity boundaries separating R 2 into three regions attracted our attention [15,16].…”

“…The case with three regions has fruitful applications in the real world. For example, such a Filippov plant disease model can help us understand disease transmission dynamics and provide economic and environmentally acceptable control strategies [15]. An avianonly model that incorporates culling off the infected and susceptible birds can also be described by a Filippov system with two discontinuity boundaries separating R 2 into three regions [16].…”

“…The inspiration for this work is the work by Chen [15] and Yang [16], where the stability of different types of equilibria of such a Filippov system is studied. However, the bifurcation analysis of such systems is rare, particularly the boundary equilibrium bifurcation.…”

Global dynamics of a non-smooth SIV system with uncertain effective vaccine protection rate

“…However, we know that vaccination presents different effects in different regions at different times, so we will consider the two-threshold policy. Recently, many scholars have proposed the two-threshold system in biomathematical modeling, such as epidemic model [11,12], predator-prey model [13], and plant disease model [14], etc.…”