Dynamic analysis of discontinuous plant disease models with a non-smooth separation line

Abstract

Based on the aim to minimize losses and maximize returns, we present a discontinuous plant disease model incorporating a threshold policy control. This control is represented by the maximum value between an infected threshold value and the product of a ratio threshold value and the number of susceptible plants, and the maximum value is chosen as an index for decisions on whether to implement control strategies or not. The threshold policy control results in a non-smooth separation line, whose structure leads to giving birth to two pieces of sliding mode regions and two pseudo-equilibria. Meanwhile, one more sliding segment or pseudo-equilibrium brings more complex and rich global dynamics. In this paper, we give a complete analysis of the discontinuous plant disease model and the analysis shows the existence of all kinds of global attractors for various parameter values. Our results reveal that the proper replanting and roguing rates can be used to design the threshold policy such that the number of infected plants can be controlled to an acceptable level.