Rich dynamics of a Filippov plant disease model with time delay

Abstract

We propose a nonsmooth Filippov plant disease model with threshold strategy of roguing of the infected plants with a saturated roguing rate. Different from traditional Filippov models, here we incorporate time delay which represents the incubation period of the plant disease to make the model more realistic. Analytical results show that the two delayed subsystems might admit Hopf bifurcations when the delay passes through some critical values. Then we investigate some key elements to our model, including the regular/virtual equilibrium, the sliding segment, sliding mode dynamics and pseudoequilibrium. The results indicate that all solutions finally converge to either the regular equilibrium, the pseudoequilibrium or a stable standard periodic solution depending on the values of the threshold level and the time delay. Furthermore, we show that the time delay plays a significant role in discontinuity-induced bifurcations. That is, as the time delay is varied, the model exhibits complex local sliding bifurcation (boundary bifurcation) and global sliding bifurcations from a standard periodic solution to a sliding bifurcation then to a crossing bifurcation. In conclusion, a Filippov system with time delay can provide some new insights for plant disease control.