Complex Dynamics of a Filippov Three-Species Food Chain Model

In order to avoid high extinction risks of prey and keep the stability of the three-species food chain model, we introduce a Filippov food chain model (FFCM) with Holling type II under threshold policy control. The threshold policy is designed to play a pivotal strategy for controlling the three species in the FFCM. With this strategy, no control is applied if the density of the prey population is less than the threshold, thus the exploitation is forbidden. However, the exploitation is permitted if the density of the prey population increases and exceeds the threshold. The dynamic behaviors and the bifurcation sets of this model including the existence and stability of different types of equilibria are discussed analytically and numerically. Moreover, the regions of sliding and crossing segments are analyzed. The dynamic behaviors of sliding mode including the bifurcation sets of pseudo-equilibria are investigated. Numerically, the bifurcation diagram and maximum Lyapunov exponents are computed and plotted to show the complex dynamics of FFCM, for instance, it has stable periodic, double periodic and chaotic solutions as well as double periodic sliding bifurcation. It is demonstrated that the threshold policy control can be easily implemented and used for stabilizing the chaotic behavior of FFCM.