Bifurcation analysis of a Filippov predator–prey model with two thresholds

Modelling integrated pest management with two thresholds can be achieved with a Filippov system. The feature of the model is that two thresholds are involved, which generates complex dynamics. By using the qualitative analysis techniques, we obtain that the Filippov system has three pieces of sliding segments as well as three pseudo-equilibria. In particular, one more sliding segment and pseudo-equilibrium brings abundant sliding mode dynamics and sliding bifurcations. On the one hand, sliding mode dynamics and the existence of all kinds of equilibria are derived. On the other hand, several kinds of sliding bifurcations such as sliding mode bifurcation, local sliding bifurcation and global sliding bifurcation are explored. The complex bifurcations reveal that the Filippov predator–prey model can admit the coexistence of multi-attractors, including real equilibria, pseudo-equilibria and (standard, sliding or crossing) periodic solutions. It is shown that control outcomes are sensitive to the threshold values, and hence it is crucial to choose an appropriate threshold level to initiate the control strategy.