Bifurcations of a Filippov ecological system with an A-type discontinuity boundary

Based on the integrated pest management strategy, this paper proposes a Filippov pest–natural enemy system with a novel threshold control strategy. We not only incorporate the changing rate into the control index of the pest population but also consider a threshold value for the natural enemy. This novel threshold policy presents the discontinuity boundary as a complicated ‘A’ type, which induces abundant and complex sliding dynamics. Through theoretical analysis, both curve boundaries could have at most six sliding segments and two pseudo-equilibria, while the other straight line boundary could have a unique stable sliding segment with two pseudo-equilibria. Numerically, the sliding mode bifurcation confirms that the system can have six sliding segments and two pseudo-equilibria simultaneously. Particularly, we discover a new global bifurcation phenomenon that may be termed as a triple limit cycle bifurcation, which reveals the coexistence of three nested limit cycles, various bistable states of two nested or independent attractors, as well as the appearance of a meaningful long transient. Our results not only demonstrate the important effect of nonlinear boundaries but also provide a new perspective on practical pest control problems.