Bifurcations analysis of a 3D Filippov pest-natural enemy system with stage structure for the prey

Pest control is an important application of the Filippov system in ecology and has attracted much attention. Many studies on Filippov pest-natural enemy systems have been done by employing the widely recognized Integrated Pest Management (IPM) strategy. However, those studies primarily focused on planar Filippov models without considering the stage structure of populations. It is well-known that almost all species have a life cycle from immature to mature. Thus, in this paper, we propose a three-dimensional (3D) Filippov system that takes into account the stage structure of the pest population. The weighted sum of two-stage pest populations is set as the control index. Since the discontinuity boundary changes from line to plane, new analytical techniques are developed to investigate the complex dynamics. In particular, abundant sliding bifurcations are detected, especially an enigmatic global bifurcation. This intriguing global bifurcation mechanism may overturn traditional bifurcation analyses, indicating the necessity for further investigations into the dynamical behavior of Filippov systems. Moreover, our results make a strong case for the critical importance of the economic threshold in the IPM strategy. Pest control is found not always effective simply by acting sooner; setting the threshold value too low would cause control failure while setting it too high would invalidate the strategy itself. Detailed biological implications are discussed in the conclusion.