Bifurcation dynamics on the sliding vector field of a Filippov ecological system

A Filippov crop-pest-natural enemy ecological system with threshold switching surface related to the pest control is developed, which has been completely analyzed by employing the qualitative techniques of non-smooth dynamical systems. The main results reveal that the proposed switching model can have multiple pseudo-equilibria in the sliding region, which result in rich bifurcations in the sliding region including saddle-node, Hopf, Bogdanov-Takens and Hopf-like boundary equilibrium bifurcations. Moreover, the pseudo-periodic solution (or pseudo-homoclinic loop) can be generated in the sliding region through a Hopf bifurcation (or a homoclinic bifurcation), which can collide with the tangential lines at the cusp singularities and finally disappears as parameter varies. This reveals that although the system can stabilize on the sliding region to achieve the purpose of pest control, there are complex dynamical behaviors and sliding bifurcations within the sliding region. Furthermore, as the threshold level varies, the model exhibits the interesting global sliding bifurcations including grazing bifurcation, buckling bifurcation, crossing bifurcation, homoclinic bifurcation to a pseudo-saddle, period-halving bifurcation and chaotic dynamics. This implies that control outcomes are sensitive to the threshold level, and hence it is crucial to choose the threshold level to initiate control strategy.