Abstract
A novel Filippov forest-pest system with threshold policy control (TPC) is established while an economic threshold (ET) is used to guide switching. The aim of our work is to address how to reasonably and successfully control pests by means of sliding dynamics for the Filippov system. On the basis of the above considerations, conditions for the existence and stability of equilibria of subsystems are addressed, and the sliding segments and several types of equilibria of the proposed system are defined. These equilibria include the regular/virtual equilibrium, pseudoequilibrium, boundary equilibrium, and tangent point. Further, not only are the relations between nullclines and equilibria of the Filippov system discussed, but the relations between pseudoequilibrium, nullclines, and the sliding segment are discussed. More importantly, four cases of sliding bifurcations of the Filippov system with respect to different types of equilibria of subsystems are investigated, and the corresponding biological implications concerning integrated pest management (IPM) are analyzed. Our results show that the points of intersection between nullclines are equilibria of the system, and the two endpoints of the sliding segment are on the nullclines. It is also verified that the pseudoequilibrium is the point of intersection of the sliding segment and nullclines of the Filippov system, and the pseudoequilibrium exists on the sliding segment. More interestingly, sliding dynamics analysis reveals that the Filippov system has sliding limit cycles, a bistable state and a stable refuge equilibrium point, and the optimal time and strategy for controlling pests are provided.
Highlights
Because insect infestations increase forest mortality, which in turn a ects the carbon cycle and causes air pollution [1], many scholars have paid much attention to the e ects of disturbances on forests [2,3,4,5,6,7]
E present paper focuses on the control of pests, and the sliding dynamics of a Filippov system based on the tree-beetle model is discussed
The relations between pseudoequilibrium, nullclines and the sliding segment are discussed. e results verify that pseudoequilibrium is the point of intersection of the sliding segment and nullclines of the Filippov system, and the pseudoequilibrium exists on the sliding segment
Summary
Because insect infestations increase forest mortality, which in turn a ects the carbon cycle and causes air pollution [1], many scholars have paid much attention to the e ects of disturbances on forests [2,3,4,5,6,7]. Filippov pest control models have been proposed to investigate sliding bifurcations [16]. Erefore, combining integrated pest management (IPM), a Filippov system with TPC (i.e., a discontinuous piecewise smooth system [25]) is established to reasonably and successfully control pests It is a challenging but important work to determine the functional response of pests [26,27,28]. E present paper focuses on the control of pests (beetles), and the sliding dynamics of a Filippov system based on the tree-beetle model is discussed.
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