Journal
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
Volume -, Issue -, Pages -Publisher
SPRINGER
DOI: 10.1007/s10884-021-10079-1
Keywords
Generalist predatory mite; Pest leafhopper; Hopf bifurcation; Bogdanov-Takens bifurcation; Nilpotent singularity of codimension 3; Nilpotent focus of codimension 4; limit cycles
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Funding
- NSERC of Canada
- York University Research Chair program
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This paper investigates the predator-prey relationship between the tea green leafhopper and the mite, and identifies some interesting nilpotent singularities and bifurcations.
The tea green leafhopper Empoasca onukii is one kind of insect pest threatening the tea production, and the mite Anystis baccarum has been used as an agent for pest control. In this paper, we introduce a generalist predator-prey model to study the dynamics for informing biological control. There have been some bifurcation studies of the generalist predator-prey model in the last few years. Except for the bifurcations include saddle-node bifurcation of codimension 1 and 2, Hopf bifurcations, and Bogdanov-Takens bifurcation of codimension 2 and 3, we also present the bifurcations of nilpotent singularities of elliptic and focus type of codimension 3. We find that the nilpotent singularities are associated with a cubic Lienard system, and the nilpotent bifurcations are three-parameter bifurcations of a codimension 4 nilpotent focus. Furthermore, we show that the nilpotent focus serves as an organizing center to connect all the codimension 3 bifurcations in the system. We also present the bifurcation diagrams to unfold the nilpotent singularities of codimension 3. One interesting observation is that we show numerically the existence of three limit cycles in the system .
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