Bifurcation of Codimension 3 in a Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response
Bifurcation of Codimension 3 in a Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response
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DOI：
10.1142/s0218127416500346
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发表时间：
2016-03
期刊：
影响因子：
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通讯作者：
Jicai Huang;Xiaojing Xia;Xinan Zhang;S. Ruan
中科院分区：
文献类型：
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作者：
Jicai Huang;Xiaojing Xia;Xinan Zhang;S. Ruan
[Li & Xiao，2007]证明了在具有简化Holling IV型功能反应的Leslie型捕食者-食饵模型中，对于某些参数值，可以同时发生一些复杂的分支，如余维1亚临界Hopf分支和余维2 Bogdanov-Takens分支.在本文中，我们表明，对于同一个模型存在一个唯一的退化的正平衡，这是一个退化的Bogdanov-Takens奇点（焦点情况）的余维3的其他参数值。证明了该模型在唯一的退化正平衡点附近存在余维为3的退化焦点型Bogdanov-Takens分支。数值模拟，包括三个双曲正平衡点的共存，两个极限环，双稳态（一个稳定平衡点和一个稳定极限环，或两个稳定平衡点），三稳态（两个稳定平衡点和一个稳定极限环），一个稳定极限环包围一个同宿环，一个同宿环...
It was shown in [Li & Xiao, 2007] that in a predator–prey model of Leslie type with simplified Holling type IV functional response some complex bifurcations can occur simultaneously for some values of parameters, such as codimension 1 subcritical Hopf bifurcation and codimension 2 Bogdanov–Takens bifurcation. In this paper, we show that for the same model there exists a unique degenerate positive equilibrium which is a degenerate Bogdanov–Takens singularity (focus case) of codimension 3 for other values of parameters. We prove that the model exhibits degenerate focus type Bogdanov–Takens bifurcation of codimension 3 around the unique degenerate positive equilibrium. Numerical simulations, including the coexistence of three hyperbolic positive equilibria, two limit cycles, bistability states (one stable equilibrium and one stable limit cycle, or two stable equilibria), tristability states (two stable equilibria and one stable limit cycle), a stable limit cycle enclosing a homoclinic loop, a homoclinic loop...