Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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BIFURCATIONS OF A PREDATOR-PREY SYSTEM WITH WEAK ALLEE EFFECTS

  • Lin, Rongzhen (The Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology) ;
  • Liu, Shengqiang (The Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology) ;
  • Lai, Xiaohong (School of Mathematical Sciences, Xiamen University)
  • Received : 2009.12.17
  • Published : 2013.07.01
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Abstract

We formulate and study a predator-prey model with non-monotonic functional response type and weak Allee effects on the prey, which extends the system studied by Ruan and Xiao in [Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math. 61 (2001), no. 4, 1445-1472] but containing an extra term describing weak Allee effects on the prey. We obtain the global dynamics of the model by combining the global qualitative and bifurcation analysis. Our bifurcation analysis of the model indicates that it exhibits numerous kinds of bifurcation phenomena, including the saddle-node bifurcation, the supercritical and the subcritical Hopf bifurcations, and the homoclinic bifurcation, as the values of parameters vary. In the generic case, the model has the bifurcation of cusp type of codimension 2 (i.e., Bogdanov-Takens bifurcation).

Keywords

Acknowledgement

Supported by : National Natural Science Foundation of China

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