J. Korean Math. Soc. 2013; 50(4): 695-713
Printed July 1, 2013
https://doi.org/10.4134/JKMS.2013.50.4.695
Copyright © The Korean Mathematical Society.
Rongzhen Lin, Shengqiang Liu, and Xiaohong Lai
Harbin Institute of Technology, Harbin Institute of Technology, Xiamen University
We formulate and study a predator-prey model with nonmonotonic 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: predator-prey, weak Allee effects, bifurcation, limit cycle
MSC numbers: 34C23, 92D25
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