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Bifurcations of a predator-prey system with weak Allee effects
Rongzhen Lin,Shengqiang Liu,Xiaohong Lai 대한수학회 2013 대한수학회지 Vol.50 No.4
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 analysisin a predator-prey system with nonmonotonic functional response, SIAMJ. Appl. Math. 61 (2001), no. 4, 1445–1472] but containing an extraterm describing weak Allee effects on the prey. We obtain the global dy-namics of the model by combining the global qualitative and bifurcationanalysis. Our bifurcation analysis of the model indicates that it exhibitsnumerous kinds of bifurcation phenomena, including the saddle-node bi-furcation, the supercritical and the subcritical Hopf bifurcations, and thehomoclinic bifurcation, as the values of parameters vary. In the genericcase, the model has the bifurcation of cusp type of codimension 2 (i.e.,Bogdanov-Takens bifurcation).
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BIFURCATIONS OF A PREDATOR-PREY SYSTEM WITH WEAK ALLEE EFFECTS
Lin, Rongzhen,Liu, Shengqiang,Lai, Xiaohong Korean Mathematical Society 2013 대한수학회지 Vol.50 No.4
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).
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