Bifurcation Of A Predator-prey System With Weak Allee Effect

We formulate and study a predator-prey model with nonmonotonic functional response type and weak Allee effects on the prey. Our model is similar to the system studied by Xiao, et al in [Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61(2001), pp. 1445-1472] but contains 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 analyses; Our bifurcation analysis of the model indicates that it exhibits numerous kinds of bifurcation phenomena, including the saddle-node bifurcation, the supercritical and 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).