Bifurcation analysis in a predator-prey model with Allee effect

In this paper, strong Allee effects on the bifur-cation of the predator-prey model with ratio-dependent Holling type III response are considered, where the prey in the model is subject to a strong Allee effect. The exis-tence and stability of equilibria and the detailed behavior of possible bifurcations are discussed. Specifically, the existence of saddle-node bifurcation is analyzed by using Sotomayor's theorem, the direction of Hopf bifurcation is determined, with two bifurcation parameters, the occur-rence of Bogdanov-Takens of codimension 2 is showed through calculation of the universal unfolding near the cusp. Comparing with the cases with a weak Allee effect and no Allee effect, the results show that the Allee effect plays a significant role in determining the stability and bifurcation phenomena of the model. It favors the coexis-tence of the predator and prey, can lead to more complex dynamical behaviors, not only the saddle-node bifurcation but also Bogdanov-Takens bifurcation. Numerical simu-lations and phase portraits are also given to verify our theoretical analysis.