Bifurcations and steady states of a predator-prey model with strong Allee and fear effects

In this paper, the predator-prey model with strong Allee and fear effects is considered. The existence of the equilibria and their stability are established. Especially it is found that there is an interesting degenerate point, which is a cusp point with codimension 2 or higher codimension, or an attracting (repelling)-type saddle-node, subject to different conditions. Then the Hopf bifurcation and its direction, the saddle-node bifurcation and the Bogdanov-Tankens bifurcation are further explored. Afterwards, with the help of the energy estimates and the Leray-Schauder degree, the nonexistence and existence of the nonconstant steady states of the model are presented. From the obtained results, we find that strong Allee effect will cause the per capita growth rate of prey species from negative to positive; both the fear and Allee effects could affect the existence of equilibria and bifurcations; meanwhile, the diffusion rates will affect the existence of the nonconstant steady states.