In this paper, we propose a Leslie-Gower predator-prey model with strong Allee effect on prey and fear effect on predator. We discuss the existence and local stability of equilibria by making full use of qualitative analytical theory. It is shown that the above system exhibits at most two positive equilibria and it can undergo a series of bifurcation phenomena. We indicate that the dynamical behavior of the model is closely related to the fear effect on predator. In detail, when the fear effect parameter p = p*, the system will undergo degenerate Hopf bifurcation. There exist two limit cycles (the inner is stable and the outer is unstable). However, when p = p(*), the system will undergo degenerate Bogdanov-Takens bifurcation. Also, by numerical simulation, we conclude that the stronger the fear effect, the bigger the density of prey species. The above shows that fear effect on predator is beneficial to the persistence of the prey species. Our results can be seen as a complement to previous works [Gonzalez-Olivares et al., 2011; Pal & Mandal, 2014].
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