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In this paper, a modified Leslie-type predator–prey system with Holling-type III functional response is formulated to investigate the dynamics in the presence of special Allee effect induced by fear factors. First, we calculate the first three focal values by the method of successor function to ensure that the system has an unstable weak focus of multiplicity 3. A focus- or center-type degenerate Bogdanov–Takens singularity of codimension 3 and a weak focus of multiplicity 1 or 2 together with a cusp of codimension 2 are derived for the system. Then multiple bifurcations are explored. It is shown that the system undergoes Hopf bifurcation of codimension 3 and hence three limit cycles are generated. There exists another large stable limit cycle enclosing these three limit cycles by Poincaré–Bendixson theorem. We demonstrate that a degenerate focus-type Bogdanov–Takens bifurcation of codimension 3 can occur by calculating the universal unfolding. It is also shown that a subcritical, supercritical or degenerate Hopf bifurcation occurs together with a Bogdanov–Takens bifurcation. Finally, numerical simulations are used to support the theoretical results. The analysis results reveal that the special Allee effect is beneficial for the persistence and diversity of ecosystem.
Shang et al. (Wed,) studied this question.