  The transition between strong and weak Allee effects in prey provides a
simple regime shift in ecology. A deteriorating environment changes weak Allee
effects into strong ones. In this paper, we study the interplay between the
functional response of Holling type IV and both strong and weak Allee effects.
The model investigated here presents complex dynamics and high codimension
bifurcations. In particular, nilpotent cusp bifurcation, nilpotent saddle
bifurcation and degenerate Hopf bifurcation of codimension 3 are completely
analyzed, and the existence of homoclinic and heteroclinic loops are proven.
Remarkably it is the first time that three limit cycles are discovered in
predator-prey models with Allee effects. It turns out that strong Allee effects
destabilize population dynamics, induce more regime shifts, decrease
establishment likelihood of both species, increase vulnerability of ecosystem
to collapse, while weak Allee effects promote sustained oscillations between
predators and preys compared to systems without Allee effects. The theory
developed here provides a sound foundation for understanding predator-prey
interactions and biodiversity of species in natural systems.
