Complex Dynamics of Predator–Prey Systems with Allee Effect

In this paper, we apply bifurcation theory to consider four predator–prey systems which include the Allee effect, and show that the species having a strong Allee effct may affect their predation and hence extinction risk. It is shown that the models with the Allee effect exhibit more complex dynamical behaviors compared with that without the Allee effect. In particular, two models with no Allee effect do not have Hopf bifurcation, but can have Hopf bifurcation with the Allee effect; and one model, which does not have Bogdanov–Takens bifurcation if no Allee effect is involved, can have Bogdanov–Takens bifurcation of codimension two. Especially, for one model with Holling type II functional response of the predator to the prey, the Allee effect not only completely changes the stability of the equilibrium at the origin, but also changes the supercritical Hopf bifurcation arising from an interior equilibrium to subcritical Hopf bifurcation with very limited parameter values to yield unstable limit cycles, and further increases the system’s stability.