Dynamics of a Predator-prey Model with Allee Effect and Prey Group Defense

Dynamical properties of a Gauss type of planar predator-prey system with Allee effect and non-monotonic response function are discussed. We are interested in persistent features lying in the first quadrant, which amount to structurally stable phase portraits. We show that all positive solutions are uniformly bounded. It is also proved that the system has at most two equilibria in the interior of the first quadrant and can exhibit interesting bifurcation phenomena, including Bogdanov-Takens, Hopf, transcritical and saddle-node bifurcations. The system may have a stable periodic orbit, or a homoclinic loop, or a heteroclinic connection, a saddle point, or a stable focus, depending on parameter values. Biologically, both populations may survive for certain values of parameters. Computer simulations are also given in support of the conclusions.