Uniqueness of limit cycles and multiple attractors in a Gause-typepredator-prey model with nonmonotonic functional response and Allee effecton prey

The main purpose of this work is to analyze a Gause type predator-prey modelin which two ecological phenomena are considered: the Allee effect affectingthe prey growth function and the formation of group defence by prey in orderto avoid the predation.
    We prove the existence of a separatrix curves in the phase plane, determinedby the stable manifold of the equilibrium point associated to the Alleeeffect, implying that the solutions are highly sensitive to the initialconditions.
    Trajectories starting at one side of this separatrix curve have theequilibrium point $(0,0)$ as their $\omega $-limit, while trajectoriesstarting at the other side will approach to one of the following threeattractors: a stable limit cycle, a stable coexistence point or the stableequilibrium point $(K,0)$ in which the predators disappear andprey attains their carrying capacity.
    We obtain conditions on the parameter values for the existence of one or twopositive hyperbolic equilibrium points and the existence of a limit cyclesurrounding one of them. Both ecological processes under study, namely thenonmonotonic functional response and the Allee effect on prey, exert astrong influence on the system dynamics, resulting in multiple domains ofattraction.
    Using Liapunov quantities we demonstrate the uniqueness of limit cycle, whichconstitutes one of the main differences with the model where the Alleeeffect is not considered. Computer simulations are also given in support ofthe conclusions.