Study of dynamical properties in a bi-dimensional model describing the prey–predator dynamics with strong Allee effect in prey.

In this work, we analyze a bi-dimensional differential equation system obtained by considering Holling type II functional response in prey–predator model with strong Allee effect in prey. One of the important consequence of this modification is the existence of separatrix curve which divides the behaviour of the trajectories in the phase plane. The results show that the origin is an attractor for any set of parameter values. Axial equilibrium points are stable or unstable according to the different parametric restrictions. The unique positive equilibrium point, if it exists, can be either an attractor or a repeller surrounded by a limit cycle, whose stability and uniqueness are also established. Therefore long-term coexistence of both populations is possible or they can go to extinction. Conditions on the parameter values are derived to show that the positive equilibrium point can be emerged or annihilated through transcritical bifurcation at axial equilibrium points. The existence of two heteroclinic curves is also established. It is also demonstrated that the origin is a global attractor in the phase plane for some parameter values, which implies that there are satisfying conditions where both populations can go to extinction. Ecological interpretations of all analytical results are provided thoroughly.