Bubbling and hydra effect in a population system with Allee effect

We present a continuous time predator-prey model and predator’s growth subjected to component Allee effect. The model also includes density dependent mortality of predator. We investigate our model both analytically and numerically, and highlighted the effect of density independent mortality and Allee effect. In our system, we find that a fixed point representing the extinction of predator is always a stable point. When coexistence equilibria exists our system is bistable. We have observed that tristability is possible for our model that includes two stable co-existence fixed point. The most important phenomena which we have observed are hydra effect and cascading effect. Due to component Allee effect in predator the system shows multiple hydra effect. We discuss the phenomenon of bubbling, which indicates increasing and decreasing of amplitudes of cycles. We have presented one-parametric as well as two-parametric bifurcation diagram and also all possible bifurcations that the system could go through.