Understanding Cannibalism Dynamics in Predator–Prey Interactions: Bifurcations and Chaos Control Strategies

The occurrence of cannibalism is common in natural colonies and can substantially affect the functional relationships between predators and prey. Despite the belief that cannibalism stabilizes or destabilizes predator–prey models, its effects on prey populations are not well-understood. In this study, we propose a discrete-time prey–predator model to examine the presence and local stability of biologically possible equilibria. We employ the center manifold theorem and normal theory to investigate the various types of bifurcations that arise in the system. The findings of our study reveal that the model exhibits transcritical bifurcation at its trivial equilibrium. In addition, the discrete-time predator–prey system demonstrates period-doubling bifurcation in the vicinity of both its boundary equilibrium and interior equilibrium. Furthermore, we analyze the existence of Neimark–Sacker bifurcation around the interior equilibrium point. We demonstrate that cannibalism in the prey population can lead to periodic outbreaks, but these outbreaks are limited to the prey population and do not affect predation. In order to regulate the periodic oscillations and other bifurcating and fluctuating behaviors of the system, various chaos control strategies are executed. Additionally, extensive numerical simulations are carried out to validate and substantiate the analytical findings. We utilized the software Mathematica 12.3, which is an efficient and effective computing tool that enables symbolic and numerical computations to carry out numerical simulations.