Stability, bifurcation, and control: Modeling interaction of the predator-prey system with Alles effect

In this study, we investigate the resilience and bifurcation properties of a discrete-time predator-prey model incorporating an Allee effect, the transfer of prey to predator energy, and a necessary threshold for prey consumption prior to predator reproduction. The discrete-time formulation involves iteratively adjusting the predator and prey populations at distinct time steps. The model encompasses equations that depict the fluctuations in population sizes, considering processes like breeding, predation, death, and the influence of the Allee effects. The investigation results indicate that the system's boundary equilibrium goes through a period-doubling bifurcation. Moreover, when the system is in close proximity to the exclusive positive equilibrium influenced by the prey-predator population, it exhibits Neimark-Sacker bifurcations. It's worth noting that our observations suggest that the predator population might approach extinction or achieve a stable equilibrium when chaotic dynamics appear in the prey population. The validity of this phenomenon is confirmed using the Lyapunov exponents calculation, which verifies the range of diverse dynamic behaviors. The outcomes and analysis of this research offer intriguing insights in both mathematical and biological realms. Implementing methods for chaos control could be beneficial in stabilizing or mitigating the occurrence of chaotic dynamics. Additionally, the theoretical results are supported by numerical simulations, phase portraits, bifurcation diagrams, graphs depicting chaos control, and maximum Lyapunov exponent graphs.