Modeling and dynamical analysis of an ecological population with the Allee effect

This article examines the complex dynamics of predator-prey models, considering the significant Allee effect. The Allee effect and its absence are considered in the stability analysis of the fixed points. Specifically, the dynamics inside the model, as well as the bifurcation between the one and two parameters, were investigated. We observe period-doubling and Neimark-Sacker bifurcations using the attracting bifurcation plots in one- and two-parameter bifurcations. Our study presents new insights into the bifurcation behavior of the model under consideration, highlighting the potential importance of periodicity and two-parameter dynamic plots in understanding parameter dependencies. The bifurcation behavior of both models is examined using bifurcation theory, and numerical examples are provided to validate our theoretical results. We observed that the Allee effect creates complexities in the model through numerical illustrations. By employing Marotto chaos criteria, we track the emergence of chaos within the model. We employed a simple control method to lessen or avoid bifurcation effects. Overall, our study contributes to understanding our ecological model's dynamic behavior and provides different perspectives on its dynamics.