Local stability, bifurcation, multistability, chaos control, and bi-parameter space analysis of a discrete population model

In this paper, we investigate the local dynamics, bifurcation phenomena, multistability, chaotic behavior, and bi-parameter space analysis of a discrete population model. Specifically, we identify three equilibrium points of the model—the trivial equilibrium, the boundary equilibrium (which exists for all parameter values), and the interior equilibrium, which is derived under specific parameter condition. Using stability theory, we analyze the local dynamic behavior around these equilibria and classify their stability properties. Based on these classifications, we determine the one-parameter bifurcation sets and carry out a detailed analysis of the resulting bifurcation phenomena. To address the emergence of complex dynamics, particularly those associated with Neimark–Sacker and flip bifurcations, we explore chaotic behavior and implement hybrid as well as OGY control strategies for chaos suppression. Furthermore, we examine multistability and conduct a comprehensive biparameter space analysis to illustrate transitions between different dynamical regimes. Finally, all theoretical findings are substantiated through numerical simulations, which confirm the validity and accuracy of the analytical results.