Bifurcation Dynamics and Complex Behavior in a Discrete-Time Predator–Prey Model With Cross-Species Interaction Incorporating Holling Type-II Response

We investigate the nonlinear dynamics of a discrete-time predator–prey model governed by a Holling Type-II functional response. Starting from a biologically motivated continuous-time system, we derive its discrete analogue via the explicit Euler method and employ nondimensionalization to reduce the number of parameters. The resulting two-dimensional nonlinear system is analyzed for the existence and local stability of fixed points. Analytical conditions are established for the occurrence of flip (period-doubling) and Neimark–Sacker bifurcations, characterizing the transition from steady states to periodic and quasi-periodic behavior as system parameters vary. Employing center manifold theory and normal form computations, we derive expressions for the first Lyapunov coefficient to determine the direction and stability of bifurcating invariant curves. To suppress chaotic dynamics induced by bifurcations, we implement a hybrid feedback control mechanism and establish sufficient conditions under which the controlled system regains local asymptotic stability. Numerical results, bifurcation diagrams, and phase portraits corroborate the theoretical results. The framework developed herein provides a rigorous foundation for analyzing and stabilizing discrete ecological models with nonlinear interaction terms.