Bifurcation and Chaos in a Nonlinear Population Model with Delayed Nonlinear Interactions

This paper investigates the dynamic behavior of a second-order nonlinear rational difference equation modeling a population system with nonlinear interactions between current and previous population states. We derive analytical conditions for the stability of fixed points, explore codim-1 bifurcations, and compute the associated topological normal forms. The analysis also establishes the existence of period-2 solutions and reveals the potential for chaotic dynamics within specific parameter ranges. To validate the theoretical findings, we conduct numerical simulations and bifurcation analysis using the MATLAB package MatContM (version 5p4). Chaotic behavior is further confirmed through the computation of the largest Lyapunov exponent. The results offer new insights into the complex dynamics of delayed population models with nonlinear feedback, extending classical models and suggesting potential applications in stochastic systems and epidemiological modeling.