Dynamics of Interaction between Two Different Populations on Two Connected Islands

This study investigates a mathematical mapping's dynamics in two dimensions, focusing on periodic oscillations in population numbers. It identifies periodic points with prime period four, requiring solving a challenging polynomial equation. The study rigorously establishes solution uniqueness and confirms the absence of certain complex roots. Numerical experiments approximate fixed points and explore periodic orbit behavior, while systematic approaches reveal attractor properties and spectral characteristics. Overall, the study enhances understanding of complex dynamical systems and their interplay with parameters, periodicity, and chaos.