Qualitative dynamics of a rational logistic-type population map with density-dependent saturation

This paper studies a discrete-time population model governed by a nonlinear rational recurrence incorporating logistic inhibition and density-dependent saturation. The main dynamical properties of the model are analyzed via equilibria, local stability, invariant intervals, periodic solutions, and bifurcation phenomena. Sufficient conditions for extinction and persistence are derived, and the corresponding asymptotic behavior is characterized.