Bifurcations in a periodic discrete-time environment

This paper investigates local and global bifurcation, as well as continuation properties for discrete-time periodic dynamical models in arbitrary (finite) dimension. Our focus is to provide explicitly verifiable conditions which guarantee or prevent bifurcations of, say ω1ω1-periodic solutions for ω0ω0-periodic difference equations. In doing so, we give concrete branching relations ensuring bifurcations of e.g. fold, transcritical, pitchfork or flip type, including information on the global branches. Beyond that we obtain formulas indicating the local behavior of mean population sizes under parameter variation or bifurcation, and furthermore tackle stability issues. Our results are applied to various real-world population models.Thus, the paper will be useful for a thorough analysis and understanding of general periodic time-discrete models in population dynamics, life sciences and beyond.