Long-lasting and slowly varying transient dynamics in discrete-time systems

Analysis of mathematical models in ecology and epidemiology often focuses on asymptotic dynamics, such as stable equilibria and periodic orbits. However, many systems exhibit long transient behaviours where certain aspects of the dynamics remain in a slowly evolving state for an extended period before undergoing rapid change. In this work, we analyse long-lasting and slowly varying transient dynamics in discrete-time systems based on extensions of previous theoretical frameworks developed for continuous-time systems. This involves clarifying the conditions under which we say an observable of the system exhibits prolonged transients, and deriving criteria for characterizing these dynamics. Our results show that specific points in the state space, analogous to previously defined transient centres in continuous-time systems, can generate and sustain long transients in discrete-time models. We demonstrate how these properties manifest in predator–prey models and epidemiological systems, particularly when populations or disease prevalence remain low for extended intervals before sudden shifts.