Long-Lasting and Slowly Varying Transient Dynamics in Discrete-Time Systems

AnthonyPasion

Queen's University

"Long-Lasting and Slowly Varying Transient Dynamics in Discrete-Time Systems"

Mathematical models of ecological and epidemiological systems often focus on asymptotic dynamics, such as equilibria and periodic orbits. However, many systems exhibit long transient behaviors where certain variables of interest remain in a slowly evolving state for an extended period before undergoing rapid change. These transient dynamics can have significant implications for population persistence, disease outbreaks, and ecosystem stability. In this work, we analyze long-lasting and slowly varying transient dynamics in discrete-time systems. We extend previous theoretical frameworks by identifying conditions under which an observable of the system can exhibit prolonged transients and derive criteria for characterizing these dynamics. Our results show that specific points in the state space, analogous to transient centers in continuous-time systems, can generate and sustain long transients in discrete-time models. We further demonstrate how these properties manifest in predator-prey models and epidemiological systems, particularly in contexts where populations or disease prevalence remain low for an extended period before experiencing a sudden shift. These findings provide a foundation for understanding and predicting long transients in discrete-time ecological and epidemiological models. (Joint Work with FMG Magpantay)
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