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

TL;DR

This paper investigates long-lasting, slowly changing transient behaviors in discrete-time ecological and epidemiological models, extending continuous-time theories to better understand prolonged states before rapid transitions.

Contribution

It introduces criteria for identifying and characterizing long transients in discrete-time systems, including conditions for transient centers that sustain these behaviors.

Findings

Long transients can be generated by specific points in the state space.

Prolonged low-level states precede rapid shifts in predator-prey and epidemiological models.

Discrete-time dynamics exhibit similar transient phenomena as continuous-time systems.

Abstract

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 behaviors where certain aspects of the dynamics remain in a slowly evolving state for an extended period before undergoing rapid change. In this work, we analyze 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 centers 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.