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

TL;DR

The paper develops a discrete-time analogue of transient dynamics by defining $v$-transient centers and $(v,s,T)$-transient points for maps $x(t+1)=f(x(t))$ with observable $v$, and proves basic invariant properties and practical criteria to identify these structures. It shows that stable fixed points cannot host long transients, while unstable fixed points with an appropriately aligned observable can act as transient centers, with extensions to nonlinear maps and PF-type systems. The authors illustrate the theory on a discrete predator–prey model and a vaccination-based epidemic model, revealing how near-constant observables persist for long periods before rapid shifts, thereby explaining honeymoon periods and temporary collapses in ecology and disease dynamics. The results provide a rigorous, observable-centered toolkit for predicting long transients in discrete ecological and epidemiological dynamics and point to future work on reachability and non-equilibrium transient centers.

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.

Figures (6)

• Figure 1: Illustration of the behaviour of $\Delta v(x(t))$ starting from a $(v, s, T)$-transient point $\xi \in \mathbb{R}^n$.
• Figure 2: (a) Trajectories of system \ref{['model: example1']} with the dashed vertical line marking $t_*=\min\{t\in \mathbb{Z}^+~:~ x_t \geq 1\}$; and (b) time series of $|\Delta v(x_t, y_t)|$ with the dashed horizontal line at $S_*=\tfrac{1}{2}\,h^2$ and the dash–dot vertical line at the $(v, S_*)$-transient time $T_{S_*}(\xi)$.
• Figure 3: Trajectories of system \ref{['model: example2']} for various initial conditions $\xi = \varepsilon w$ with $a =1.5$ and $b =1.3$. The lowest panel shows the magnitude of the observable increment $|\Delta v(x,y)|$ where $v(x,y) = x+y$.
• Figure 4: (a) Sample trajectories of model system \ref{['model: predator-prey']} subject to several initial value $x_0$. The other parameters are $r = 0.5$, $K = 1.0$, $\alpha = 1.0$, $\gamma = 4.0$, $d= 1.0$, and $y_0 = 10^{-4}$. (b) The augmented phase portrait for the same predator-prey model. The dashed red and blue lines are the prey ($x$) and predator ($y$) nullclines. The solid red and blue curves are their respective next-iterate root curves. The $'+'$ and $'-'$ symbols indicate the sign of the next-iterate operator in various regions, and the black arrows depict the direction field. The definitions of $D$, $N$, $L$, and $J$ are in the proof of Theorem \ref{['thm: pp']}.
• Figure 6: Trajectories of model system \ref{['model: epidemic_simple']} subject to $b = 115$, $p = 0.3 \times 10^{-2}$, $\alpha = 0.4 \times 10^{-4}$, $S(0) =2.4 \times 10^4$ and $I(0) =250$.

Theorems & Definitions (18)

• Definition 2.1: Transient Points
• Definition 2.2: Transient Centers
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Example 2.6
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4