Noise-induced excitability: bloom, bust and extirpation in autotoxic population dynamics

TL;DR

This work tackles finite-time extinction in boom-bust population dynamics driven by slow environmental feedback. By deriving a mesoscopic stochastic model from an agent-based autotoxicity setup via Van Kampen expansion, it yields a two-variable Langevin system with a slow toxin dynamic and demographic noise. The authors uncover a noise-induced excitability regime and classify extinction pathways into short-lived, excitable, and persistent trajectories, providing analytical formulas for extinction probabilities that depend on the diffusion strength $D$ and timescale ratio $\rho$. The framework demonstrates how intrinsic fluctuations shape irreversible transitions to extinction and offers a broadly applicable approach for ecological and microbial systems where toxin–population feedbacks operate on disparate timescales.

Abstract

Species populations often modify their environment as they grow. When environmental feedback operates more slowly than population growth, the system can undergo boom-bust dynamics, where the population overshoots its carrying capacity and subsequently collapses. In extreme cases, this collapse leads to total extinction. While deterministic models typically fail to capture these finite-time extinction events, we propose a stochastic framework, derived from an individual-based model, to describe boom-bust-extirpation dynamics. We identify a noise-driven, threshold-like behavior where, depending on initial conditions, the population either undergoes a "boom" or is extirpated before the expansion occurs. Furthermore, we characterize a transition between an excitable regime, where most trajectories are captured by the absorbing state immediately after the first bust, and a persistent regime, where most populations reach a metastable state. We show that this transition is governed by the diffusion strength and the ratio of environmental-to-population timescales. This framework provides a theoretical basis for understanding irreversible transitions in invasive species, plant succession, and microbial dynamics.

Figures (3)

• Figure 1: Deterministic behavior. (a) Example of a deterministic trajectory exhibiting a boom--bust cycle (highlighted by shaded rectangles) followed by oscillations around the coexistence fixed point (horizontal solid line). The figure shows the quantities plotted in (b) and (c). (b) Minimum population value after the initial boom--bust cycle, which tends to $S_0$ as $\rho$ decreases. For large $\rho$, oscillatory behavior fades out, and the minimum coincides with the stable populated fixed point $S_p$, indicated by a horizontal solid line. (c) Time interval between the first two peaks in population density.
• Figure 2: Stochastic behavior: extinction pathways. Instances of trajectories for fixed $\rho = 0.08$ and $D=0$ (black), $D=0.1$ (blue), $D=0.2$ (orange), and $D=0.4$ (green). Panels (a) and (b) show the components $x$ and $y$, respectively, of the trajectories as functions of time in log-log scale. Panel (c) shows trajectories in the phase space, while zoom in close to the (0,0) is shown in (d) in log-log scale. Different extinction pathways are observed as $D$ changes. For $D=0$ the trajectory follows the deterministic path and no extinction occurs (black line). For small noise ($D=0.1$, blue curve), stochastic trajectories fluctuate about the deterministic path ($D=0$), reaching a meta-stable state and experiencing extinction through a rare fluctuation at very long times. For intermediate noise strength ($D=0.2$, orange curve), trajectories are likely to touch the absorbing state after reaching the macroscopic maximum. Extinction times in this phase will be of the order of the time to reach the first minimum of the deterministic trajectory. For big noise ($D=0.4$, green curve), trajectories are likely to become extinct before reaching the first maximum of the deterministic trajectory. Trajectories were obtained integrating Eq. \ref{['Eq:langevin']} using Milstein algorithm with discretization $\Delta t=10^{-4}$ and initial conditions $x_0=0.1$, and $y_0=0$.
• Figure 3: Extinction probabilities In (a), sketch regions in parameter space where absorption probability is dominated by short-lived trajectories ($p_{sl}$), excited trajectories ($p_e$), and persistent trajectories ($p_p$). Panels (b–d) show, respectively, the probabilities of extinction via short-lived, excited, and persistent trajectories as functions of $\rho$ and $D$. Panels (e) show the numerically computed extinction probabilities and analytical approximation for fixed $D=10^{-3}$ as a function of $\rho$. Panel (f) shows probabilities and approximations for fixed $\rho=10^{-1.5}$ as a function of $D$. Panels (g-i) show the same information as (b-d), but using our approximated analytical probabilities [Eq. \ref{['Eq:Anal_prob']}], instead of Monte Carlo estimators. In all cases we set $x(0)=0.01$ and $y(0)=0$.