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

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 noise strength and the ratio of environmental-to-population timescales. This framework provides a theoretical basis for understanding irreversible transitions in invasive species, plant succession, microbial dynamics, and the elimination of cancerous tumors.