Dynamical and phenomenological bifurcations induced by multiplicative noise in a stochastic single-species model with double Allee effect

Liang Hong; Yanhua Yang; Liang Hong; Yanhua Yang

doi:10.3934/math.2025887

AIMS Mathematics

2025, Volume 10, Issue 8: 19878-19895. doi: 10.3934/math.2025887

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Research article

Dynamical and phenomenological bifurcations induced by multiplicative noise in a stochastic single-species model with double Allee effect

Liang Hong ¹,
Yanhua Yang ^{2
,
,}

1.
College of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China
2.
Department of Information Engineering, Xinyang Agriculture and Forestry University, Xinyang 464000, China

Received: 08 July 2025 Revised: 16 August 2025 Accepted: 22 August 2025 Published: 28 August 2025
MSC : 34F05, 37H10, 60J70, 92B05

The interplay between system nonlinearity and environmental noise can lead to counterintuitive phenomena that deterministic models are unable to explain. However, the underlying mechanisms are not fully understood. In this study, we proposed a stochastic single-species model driven by multiplicative noise that incorporates a double Allee effect to investigate such counter-intuitive phenomena, specifically dynamical and phenomenological bifurcations induced by noise. Our results underscore the critical role of parameter $ \pi $, which governs both the persistence and extinction of the species and determines the dynamical and phenomenological bifurcations in the system. Furthermore, our study offers significant biological insights, including: (i) Noise-induced state transitions that alter population density distributions and elevate extinction risk; and (ii) variations in the intensity of the Allee effect, which result in four distinct modifications in the shape of the density function and accelerate the extinction process for endangered species. These findings may have important implications for the management of ecological resources.
- dynamical and phenomenological bifurcations,
- stochastic single-species model,
- double Allee effect,
- stationary probability density
Citation: Liang Hong, Yanhua Yang. Dynamical and phenomenological bifurcations induced by multiplicative noise in a stochastic single-species model with double Allee effect[J]. AIMS Mathematics, 2025, 10(8): 19878-19895. doi: 10.3934/math.2025887

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Abstract

The interplay between system nonlinearity and environmental noise can lead to counterintuitive phenomena that deterministic models are unable to explain. However, the underlying mechanisms are not fully understood. In this study, we proposed a stochastic single-species model driven by multiplicative noise that incorporates a double Allee effect to investigate such counter-intuitive phenomena, specifically dynamical and phenomenological bifurcations induced by noise. Our results underscore the critical role of parameter $ \pi $, which governs both the persistence and extinction of the species and determines the dynamical and phenomenological bifurcations in the system. Furthermore, our study offers significant biological insights, including: (i) Noise-induced state transitions that alter population density distributions and elevate extinction risk; and (ii) variations in the intensity of the Allee effect, which result in four distinct modifications in the shape of the density function and accelerate the extinction process for endangered species. These findings may have important implications for the management of ecological resources.

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