Stochastic bifurcation in single-species model induced by α-stable Lévy noise

Tesfay, Almaz; Tesfay, Daniel; Yuan, Shenglan; Brannan, James R.; Duan, Jinqiao

doi:10.1088/1742-5468/ac2255

Cited by 8 publications

(5 citation statements)

References 42 publications

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“…It is a challenge to plot the bifurcation diagram in the (k, λ)-plane of the random slow flow. Luckily, we can explore stochastic bifurcations in dynamical systems driven by Lévy noises with support for statistical modeling and computation [18,19,38].…”

Section: Conclusion and Future Challengesmentioning

confidence: 99%

Slow Manifolds for Stochastic Koper Models with Stable Lévy Noises

Zulfiqar

Yuan

Saleem

2023

Axioms

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The Koper model is a vector field in which the differential equations describe the electrochemical oscillations appearing in diffusion processes. This work focuses on the understanding of the slow dynamics of a stochastic Koper model perturbed by stable Lévy noise. We establish the slow manifold for a stochastic Koper model with stable Lévy noise and verify exponential tracking properties. We also present two practical examples to demonstrate the analytical results with numerical simulations.

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Section: Conclusion and Future Challengesmentioning

confidence: 99%

Slow Manifolds for Stochastic Koper Models with Stable Lévy Noises

Zulfiqar

Yuan

Saleem

2023

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“…These sudden environmental perturbations may bring substantial social and economic losses. Stochastic single-species models perturbed by Brownian motion have been extensively researched by many scholars [21][22][23][24][25][26]. However, stochastic extension of population process driven by Gaussian noise cannot explain the aforementioned random and intermittent environmental perturbations.…”

Section: Introductionmentioning

confidence: 99%

Most Probable Dynamics of the Single-Species with Allee Effect under Jump-Diffusion Noise

Abebe,

Yuan,

Tesfay

et al. 2024

Mathematics

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We explore the most probable phase portrait (MPPP) of a stochastic single-species model incorporating the Allee effect by utilizing the nonlocal Fokker–Planck equation (FPE). This stochastic model incorporates both non-Gaussian and Gaussian noise sources. It has three fixed points in the deterministic case. One is the unstable state, which lies between the two stable equilibria. Our primary focus is on elucidating the transition pathways from extinction to the upper stable state in this single-species model, particularly under the influence of jump-diffusion noise. This helps us to study the biological behavior of species. The identification of the most probable path relies on solving the nonlocal FPE tailored to the population dynamics of the single-species model. This enables us to pinpoint the corresponding maximum possible stable equilibrium state. Additionally, we derive the Onsager–Machlup function for the stochastic model and employ it to determine the corresponding most probable paths. Numerical simulations manifest three key insights: (i) when non-Gaussian noise is present in the system, the peak of the stationary density function aligns with the most probable stable equilibrium state; (ii) if the initial value rises from extinction to the upper stable state, then the most probable trajectory converges towards the maximally probable equilibrium state, situated approximately between 9 and 10; and (iii) the most probable paths exhibit a rapid ascent towards the stable state, then maintain a sustained near-constant level, gradually approaching the upper stable equilibrium as time goes on. These numerical findings pave the way for further experimental investigations aiming to deepen our comprehension of dynamical systems within the context of biological modeling.

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“…These sudden environmental perturbations may bring substantial social and economic losses. Stochastic single-species model perturbed by Brownian motion has been extensively researched by many scholars [17,26,37,41,43]. However, stochastic extension of population process driven by Gaussian noise cannot explain the aforementioned random and intermittent environmental perturbations.…”

Section: Introductionmentioning

confidence: 99%

Most Probable Dynamics of the Single-Species with Allee Effect under Jump-Diffusion Noise

Tesfay,

Yuan,

Tesfay

et al. 2023

Preprint

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We investigate the most probable phase portrait (MPPP) of a stochastic single-species model with the Alleeeffect using the non-local Fokker-Planck equation (FPE). This stochastic model is driven by non-Gaussianas well as Gaussian noise. It has three fixed points in the deterministic case. One of them is the unstablestate which lies between the two stable equilibria. We focus on the transition pathways from the extinctionstate to the upper fixed stable state for the transcription factor activator in a single-species model. This helpsus to study the biological behavior of species. The most probable path is obtained from the solution of thenon-local Fokker-Planck equation corresponding to the population system of the single-species model, andthe corresponding maximum possible stable equilibrium state is determined. We also acquire the Onsager-Machlup function for the stochastic model and solve the corresponding most probable paths. The numericalsimulation manifests that: (i) When non-Gaussian noise is presented in the system, the maximum of thestationary density function is located at the most probable stable equilibrium state; (ii) If the initial valueincreases from extinction state to the upper stable state, the most probable trajectory goes to the maximallikely equilibrium state, in our case it lies between 9 and 10; (iii) The most probable paths increase to stablestate quickly, then maintain a nearly constant level, and approach to the upper stable equilibrium state astime goes on. These numerical experiment findings accelerate growth for further experimental study, inorder to achieve good knowledge about dynamical systems in biology. 2020 MSC: 39A50, 45K05, 65N12

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Stochastic bifurcation in single-species model induced by α-stable Lévy noise

Cited by 8 publications

References 42 publications

Slow Manifolds for Stochastic Koper Models with Stable Lévy Noises

Slow Manifolds for Stochastic Koper Models with Stable Lévy Noises

Most Probable Dynamics of the Single-Species with Allee Effect under Jump-Diffusion Noise

Most Probable Dynamics of the Single-Species with Allee Effect under Jump-Diffusion Noise

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