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

Author

Listed:
  • Almaz T. Abebe

    (Department of Mathematics, College of Art and Science, Howard University, Washington, DC 20059, USA)

  • Shenglan Yuan

    (Department of Mathematics, School of Sciences, Great Bay University, Dongguan 523000, China)

  • Daniel Tesfay

    (Department of Mathematics, Mekelle University, Mekelle P.O. Box 231, Ethiopia)

  • James Brannan

    (Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA)

Abstract

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.

Suggested Citation

  • Almaz T. Abebe & Shenglan Yuan & Daniel Tesfay & James Brannan, 2024. "Most Probable Dynamics of the Single-Species with Allee Effect under Jump-Diffusion Noise," Mathematics, MDPI, vol. 12(9), pages 1-18, April.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:9:p:1377-:d:1386899
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    References listed on IDEAS

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    1. Gao, Ting & Duan, Jinqiao & Li, Xiaofan, 2016. "Fokker–Planck equations for stochastic dynamical systems with symmetric Lévy motions," Applied Mathematics and Computation, Elsevier, vol. 278(C), pages 1-20.
    2. Liu, Meng & Bai, Chuanzhi, 2015. "A remark on a stochastic logistic model with Lévy jumps," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 521-526.
    3. Mao, Xuerong & Marion, Glenn & Renshaw, Eric, 2002. "Environmental Brownian noise suppresses explosions in population dynamics," Stochastic Processes and their Applications, Elsevier, vol. 97(1), pages 95-110, January.
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