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Threshold dynamics of a chemostat model with Ornstein–Uhlenbeck process and single-species growth on two nutrients

Author

Listed:
  • Gao, Miaomiao
  • Jiang, Yanhui
  • Jiang, Daqing

Abstract

This paper aims to study the dynamic properties of a stochastic chemostat model with single-species growth on two perfectly substitutable nutrients, in which the maximal growth rates of the microorganisms satisfy the log-normal Ornstein–Uhlenbeck process. We first show the existence and uniqueness of the global positive solution. Then, sufficient condition for the extinction of the microorganisms is given. And by constructing suitable Lyapunov functions, we obtain the condition for the existence of stationary distribution, which implies the persistence of the microorganisms. We find that there is a threshold between persistence and extinction for the considered model. Moreover, the exact expression of the probability density function of the distribution is further derived. Finally, numerical simulations are provided to demonstrate the theoretical results.

Suggested Citation

  • Gao, Miaomiao & Jiang, Yanhui & Jiang, Daqing, 2026. "Threshold dynamics of a chemostat model with Ornstein–Uhlenbeck process and single-species growth on two nutrients," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 239(C), pages 917-949.
    • Handle: RePEc:eee:matcom:v:239:y:2026:i:c:p:917-949
    DOI: 10.1016/j.matcom.2025.07.063
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    References listed on IDEAS

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    1. Zhang, Xiaofeng & Yuan, Rong, 2021. "A stochastic chemostat model with mean-reverting Ornstein-Uhlenbeck process and Monod-Haldane response function," Applied Mathematics and Computation, Elsevier, vol. 394(C).
    2. Gao, Miaomiao & Jiang, Daqing & Ding, Jieyu, 2023. "Dynamical behavior of a nutrient–plankton model with Ornstein–Uhlenbeck process and nutrient recycling," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    3. Jian Zhang & Anqi Miao & Tongqian Zhang, 2017. "Threshold Dynamics of a Stochastic Chemostat Model with Two Nutrients and One Microorganism," Mathematical Problems in Engineering, Hindawi, vol. 2017, pages 1-11, September.
    4. Wang, Weiming & Cai, Yongli & Ding, Zuqin & Gui, Zhanji, 2018. "A stochastic differential equation SIS epidemic model incorporating Ornstein–Uhlenbeck process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 921-936.
    5. Jiao, Jianjun & Ye, Kaili & Chen, Lansun, 2011. "Dynamical analysis of a five-dimensioned chemostat model with impulsive diffusion and pulse input environmental toxicant," Chaos, Solitons & Fractals, Elsevier, vol. 44(1), pages 17-27.
    6. David Adam, 2020. "A guide to R — the pandemic’s misunderstood metric," Nature, Nature, vol. 583(7816), pages 346-348, July.
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