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A stochastic turbidostat model with Ornstein-Uhlenbeck process: dynamics analysis and numerical simulations

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Abstract

Many turbidostat models are affected by environmental noise due to various complicated and uncertain factors, and Ornstein-Uhlenbeck process is a more effective and precise way. We formulate a stochastic turbidostat system incorporating Ornstein-Uhlenbeck process in this paper and develop dynamical behavior for the stochastic model, which includes the existence and uniqueness of globally positive equilibrium, sufficient conditions of the extinction, the existence of a unique stationary distribution and an expression of density function of quasi-stationary distribution around the positive solution of the deterministic model. The results indicate that the weaker volatility intensity can ensure the existence and uniqueness of the stationary distribution, and the stronger reversion speed can lead to the extinction of microorganisms. Numerical simulations verify the validity of the analysis results, which assess the influence of the speed of reversion and the intensity of volatility on the long-term behavior of microorganisms.

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Data availablity statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

The research is supported by the Natural Science Foundation of China (No. 11871473), Shandong Provincial Natural Science Foundation (Nos. ZR2019MA010, ZR2019MA006, ZR2020MA039) and the Fundamental Research Funds for the Central Universities (No. 19CX02055A).

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Authors and Affiliations

  1. School of Petroleum Engineering, Key Laboratory of Unconventional Oil & Gas Development, China University of Petroleum (East China), Ministry of Education, Qingdao, 266580, People’s Republic of China

    Xiaojie Mu & Daqing Jiang

  2. College of Science, China University of Petroleum (East China), Qingdao, 266580, People’s Republic of China

    Daqing Jiang & Yunhui Liao

  3. Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, King Abdulaziz University, Jeddah, 121589, Saudi Arabia

    Daqing Jiang, Tasawar Hayat & Ahmed Alsaedi

  4. Department of Mathematics, Quaid-I-Azam University 45320, Islamabad, 44000, Pakistan

    Tasawar Hayat

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  1. Xiaojie Mu
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  2. Daqing Jiang
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  3. Tasawar Hayat
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  4. Ahmed Alsaedi
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  5. Yunhui Liao
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Contributions

DJ designed the research and methodology. XM wrote the original draft. All authors read and approved the final manuscript.

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Mu, X., Jiang, D., Hayat, T. et al. A stochastic turbidostat model with Ornstein-Uhlenbeck process: dynamics analysis and numerical simulations. Nonlinear Dyn 107, 2805–2817 (2022). https://doi.org/10.1007/s11071-021-07093-9

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  • DOI: https://doi.org/10.1007/s11071-021-07093-9

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