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Dynamical Behavior of a Stochastic Microorganism Flocculation Model with Nonlinear Perturbation

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Abstract

Due to many uncertain factors, the microorganism flocculation models could be affected by environmental noise. The paper aims to discuss the dynamical behavior of a stochastic microorganism flocculation model, including the extinction and the persistence. Moreover, the expression of density function near the positive equilibrium point is explicitly obtained. Our results indicate that a larger white noise can accelerate the extinction of microorganism, while a weaker white noise can guarantee the existence of stationary distribution. In addition, our theory is confirmed by some numerical examples.

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References

  1. Tai, X., Ma, W., Guo, S., et al.: A class of dynamic delayed model describing flocculation of micoorganism and its theoretical analysis. Math. Pract. Theory 45(13), 198–209 (2015). (in Chinese)

    MathSciNet  Google Scholar 

  2. Guo, S., Ma, W.: Global dynamics of a microorganism flocculation model with time delay. Commun. Pure Appl. Anal. 16(5), 1883–1891 (2017)

    Article  MathSciNet  Google Scholar 

  3. Guo, S., Ma, W., Zhao, X.: Global dynamics of a time-delayed microorganism flocculation model with saturated functional responses. J. Dyn. Differ. Equ. 30(5), 1247–1271 (2018)

    Article  MathSciNet  Google Scholar 

  4. Aldajani, M., Alipoormazandarani, N., Kong, F., et al.: Acid hydrolysis of kraft lignin-acrylamide polymer to improve its flocculation affinity. Sep. Purif. Technol. 258, 117964 (2021)

    Article  Google Scholar 

  5. Zhang, T., Ma, W., Meng, X.: Dynamical analysis of a continuous-culture and harvest chemostat model with impulsive effect. J. Biol. Syst. 23(04), 715–716 (2015)

    Article  MathSciNet  Google Scholar 

  6. Zhang, H., Zhang, T.: Asymptotic behavior of a stochastic microorganism flocculation model with time delay. Appl. Math. Lett. 121(1), 107384 (2021)

    Article  MathSciNet  Google Scholar 

  7. Guo, S., Cui, J., Ma, W.: An analysis approach to permanence of a delay differential equations model of microorganism flocculation. Discrete Contin. Dyn. Syst. - Ser. B (2021). https://doi.org/10.3934/dcdsb.2021208

    Article  Google Scholar 

  8. Zhao, Z., Zhang, X., Chen, L.: Nonlinear modelling of chemostat model with time delay and impulsive effect. Nonlinear Dyn. 63(1), 95–104 (2011)

    Article  MathSciNet  Google Scholar 

  9. Zhang, H., Zhang, T.: The stationary distribution of a microorganism flocculation model with stochastic perturbation. Appl. Math. Lett. (2020). https://doi.org/10.1016/j.aml.2020.106217

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu, Q., Jiang, D., Hayat, T., et al.: Periodic solution and stationary distribution of stochastic SIR epidemic models with higher order perturbation. Phys. A 482, 209–217 (2017)

    Article  MathSciNet  Google Scholar 

  11. Liu, Q., Jiang, D.: Stationary distribution and extinction of a stochastic SIR model with nonlinear perturbation. Appl. Math. Lett. 73, 8–15 (2017)

    Article  MathSciNet  Google Scholar 

  12. Han, B., Jiang, D., Hayat, T., et al.: Stationary distribution and extinction of a stochastic staged progression AIDS model with staged treatment and second-order perturbation. Chaos Solitons Fractals (2020). https://doi.org/10.1016/j.chaos.2020.110238

    Article  MathSciNet  Google Scholar 

  13. Ikeda, N., Watanabe, S.: A comparison theorem for solutions of stochastic differential equations and applications. J. Math. 14, 619–633 (1977)

    MathSciNet  MATH  Google Scholar 

  14. Zhou, B., Jiang, D., Dai, Y., et al.: Stationary distribution and probability density function of a stochastic SVIS epidemic model with standard incidence and vaccination strategies. Chaos Solitons Fractals (2021). https://doi.org/10.1016/j.chaos.2020.110601

    Article  MathSciNet  Google Scholar 

  15. Mao, X.: Stochastic Differential Equations and Applications. Horwood Publishing, second edition, (1997)

  16. Bao, K., Rong, L., Zhang, Q.: Analysis of a stochastic SIRS model with interval parameters. Discrete Contin. Dyn. Syst. Ser. B 24(9), 4827–4849 (2019)

    MathSciNet  MATH  Google Scholar 

  17. Sadovsky, M., Senashova, M.: Model of prey-predator dynamics with reflexive spatial behaviour of species based on optimal migration. Bull. Math. Biol. 78(4), 736–753 (2016)

    Article  MathSciNet  Google Scholar 

  18. Has’miniskii, R.: Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, (1980)

  19. Rudnicki, R.: Asymptotic properties of the Fokker-Planck equation, pp. 517–521. Springer, Berlin, Heidelberg (1995)

    MATH  Google Scholar 

  20. Higham, D.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 433, 525–546 (2001)

    Article  MathSciNet  Google Scholar 

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Funding

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

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Authors and Affiliations

  1. Key Laboratory of Unconventional Oil and Gas Development, School of Petroleum Engineering, 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

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

    Daqing Jiang & Ahmed Alsaedi

Authors
  1. Xiaojie Mu
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  2. Daqing Jiang
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  3. Ahmed Alsaedi
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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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We declared that we have no conflict of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

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Mu, X., Jiang, D. & Alsaedi, A. Dynamical Behavior of a Stochastic Microorganism Flocculation Model with Nonlinear Perturbation. Qual. Theory Dyn. Syst. 21, 42 (2022). https://doi.org/10.1007/s12346-022-00566-1

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