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Survival and ergodicity of a stochastic microorganism flocculation model with nonlinear response functionals

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Abstract

This paper aims to study the long-time dynamic of a stochastic microorganism flocculation model with Monod response functionals. By introducing the flocculant into the classical chemostat, the studied model is extended to a stochastic one with nonlinear reaction terms. For the corresponding deterministic model, several sufficient conditions are given for the stability of equilibrium points. Considering the influence of noises, a unique threshold for determining persistence or not of the microorganism is derived firstly; and then influences of the flocculation on the output of chemostat is investigated. Results show that the input concentration of flocculants has significant influences on the dynamic and output of the chemostat. Finally, theoretical conclusions are illustrated by numerical method for wastewater treatment.

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References

  1. Monod J, La technique de culture continue thorie et applications. 1978

  2. Novick, A., Szilard, L.: Description of the chemostat. Science 112(2920), 715–716 (1950)

    Article  Google Scholar 

  3. Mohammed, A.: The Numerical simulation of the rivalry between aerobic and anaerobic bacteria species in a chemostat model. Journal of Physics: Conference Series, 1897(1) (2021)

  4. Ireri, J., Pokhariyal, G., Moindi, S.: Chemostat model with periodic nutrient input described by Fourier series. Asian J. Math. 16(8), 16–27 (2020)

    Article  Google Scholar 

  5. Lai, C., Dong, Q., et al.: Role of extracellular polymeric substances in a methane based membrane biofilm reactor reducing vanadate. Environ. Sci. Technol. 52(18), 10680–10688 (2018)

    Article  Google Scholar 

  6. Yang, J., Wu, D., Li, A., et al.: The addition of N-Hexanoyl-Homoserine Lactone to improve the microbial flocculant production of Agrobacterium tumefaciens strain F2, an Exopolysaccharide Bioflocculant-Producing Bacterium. Appl. Biochem. Biotechnol. 179(5), 728–739 (2016)

    Article  Google Scholar 

  7. Salehizadeh, H.: Extracellular biopolymeric flocculants-recent trends and biotechnological importance. Biotechnol. Adv. 19(5), 371–385 (2001)

    Article  Google Scholar 

  8. Tang, X., Wang, T., Shang, S., et al.: Enhanced performance of a novel flocculant containing rich fluorine groups in refractory dyeing wastewater treatment Removal mechanisms. Separ. Purif. Technol. 263, 118411 (2021)

    Article  Google Scholar 

  9. Zhang, X., Yuan, R.: A stochastic chemostat model with mean-reverting Ornstein-Uhlenbeck process and Monod-Haldane response function. Appl. Math. Comput. 394, 125833 (2021)

    MathSciNet  MATH  Google Scholar 

  10. Du, N.H., Nhu, N.N.: Permanence and extinction for the stochastic SIR epidemic model. J. Diff. Eq. 269(11), 9619–9652 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jin, M., Lin, Y.: Classification of asymptotic behavior in a stochastic SEIR epidemic model. Appl. Math. Lett. 118, 107184 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhang, L., Wang, Z., Zhao, X.: Threshold dynamics of a time periodic reaction-diffusion epidemic model with latent period. J. Diff. Eq. 258(9), 3011–3036 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhang, T., Ma, W., Meng, X.: Dynamical analysis of a continuous-culture and harvest chemostat model with impulsive effect. J. Biol. Syst. 23(04), 555–575 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ruan, S., Wang, W.: Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Diff. Eq. 188(1), 135–163 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huo, L., Jiang, J.: Dynamical behavior of a rumor transmission model with Holling-type II functional response in emergency event. Phys., A. Stat. Mech. Appl. 450, 228–240 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yang, Q., Jiang, D., Shi, N., et al.: The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence. J. Math. Anal. Appl. 388(1), 248–271 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ruan, S., Wang, W.: Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Diff. Eq. 188(1), 135–163 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Du, N.H., Thanh, D.N., Ngoc, N.N.: Conditions for permanence and ergodicity of certain SIR epidemic models. Acta Applicandae Mathematicae 160(1), 81–89 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Du, N.H., Nhu, N.N.: Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises. Appl. Math. Lett. 64, 223–230 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, R., Ma, W.: Noise-induced stochastic transition: a stochastic chemostat model with two complementary nutrients and flocculation effect. Chaos, Solitons Fract. 147, 110951 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Campillo, F., Joannides, M., Larramendy-Valverde, I.: Stochastic modeling of the chemostat. Ecol. Modell. 222(15), 2676–2689 (2011)

  22. Xu, C., Yuan, S., Zhang, T.: Competitive exclusion in a general multi-species chemostat model with stochastic perturbations. Bull. Math. Biol. 83(1), 4 (2021). https://doi.org/10.1007/s11538-020-00843-7

    Article  MathSciNet  MATH  Google Scholar 

  23. Gao, M., Jiang, D., Hayat, T., et al.: Stationary distribution and extinction for a food chain chemostat model with random perturbation. Math. Methods Appl. Sci. 44(1), 1013–1028 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Imhof, L., Walcher, S.: Exclusion and persistence in deterministic and stochastic chemostat models. J. Diff. Eq. 217(1), 26–53 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Maia, M.: An introduction to mathematical epidemiology. Springer, Boston (2015)

    MATH  Google Scholar 

  26. Lahrouz, A., Settati, A., Akharif, A.: Effects of stochastic perturbation on the SIS epidemic system. J. Math. Biol. 74(1–2), 469–498 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Khasminskii, R.: Stochastic stability of differential equations, 2nd edn. Springer-Berlin Heidelberg, Berlin (2012)

    Book  MATH  Google Scholar 

  28. Nguyen, D.H., Nguyen, N.N., Yin, G.: General nonlinear stochastic systems motivated by chemostat models: complete characterization of long-time behavior, optimal controls, and applications to wastewater treatment. Stoch. Process. Appl. 130(8), 4608–4642 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mao, X.: Stochastic differential equations and applications, 2nd edn. Academic Press, Cambridge (2006)

    Google Scholar 

  30. Zhu, C., Yin, G.: On strong Feller, recurrence, and weak stabilization of regime-switching diffusions. Siam J. Control Optim. 48(3), 2003–2031 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhao, D., Yuan, S.: Stochastic dynamics of the delayed chemostat with Lévy noises. Int. J. Biomath. 12(5), 1950056 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Stettner, L.: On the existence and uniqueness of invariant measure for continuous time Markov processes, Technical Report, LCDS 86-18, Brown University, Providence, RI, April (1986)

  33. Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes III: Foster. Lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 25, 518–548 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  34. Chang, H., Parulekar, S.J., Ahmed, M.: A dual-growth kinetic model for biological wastewater reactors. Biotechnol. Progr. 21(2), 423–431 (2005)

    Article  Google Scholar 

  35. Ahmed, M.E., Abusam, A., Mydlarczyk, A.: Kinetic modeling of GAC-IFAS chemostat for petrochemical wastewater treatment. J. Water Resour. Hydraulic Eng. 6(2), 27–33 (2017)

    Article  Google Scholar 

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Acknowledgements

This work was supported by NSFC(No.12071293, 11671260) and Shanghai Leading Academic Discipline Project (No. XTKX2012).

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The authors declare that they do not have any financial interests which may be considered as potential competing interests.

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Authors and Affiliations

  1. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Qiuya Li & Dianli Zhao

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  1. Qiuya Li
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  2. Dianli Zhao
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Contributions

QL: Methodology, Software, Writing–original draft. DZ: Conceptualization, Supervision, Writing Creview & editing.

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Correspondence to Dianli Zhao.

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Li, Q., Zhao, D. Survival and ergodicity of a stochastic microorganism flocculation model with nonlinear response functionals. Nonlinear Dyn 111, 2663–2680 (2023). https://doi.org/10.1007/s11071-022-07933-2

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  • DOI: https://doi.org/10.1007/s11071-022-07933-2

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