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A Note on the Stationary Probability Density Function and Covariance Matrix of a Stochastic Chemostat Model with Distributed Delay

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Abstract

In this paper, we will focus on the stochastic chemostat model with distributed delay and Monod-type nutrient absorption function in Stratonovich form. We study some probabilistic properties of this stochastic chemostat model. By solving the corresponding Fokker-Planck equation of this model, we obtain the expression of joint density function near the positive equilibrium point of the deterministic chemostat system, and we also obtain the covariance matrix of the random variables.

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References

  1. Smith, H., Waltman, P.: The theory of the chemostat: dynamics of microbial competition. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  2. Butler, G., Wolkowicz, G.: A mathematical model of the chemostat with a general class of functions describing nutrient uptake. SIAM J. Appl. Math. 45, 138–151 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Wolkowicz, G., Lu, Z.: Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates. SIAM J. Appl. Math. 52, 222–233 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Li, B.: Global asymptotic behavior of the chemostat: general response functions and different removal rates. SIAM J. Appl. Math. 59, 411–422 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Droop, M.: Vitamin \({{\rm B}} _{12}\) and marine ecology. IV. The kinetics of uptake, growth, and inhibition in Monochrysis lutheri. J. Mar. Biol. Assoc. U.K. 48, 689–733 (1968)

    Article  Google Scholar 

  6. Finn, R.K., Wilson, R.E.: Population dynamics of a continuous propagator for micro-organisms. J. Agric. Food Chem. 2, 66–69 (1953)

    Article  Google Scholar 

  7. Bush, A., Cook, A.: The effect of time delay and growth inhibition in the bacterial treatment of wastewater. J. Theor. Biol. 63, 385–395 (1975)

    Article  Google Scholar 

  8. Caperon, J.: Time lag in population growth response of isochrysis galbana to a variable nitrate environment. Ecology 50, 188–192 (1969)

    Article  Google Scholar 

  9. Han, B., Zhou, B., Jiang, D., Hayat, T., Alsaedi, A.: Stationary solution, extinction and density function for a high-dimensional stochastic SEI epidemic model with general distributed delay. Appl. Math. Comput. 405, 126236 (2021)

    MathSciNet  MATH  Google Scholar 

  10. Zhang, X., Yuan, R.: The existence of stationary distribution of a stochastic delayed chemostat model. Appl. Math. Lett. 93, 15–21 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhang, X., Yuan, R.: Stochastic properties of solution for a chemostat model with a distributed delay and random disturbance. Int. J. Biomath. 13, 2050066 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhang, X., Yuan, R.: Forward attractor for stochastic chemostat model with multiplicative noise. Chaos Solitons Fractals 153, 111585 (2021)

    Article  MathSciNet  Google Scholar 

  13. Zhang, X., Yuan, R.: Pullback attractor for random chemostat model driven by colored noise. Appl. Math. Lett. 112, 106833 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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 

  15. Liu, R., Ma, W., Guo, K.: The general chemostat model with multiple nutrients and flocculating agent: from deterministic behavior to stochastic forcing transition. Commun. Nonlinear Sci. Numer. Simul. 117, 106910 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yan, R., Sun, S.: Stochastic characteristics of a chemostat model with variable yield, physica A. Stat. Mech. Appl. 537, 122681 (2020)

    Article  MATH  Google Scholar 

  17. Wang, L., Jiang, D., Wolkowicz, G.S.K.: Global asymptotic behavior of a multi-species stochastic chemostat model with discrete delays. J. Dyn. Differ. Equ. 32(2), 849–872 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhang, X., Yuan, R.: Stochastic properties of solution for a chemostat model with a distributed delay and random disturbance. Int. J. Biomath. 13, 2050066 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Macdonald, N.: Time Lags in Biological Models, Lecture Notes in Biomathematics. Springer-Verlag, Berlin-New York (1978)

    Book  MATH  Google Scholar 

  20. Liu, Q., Jiang, D., Hayat, T., Alsaedi, A.: Long-time behaviour of a stochastic chemostat model with distributed delay. Stochastics 91(8), 1141–1163 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wolkowicz, G., Xia, H., Ruan, S.: Competition in the chemostat: a distributed delay model and its global asymptotic behavior. SIAM J. Appl. Math. 57, 1281–1310 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Arnold, L.: Random Dynamical Systems. Springer, Berlin Heidelberg (1998)

    Book  MATH  Google Scholar 

  23. Xu, C.: Phenomenological bifurcation in a stochastic logistic model with correlated colored noises. Appl. Math. Lett. 101, 106064 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ji, W., Zhang, Y., Liu, M.: Dynamical bifurcation and explicit stationary density of a stochastic population model with Allee effects. Appl. Math. Lett. 111, 106662 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, X., Yuan, R.: Stochastic bifurcation and density function analysis of a stochastic logistic equation with distributed delay and strong kernel. Int. J. Biomath. 16, 2250085 (2023)

    Article  MathSciNet  Google Scholar 

  26. Zhang, X., Yuan, R.: Stochastic bifurcation and density function analysis of a stochastic logistic equation with distributed delay and weak kernel. Math. Comput. Simul. 195, 56–70 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  27. Chen, Q.: A new idea on density function and covariance matrix analysis of a stochastic SEIS epidemic model with degenerate diffusion. Appl. Math. Lett. 103, 106200 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, S.: Solutions of Fokker-Planck equation with applications in nonlinear random vibration. Bell Lab. Tech. J. 48, 2031–2051 (1969)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 12171039) and the Fundamental Research Funds for the Central Universities (No. 2021NTST03).

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Correspondence to Xiaofeng Zhang.

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Zhang, X. A Note on the Stationary Probability Density Function and Covariance Matrix of a Stochastic Chemostat Model with Distributed Delay. Qual. Theory Dyn. Syst. 22, 114 (2023). https://doi.org/10.1007/s12346-023-00816-w

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