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

• Xiaofeng Zhang ^[1]
  1. [1] Beijing Normal University

     Beijing Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 3, 2023
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • 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.

• Referencias bibliográficas
  • 1. Smith, H., Waltman, P.: The theory of the chemostat: dynamics of microbial competition. Cambridge University Press, Cambridge (1995)
  • 2. Butler, G., Wolkowicz, G.: A mathematical model of the chemostat with a general class of functions describing nutrient uptake. SIAM J....
  • 3. Wolkowicz, G., Lu, Z.: Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential...
  • 4. Li, B.: Global asymptotic behavior of the chemostat: general response functions and different removal rates. SIAM J. Appl. Math. 59, 411–422...
  • 5. Droop, M.: Vitamin B12 and marine ecology. IV. The kinetics of uptake, growth, and inhibition in Monochrysis lutheri. J. Mar. Biol. Assoc....
  • 6. Finn, R.K., Wilson, R.E.: Population dynamics of a continuous propagator for micro-organisms. J. Agric. Food Chem. 2, 66–69 (1953)
  • 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...
  • 8. Caperon, J.: Time lag in population growth response of isochrysis galbana to a variable nitrate environment. Ecology 50, 188–192 (1969)
  • 9. Han, B., Zhou, B., Jiang, D., Hayat, T., Alsaedi, A.: Stationary solution, extinction and density function for a high-dimensional stochastic...
  • 10. Zhang, X., Yuan, R.: The existence of stationary distribution of a stochastic delayed chemostat model. Appl. Math. Lett. 93, 15–21 (2019)
  • 11. Zhang, X., Yuan, R.: Stochastic properties of solution for a chemostat model with a distributed delay and random disturbance. Int. J....
  • 12. Zhang, X., Yuan, R.: Forward attractor for stochastic chemostat model with multiplicative noise. Chaos Solitons Fractals 153, 111585 (2021)
  • 13. Zhang, X., Yuan, R.: Pullback attractor for random chemostat model driven by colored noise. Appl. Math. Lett. 112, 106833 (2021)
  • 14. Zhang, X., Yuan, R.: A stochastic chemostat model with mean-reverting Ornstein-Uhlenbeck process and Monod-Haldane response function....
  • 15. Liu, R., Ma, W., Guo, K.: The general chemostat model with multiple nutrients and flocculating agent: from deterministic behavior to stochastic...
  • 16. Yan, R., Sun, S.: Stochastic characteristics of a chemostat model with variable yield, physica A. Stat. Mech. Appl. 537, 122681 (2020)
  • 17. Wang, L., Jiang, D., Wolkowicz, G.S.K.: Global asymptotic behavior of a multi-species stochastic chemostat model with discrete delays....
  • 18. Zhang, X., Yuan, R.: Stochastic properties of solution for a chemostat model with a distributed delay and random disturbance. Int. J....
  • 19. Macdonald, N.: Time Lags in Biological Models, Lecture Notes in Biomathematics. Springer-Verlag, Berlin-New York (1978)
  • 20. Liu, Q., Jiang, D., Hayat, T., Alsaedi, A.: Long-time behaviour of a stochastic chemostat model with distributed delay. Stochastics 91(8),...
  • 21. Wolkowicz, G., Xia, H., Ruan, S.: Competition in the chemostat: a distributed delay model and its global asymptotic behavior. SIAM J....
  • 22. Arnold, L.: Random Dynamical Systems. Springer, Berlin Heidelberg (1998)
  • 23. Xu, C.: Phenomenological bifurcation in a stochastic logistic model with correlated colored noises. Appl. Math. Lett. 101, 106064 (2020)
  • 24. Ji, W., Zhang, Y., Liu, M.: Dynamical bifurcation and explicit stationary density of a stochastic population model with Allee effects....
  • 25. Zhang, X., Yuan, R.: Stochastic bifurcation and density function analysis of a stochastic logistic equation with distributed delay and...
  • 26. Zhang, X., Yuan, R.: Stochastic bifurcation and density function analysis of a stochastic logistic equation with distributed delay and...
  • 27. Chen, Q.: A new idea on density function and covariance matrix analysis of a stochastic SEIS epidemic model with degenerate diffusion....
  • 28. Liu, S.: Solutions of Fokker-Planck equation with applications in nonlinear random vibration. Bell Lab. Tech. J. 48, 2031–2051 (1969)