Abstract
The global dynamics of a deterministic model in wastewater treatment has been investigated in Zhang (J Math Chem 50:2239–2247, 2012). The stochastic version, which can be used for continuous flow bioreactor and membrane reactor is presented in this study. Precisely, we assume there is some uncertainty in the part describing the recycle, which results in a set of stochastic differential equations with white noise. We first show that the stochastic model has always a unique positive solution. Then long time behavior of the model is studied. Our study shows that both the washout equilibrium and non-washout equilibrium are stochastically stable. At the end, we carry out some numerical simulations, which support our theoretical conclusions well.
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References
Ebeling W., Schimansky-Geier L., Romanovsky Y.M.: Stochastic Dynamics of Reacting Biomolecules. World Scientific, Singapore (2002)
Hu G., Wang K.: Stability in distribution of neutral stochastic functional differential equations with Markovian switching. J. Math. Anal. Appl. 385, 757–769 (2012)
Huang X., Wang Y., Zhu L.: Competition in the bioreactor with general quadratic yields when one competitor produces a toxin. J. Math. Chem. 39, 281–294 (2006)
Huang X., Zhu L.: A note on competition in the bioreactor with toxin. J. Math. Chem. 42, 645–659 (2007)
Imhofa L., Walcherb S.: Exclusion and persistence in deterministic and stochastic chemostat models. J. Differ. Equ. 217, 26–53 (2005)
Jiang D., Ji C.: The long time behavior of DI SIR epidemic model with stochastic perturbation. J. Math. Anal. Appl. 372, 162–180 (2010)
Kas’minskii R.: Stochastic Stability of Differential Equations. Alphen an den Rijn, The Netherlands (1980)
Lemons D.S.: An Introduction to Stochastic Processes in Physics. The Johns Hopkins University Press, London (2002)
Mao X.: Stochastic Different Equations and Application. Horwood, Chichester (1997)
Mao X., Yuan C., Zou J.: Stochastic differential delay equations of population dynamics. J. Math. Anal. Appl. 304, 296–320 (2005)
Nelson M.I., Kerr T.B., Chen X.D.: A fundamental analysis of continuous flow ioreactor and membrane reactor models with death and maintenance included. Asis-Pac. J. Chem. Eng. 3, 70–80 (2008)
Nelson M.I., Sidhu H.S.: Analysis of a chemostat model with variable yield coefficient: Tessier kinetics. J. Math. Chem. 46, 303–321 (2009)
Nelson M.I., Holder A.: A fundamental analysis of continuous flow bioreactor models governed by Contois kinetics. II. Reactor cascades. Chem. Eng. J. 149, 406–416 (2009)
Zhang T.: Global analysis of continuous flow bioreactor and membrane reactor models with death and maintenance. J. Math. Chem. 50, 2239–2247 (2012)
Zhu C., Yin G.: On competitive Lotka-Volerra model in random environment. J. Math. Anal. Appl. 357, 154–170 (2009)
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Chen, Z., Zhang, T. Long time behaviour of a stochastic model for continuous flow bioreactor. J Math Chem 51, 451–464 (2013). https://doi.org/10.1007/s10910-012-0095-6
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DOI: https://doi.org/10.1007/s10910-012-0095-6