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2015, 5(4): 381-392. doi: 10.3934/naco.2015.5.381

A stochastic model for microbial fermentation process under Gaussian white noise environment

Yan Wang 1, , Lei Wang 2, , Yanxiang Zhao 3, , Aimin Song 1, and  Yanping Ma 4,

1. 

School of Science, Dalian Jiaotong University,Dalian, MO 116028, China, China
2. 

School of Mathematical Sciences, Dalian University of Technology, Dalian, MO 116023, China
3. 

Department of Mathematics, The George Washington University,Washington DC 20052, United States
4. 

Department of Mathematics, Loyola Marymount University, Los Angeles CA 90045, United States

Received  March 2015 Revised  October 2015 Published  October 2015

In this paper, we propose a stochastic model for the microbial fermentation process under the framework of white noise analysis, where Gaussian white noises are used to model the environmental noises and the specific growth rate is driven by Gaussian white noises. In order to keep the regularity of the terminal time, the adjustment factors are added in the volatility coefficients of the stochastic model. Then we prove some fundamental properties of the stochastic model: the regularity of the terminal time, the existence and uniqueness of a solution and the continuous dependence of the solution on the initial values.
Keywords: stochastic model, stochastic differential equation, Gaussian white noise, microbial fermentation process..
Mathematics Subject Classification: Primary: 60H40, 91B70, 91B74 Secondary: 60H1.
Citation: Yan Wang, Lei Wang, Yanxiang Zhao, Aimin Song, Yanping Ma. A stochastic model for microbial fermentation process under Gaussian white noise environment. Numerical Algebra, Control and Optimization, 2015, 5 (4) : 381-392. doi: 10.3934/naco.2015.5.381
References:
[1]

I. Albert, R. Pouillot and J.-B. Denis, Stochastically modeling listeria monocytogenes growth in farm tank milk, Risk Analysis, 25 (2005), 1171-1185.

[2]

H. Biebl, K. Menzel, A. P. Zeng and W. Deckwer, Microbial production of 1,3-propanediol, Applied Microbiology and Biotechnology, 52 (1999), 297-298.

[3]

R. Bona and A. Moser, Modeling of l-glutamic acid production with Corynebacterium glutamicum under biotin limitation, Bioprocess Engineering, 17 (1997), 139-142.

[4]

C. Hartmann and A. Delgado, Numerical simulation of the mechanics of a yeast cell under high hydrostatic pressure, Journal of Biomechanics, 37 (2004), 977-987.

[5]

H. J. Henzler, Particle stress in bioreactors, Advances in Biochemical Engineering, 67 (2000), 35-82.

[6]

H. Holden, B. Øksendal, J. Ubøe and T. S. Zhang, Stochastic Partial Differential Equations-A Modeling, White Noise Functional Approach, 2nd edition, Springer-Verlag, New York, 2010. doi: 10.1007/978-0-387-89488-1.

[7]

A. Kasperski, Modelling of cells bioenergetics, Acta Biotheoretica, 56 (2008), 233-247.

[8]

A. Kasperski and T. Miskiewicz, Optimization of pulsed feeding in a Baker's yeast process with dissolved oxygen concentration as a control parameter, Biochemical Engineering Journal, 40 (2008), 321-327.

[9]

Z. Kutalik, M. Razaz and J. Baranyi, Connection between stochastic and deterministic modelling of microbial growth, Journal of Theoretical Biology, 232 (2005), 285-299. doi: 10.1016/j.jtbi.2004.08.013.

[10]

X. Li and X. Mao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete and Continuous Dynamical Systems, 24 (2009), 523-545. doi: 10.3934/dcds.2009.24.523.

[11]

B. Ø ksendal and A. Sulem, Applied Stochastic Control of Jump Diffusion, 2nd edition, Springer, Berlin, 2007. doi: 10.1007/978-3-540-69826-5.

[12]

B. Ø ksendal, Stochastic Differential Equations, 6nd edition, Springer, Berlin, Heidelberg, New York, 2005.

[13]

H. J. Rehm and G. Reed, Microbial Fundamentals, Verlag Chemie, Weinheim, 1981.

[14]

K. Schügerl, Bioreaction Engineering: Reactions Involving Microorganisms and Cells: Fundamentals, Thermodynamics, Formal Kinetics, Idealized Reactor Types and Operation, Wiley, Chichester, 1987.

[15]

T. K. Soboleva, A. E. Filippov, A. B. Pleasants, R. J. Jones and G. A. Dykes, Stochastic modelling of the growth of a microbial population under changing temperature regimes, International Journal of Food Microbiology, 64 (2001), 317-323.

[16]

S. Suresh, N. S. Khan, V. C. Srivastava and I. M. Mishra, Kinetic modeling and sensitivity analysis of kinetic parameters for $l$-glutamic acid production using Corynebacterium glutamicum, International Journal of Chemical Reactor Engineering, 7 (2009), Article A89.

[17]

Y. Tian, A. Kasperski, K. Sun and Lansun Chen, Theoretical approach to modelling and analysis of the bioprocess with product inhibition and impulse effect, BioSystems, 104 (2011), 77-86.

[18]

M. K. Toma, M. P. Rukilisha, J. J. Vanags, M. O. Zeltina, M. P. Leite, N. I. Galinina, U. E. Viesturs and R. P. Tengerdy, Inhibition of microbial growth and metabolism by excess turbulence, Biotechnology and Bioengineering, 38 (2000), 552-556.

[19]

L. Wang, Z. Xiu and E. Feng, A stochastic model of microbial bioconversion process in batch culture, International journal of Chemical reactor engineering, 9 (2011), Article A82.

[20]

L. Wang, Z. Xiu and E. Feng, Modeling nonlinear stochastic kinetic system and stochastic optimal control of microbial bioconversion process in batch culture, Nonlinear Analysis: Modelling and Control, 18 (2013), 99-111.

show all references

References:
[1]

I. Albert, R. Pouillot and J.-B. Denis, Stochastically modeling listeria monocytogenes growth in farm tank milk, Risk Analysis, 25 (2005), 1171-1185.

[2]

H. Biebl, K. Menzel, A. P. Zeng and W. Deckwer, Microbial production of 1,3-propanediol, Applied Microbiology and Biotechnology, 52 (1999), 297-298.

[3]

R. Bona and A. Moser, Modeling of l-glutamic acid production with Corynebacterium glutamicum under biotin limitation, Bioprocess Engineering, 17 (1997), 139-142.

[4]

C. Hartmann and A. Delgado, Numerical simulation of the mechanics of a yeast cell under high hydrostatic pressure, Journal of Biomechanics, 37 (2004), 977-987.

[5]

H. J. Henzler, Particle stress in bioreactors, Advances in Biochemical Engineering, 67 (2000), 35-82.

[6]

H. Holden, B. Øksendal, J. Ubøe and T. S. Zhang, Stochastic Partial Differential Equations-A Modeling, White Noise Functional Approach, 2nd edition, Springer-Verlag, New York, 2010. doi: 10.1007/978-0-387-89488-1.

[7]

A. Kasperski, Modelling of cells bioenergetics, Acta Biotheoretica, 56 (2008), 233-247.

[8]

A. Kasperski and T. Miskiewicz, Optimization of pulsed feeding in a Baker's yeast process with dissolved oxygen concentration as a control parameter, Biochemical Engineering Journal, 40 (2008), 321-327.

[9]

Z. Kutalik, M. Razaz and J. Baranyi, Connection between stochastic and deterministic modelling of microbial growth, Journal of Theoretical Biology, 232 (2005), 285-299. doi: 10.1016/j.jtbi.2004.08.013.

[10]

X. Li and X. Mao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete and Continuous Dynamical Systems, 24 (2009), 523-545. doi: 10.3934/dcds.2009.24.523.

[11]

B. Ø ksendal and A. Sulem, Applied Stochastic Control of Jump Diffusion, 2nd edition, Springer, Berlin, 2007. doi: 10.1007/978-3-540-69826-5.

[12]

B. Ø ksendal, Stochastic Differential Equations, 6nd edition, Springer, Berlin, Heidelberg, New York, 2005.

[13]

H. J. Rehm and G. Reed, Microbial Fundamentals, Verlag Chemie, Weinheim, 1981.

[14]

K. Schügerl, Bioreaction Engineering: Reactions Involving Microorganisms and Cells: Fundamentals, Thermodynamics, Formal Kinetics, Idealized Reactor Types and Operation, Wiley, Chichester, 1987.

[15]

T. K. Soboleva, A. E. Filippov, A. B. Pleasants, R. J. Jones and G. A. Dykes, Stochastic modelling of the growth of a microbial population under changing temperature regimes, International Journal of Food Microbiology, 64 (2001), 317-323.

[16]

S. Suresh, N. S. Khan, V. C. Srivastava and I. M. Mishra, Kinetic modeling and sensitivity analysis of kinetic parameters for $l$-glutamic acid production using Corynebacterium glutamicum, International Journal of Chemical Reactor Engineering, 7 (2009), Article A89.

[17]

Y. Tian, A. Kasperski, K. Sun and Lansun Chen, Theoretical approach to modelling and analysis of the bioprocess with product inhibition and impulse effect, BioSystems, 104 (2011), 77-86.

[18]

M. K. Toma, M. P. Rukilisha, J. J. Vanags, M. O. Zeltina, M. P. Leite, N. I. Galinina, U. E. Viesturs and R. P. Tengerdy, Inhibition of microbial growth and metabolism by excess turbulence, Biotechnology and Bioengineering, 38 (2000), 552-556.

[19]

L. Wang, Z. Xiu and E. Feng, A stochastic model of microbial bioconversion process in batch culture, International journal of Chemical reactor engineering, 9 (2011), Article A82.

[20]

L. Wang, Z. Xiu and E. Feng, Modeling nonlinear stochastic kinetic system and stochastic optimal control of microbial bioconversion process in batch culture, Nonlinear Analysis: Modelling and Control, 18 (2013), 99-111.

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