Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2016 (2016), Article ID 3685941, 13 pages
http://dx.doi.org/10.1155/2016/3685941
Research Article

Dynamics Analysis and Biomass Productivity Optimisation of a Microbial Cultivation Process through Substrate Regulation

1School of Control Science and Engineering, Dalian University of Technology, Dalian 116024, China
2Faculty of Biological Sciences, Department of Biotechnology, University of Zielona Góra, Ulica Szafrana 1, 65-516 Zielona Góra, Poland
3School of Information Engineering, Dalian University, Dalian 116622, China

Received 24 February 2016; Revised 1 April 2016; Accepted 13 April 2016

Academic Editor: Carmen Coll

Copyright © 2016 Kaibiao Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. K. Schugerl, Bioreaction Engineering: Reactions Involving Microorganisms and Cells: Fundamentals, Thermodynamics, Formal Kinetics, Idealized Reactor Types and Operation, John Wiley & Sons, Chichester, UK, 1987.
  2. D. Krishnaiah, “Chemical and bioprocess engineering: a special issue of journal of applied sciences,” Journal of Applied Sciences, vol. 7, no. 15, pp. 1989–1990, 2007. View at Publisher · View at Google Scholar · View at Scopus
  3. G. J. Butler and G. S. K. Wolkowicz, “A mathematical model of the chemostat with a general class of functions describing nutrient uptake,” SIAM Journal on Applied Mathematics, vol. 45, no. 1, pp. 138–151, 1985. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. G. F. Fu, W. B. Ma, and S. G. Ruan, “Qualitative analysis of a chemostat model with inhibitory exponential substrate uptake,” Chaos, Solitons & Fractals, vol. 23, no. 3, pp. 873–886, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. P. De Leenheer and H. Smith, “Feedback control for chemostat models,” Journal of Mathematical Biology, vol. 46, no. 1, pp. 48–70, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. J.-L. Gouzé and G. Robledo, “Feedback control for nonmonotone competition models in the chemostat,” Nonlinear Analysis: Real World Applications, vol. 6, no. 4, pp. 671–690, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. H. Zhang, P. Georgescu, and L. Zhang, “Periodic patterns and Pareto efficiency of state dependent impulsive controls regulating interactions between wild and transgenic mosquito populations,” Communications in Nonlinear Science and Numerical Simulation, vol. 31, no. 1–3, pp. 83–107, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. G. R. Jiang, Q. S. Lu, and L. N. Qian, “Complex dynamics of a Holling type II prey-predator system with state feedback control,” Chaos, Solitons & Fractals, vol. 31, no. 2, pp. 448–461, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. S. Y. Tang and R. A. Cheke, “State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences,” Journal of Mathematical Biology, vol. 50, no. 3, pp. 257–292, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. S. Y. Tang, J. H. Liang, Y. N. Xiao, and R. A. Cheke, “Sliding bifurcations of Filippov two stage pest control models with economic thresholds,” SIAM Journal on Applied Mathematics, vol. 72, no. 4, pp. 1061–1080, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. S. Y. Tang, W. H. Pang, R. A. Cheke, and J. H. Wu, “Global dynamics of a state-dependent feedback control system,” Advances in Difference Equations, vol. 2015, article 322, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. G. R. Jiang, Q. S. Lu, and L. P. Peng, “Impulsive ecological control of a stage-structured pest management system,” Mathematical Biosciences and Engineering, vol. 2, no. 2, pp. 329–344, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  13. Y. Tian, K. B. Sun, and L. S. Chen, “Geometric approach to the stability analysis of the periodic solution in a semi-continuous dynamic system,” International Journal of Biomathematics, vol. 7, no. 2, Article ID 1450018, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. J. Jiao, L. Chen, and W. Long, “Pulse fishing policy for a stage-structured model with state-dependent harvesting,” Journal of Biological Systems, vol. 15, no. 3, pp. 409–416, 2007. View at Publisher · View at Google Scholar · View at Scopus
  15. M. Z. Huang, J. X. Li, X. Y. Song, and H. J. Guo, “Modeling impulsive injections of insulin: towards artificial pancreas,” SIAM Journal on Applied Mathematics, vol. 72, no. 5, pp. 1524–1548, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. R. Smith, “Impulsive differential equations with applications to self-cycling fermentation,” Open Access Dissertations and Theses, Paper 1526, 2001. View at Google Scholar
  17. R. Smith and G. Wolkowicz, “A size-structured model for the nutrient-driven self-cycling fermentation process,” Discrete and Continuous Dynamical Systems B, vol. 10, pp. 207–219, 2003. View at Google Scholar
  18. R. J. Smith and G. S. Wolkowicz, “Analysis of a model of the nutrient driven self-cycling fermentation process,” Dynamics of Continuous, Discrete & Impulsive Systems. Series B: Applications & Algorithms, vol. 11, no. 3, pp. 239–265, 2004. View at Google Scholar · View at MathSciNet
  19. G. H. Fan and G. Wolkowicz, “Analysis of a model of nutrient driven self-cycling fermentation allowing unimodal response function,” Discrete and Continuous Dynamical Systems B, vol. 8, no. 4, pp. 801–831, 2007. View at Publisher · View at Google Scholar
  20. H. J. Guo and L. S. Chen, “Periodic solution of a chemostat model with Monod growth rate and impulsive state feedback control,” Journal of Theoretical Biology, vol. 260, no. 4, pp. 502–509, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. B. M. Wincure, D. G. Cooper, and A. Rey, “Mathematical model of self-cycling fermentation,” Biotechnology and Bioengineering, vol. 46, no. 2, pp. 180–183, 1995. View at Publisher · View at Google Scholar · View at Scopus
  22. D. Dochain, “State and parameter estimation in chemical and biochemical processes: A tutorial,” Journal of Process Control, vol. 13, no. 8, pp. 801–818, 2003. View at Publisher · View at Google Scholar · View at Scopus
  23. J. Gonzalez, G. Fernandez, R. Aguilar, M. Barron, and J. Alvarez-Ramirez, “Sliding mode observer-based control for a class of bioreactors,” Chemical Engineering Journal, vol. 83, no. 1, pp. 25–32, 2001. View at Publisher · View at Google Scholar · View at Scopus
  24. S. Biagiola and J. Solsona, “State estimation in batch processes using a nonlinear observer,” Mathematical and Computer Modelling, vol. 44, no. 11-12, pp. 1009–1024, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  25. P. S. Crooke, C. J. Wei, and R. D. Tanner, “Effect of the specific growth rate and yield expressions on the existence of oscillatory behavior of a continuous fermentation model,” Chemical Engineering Communications, vol. 6, no. 6, pp. 333–347, 1980. View at Publisher · View at Google Scholar · View at Scopus
  26. G. Tessier, “Les lois quantitatives de la croissance,” Annales de Physiologie et de Physiochimie Biologique, vol. 12, pp. 527–573, 1936. View at Google Scholar
  27. J. Monod, “The growth of bacterial culture,” Annual Review of Microbiology, vol. 3, no. 1, pp. 371–394, 1949. View at Publisher · View at Google Scholar
  28. H. Moser, The Dynamics of Bacterial Populations Maintained in the Chemostat, Publication 614, Carnegie Institution of Washington, Washington, DC, USA, 1958.
  29. D. E. Contois, “Kinetics of bacterial growth: relationship between population density and specific growth rate of continuous cultures,” Journal of General Microbiology, vol. 21, no. 1, pp. 40–50, 1959. View at Publisher · View at Google Scholar
  30. J. F. Andrews, “A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates,” Biotechnology and Bioengineering, vol. 10, no. 6, pp. 707–723, 1968. View at Publisher · View at Google Scholar
  31. A. Dorofeev, M. Glagolev, T. Bondarenko, and N. Panikov, “Observation and explanation of the unusual growth-kinetics of arthrobacter-globiformis,” Microbiology, vol. 61, no. 1, pp. 24–31, 1992. View at Google Scholar
  32. F. Menkel and A. J. Knights, “A biological approach on modelling a variable biomass yield,” Process Biochemistry, vol. 30, no. 6, pp. 485–495, 1995. View at Publisher · View at Google Scholar · View at Scopus
  33. X. C. Huang, “Limit cycles in a continuous fermentation model,” Journal of Mathematical Chemistry, vol. 5, no. 3, pp. 287–296, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. S. S. Pilyugin and P. Waltman, “Multiple limit cycles in the chemostat with variable yield,” Mathematical Biosciences, vol. 182, no. 2, pp. 151–166, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  35. R. T. Alqahtani, M. I. Nelson, and A. L. Worthy, “Analysis of a chemostat model with variable yield coefficient and substrate inhibition: contois growth kinetics,” Chemical Engineering Communications, vol. 202, no. 3, pp. 332–344, 2015. View at Publisher · View at Google Scholar · View at Scopus
  36. P. S. Simeonov and D. D. Bainov, “Orbital stability of periodic solutions of autonomous systems with impulse effect,” International Journal of Systems Science, vol. 19, no. 12, pp. 2561–2585, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  37. L. S. Chen, “Pest control and geometric theory of semicontinuous dynamical system,” Journal of Beihua University, vol. 12, pp. 1–9, 2011. View at Google Scholar
  38. P. Holmes and E. Shea-Brown, “Stability,” Scholarpedia, vol. 1, no. 10, article 1838, 2006. View at Publisher · View at Google Scholar