Advertisement

Dynamics of Nonautonomous Chemostat Models

  • Tomás CaraballoEmail author
  • Xiaoying Han
  • Peter E. Kloeden
  • Alain Rapaport
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 30)

Abstract

Chemostat models have a long history in the biological sciences as well as in biomathematics. Hitherto most investigations have focused on autonomous systems, that is, with constant parameters, inputs, and outputs. In many realistic situations these quantities can vary in time, either deterministically (e.g., periodically) or randomly. They are then nonautonomous dynamical systems for which the usual concepts of autonomous systems do not apply or are too restrictive. The newly developing theory of nonautonomous dynamical systems provides the necessary concepts, in particular that of a nonautonomous pullback attractor. These will be used here to analyze the dynamical behavior of nonautonomous chemostat models with or without wall growth, time-dependent delays, variable inputs and outputs. The possibility of overyielding in nonautonomous chemostats will also be discussed.

Keywords

Delay Differential Equation Random Dynamical System Random Attractor Wall Growth Pullback Attractor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

This work has been partially supported by the Spanish Ministerio de Economía y Competitividad project MTM2011-22411 and the Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under grant 2010/FQM314 and Proyecto de Excelencia P12-FQM-1492.

References

  1. 1.
    Beretta, E., Takeuchi, Y.: Qualitative properties of chemostat equations with time delays: boundedness, local and global asymptotic stability. Differ. Equ. Dyn. Syst. 2, 19–40 (1994)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Beretta, E., Takeuchi, Y.: Qualitative properties of chemostat equations with time delays II. Differ. Equ. Dyn. Syst. 2, 263–288 (1994)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Ballyk, M., Jones, D., Smith, H.: The biofilm model of Freter: a review. In: Magal, P., Ruan, S. (eds.) Structured Population Models in Biology and Epidemiology, pp. 265–302. Springer, Berlin (2008)CrossRefGoogle Scholar
  4. 4.
    Jones, D., Kojouharov, H., Le, D., Smith, H.L.: The Freter model: a simple model of biofilm formation. J. Math. Biol. 47, 137–152 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Pilyugin, S.S., Waltman, P.: The simple chemostat with wall growth. SIAM J. Appl. Math. 59, 1552–1572 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Sree Hari Rao, V., Raja Sekhara Rao, P.: Dynamic Models and Control of Biological Systems. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  7. 7.
    Topiwala, H., Hamer, G.: Effect of wall growth in steady state continuous culture. Biotech. Bioeng. 13, 919–922 (1971)CrossRefGoogle Scholar
  8. 8.
    Butler, G.J., Hsu, S.B., Waltman, P.: A mathematical model of the chemostat with periodic washout rate. SIAM J. Appl. Math. 45, 435–449 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Caraballo, T., Han, X., Kloeden, P.E.: Chemostats with time-dependent inputs and wall growth. Appl. Math. Inf. Sci. (to appear)Google Scholar
  10. 10.
    Caraballo, T., Han, X., Kloeden, P. E.: Chemostats with random inputs and wall growth. Math. Methods Appl. Sci. (to appear). doi: 10.1002/mma.3437
  11. 11.
    Caraballo, T., Han X., Kloeden, P. E.: Non-autonomous chemostats with variable delays. SIAM J. Math. Anal. (to appear). doi: 10.1137/14099930X
  12. 12.
    Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems. American Mathematical Society, Providence (2011)zbMATHCrossRefGoogle Scholar
  13. 13.
    Caraballo, T., Langa, J.A., Robinson, J.C.: Attractors for differential equations with variable delays. J. Math. Anal. Appl. 260(2), 421–438 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Kloeden, P.E., Lorenz, T.: Pullback incremental stability. Nonauton. Random Dyn. Sys. 53–60 (2013). doi: 10.2478/msds-2013-0004
  15. 15.
    Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional-Differential Equations. Applied Mathematical Sciences, vol. 99. Springer, New York (1993)zbMATHCrossRefGoogle Scholar
  16. 16.
    Smith, H.L., Waltman, P.: The Theory of the Chemostat: Dynamics of Microbial Competition. Cambridge University Press, Cambridge (1995)zbMATHCrossRefGoogle Scholar
  17. 17.
    Asai, Y., Kloeden, P.E.: Numerical schemes for random ODEs via stochastic differential equations. Commun. Appl. Anal. 17(3 and 4), 521–528 (2013)MathSciNetGoogle Scholar
  18. 18.
    Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)zbMATHCrossRefGoogle Scholar
  19. 19.
    Caraballo, T., Kloeden, P.E., Real, J.: Discretization of asymptotically stable stationary solutions of delay differential equations with a random stationary delay. J. Dyn. Differ. Equ. 18(4), 863–880 (2006)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Tomás Caraballo
    • 1
    Email author
  • Xiaoying Han
    • 2
  • Peter E. Kloeden
    • 3
    • 4
  • Alain Rapaport
    • 5
    • 6
  1. 1.Dpto. Ecuaciones Diferenciales y Análisis NuméricoUniversidad de SevillaSevillaSpain
  2. 2.Department of Mathematics and StatisticsAuburn UniversityAuburnUSA
  3. 3.School of Mathematics and StatisticsHuazhong University of Science & TechnologyWuhanChina
  4. 4.Felix-Klein-Zentrum Für MathematikTU KaiserslauternKaiserslauternGermany
  5. 5.UMR INRA/SupAgro MISTEAMontpellierFrance
  6. 6.MODEMIC TeamINRIASophia-AntipolisFrance

Personalised recommendations