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Dynamics of Nonautonomous Chemostat Models

Part of the Studies in Systems, Decision and Control book series (SSDC,volume 30)


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.


  • Delay Differential Equation
  • Random Dynamical System
  • Random Attractor
  • Wall Growth
  • Pullback Attractor

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  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)

    MATH  MathSciNet  Google Scholar 

  2. Beretta, E., Takeuchi, Y.: Qualitative properties of chemostat equations with time delays II. Differ. Equ. Dyn. Syst. 2, 263–288 (1994)

    MATH  MathSciNet  Google Scholar 

  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)

    CrossRef  Google Scholar 

  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)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. Pilyugin, S.S., Waltman, P.: The simple chemostat with wall growth. SIAM J. Appl. Math. 59, 1552–1572 (1999)

    CrossRef  MATH  MathSciNet  Google Scholar 

  6. Sree Hari Rao, V., Raja Sekhara Rao, P.: Dynamic Models and Control of Biological Systems. Springer, Heidelberg (2009)

    MATH  Google Scholar 

  7. Topiwala, H., Hamer, G.: Effect of wall growth in steady state continuous culture. Biotech. Bioeng. 13, 919–922 (1971)

    CrossRef  Google Scholar 

  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)

    CrossRef  MATH  MathSciNet  Google Scholar 

  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. 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. Caraballo, T., Han X., Kloeden, P. E.: Non-autonomous chemostats with variable delays. SIAM J. Math. Anal. (to appear). doi:10.1137/14099930X

  12. Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems. American Mathematical Society, Providence (2011)

    CrossRef  MATH  Google Scholar 

  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)

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. Kloeden, P.E., Lorenz, T.: Pullback incremental stability. Nonauton. Random Dyn. Sys. 53–60 (2013). doi:10.2478/msds-2013-0004

  15. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional-Differential Equations. Applied Mathematical Sciences, vol. 99. Springer, New York (1993)

    CrossRef  MATH  Google Scholar 

  16. Smith, H.L., Waltman, P.: The Theory of the Chemostat: Dynamics of Microbial Competition. Cambridge University Press, Cambridge (1995)

    CrossRef  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  18. Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)

    CrossRef  MATH  Google Scholar 

  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)

    CrossRef  MATH  MathSciNet  Google Scholar 

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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.

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Correspondence to Tomás Caraballo .

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Caraballo, T., Han, X., Kloeden, P.E., Rapaport, A. (2015). Dynamics of Nonautonomous Chemostat Models. In: Sadovnichiy, V., Zgurovsky, M. (eds) Continuous and Distributed Systems II. Studies in Systems, Decision and Control, vol 30. Springer, Cham.

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