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Mathematical Biosciences and Engineering

2007, Volume 4, Issue 2: 319-338. doi: 10.3934/mbe.2007.4.319
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On the stability of periodic solutions in the perturbed chemostat

  • 1.
    Projet MERE INRIA-INRA, UMR Analyse des Systèmes et Biométrie INRA, 2, pl. Viala, 34060 Montpellier
  • 2.
    Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918
  • 3.
    Department of Mathematics, University of Florida, Gainesville, FL 32611-8105
  • Received: 01 May 2006 Accepted: 29 June 2018 Published: 01 February 2007

  • MSC : 93D20.

  • We study the chemostat model for one species competing for one nutrient using a Lyapunov-type analysis. We design the dilution rate function so that all solutions of the chemostat converge to a prescribed periodic solution. In terms of chemostat biology, this means that no matter what positive initial levels for the species concentration and nutrient are selected, the long-term species concentration and substrate levels closely approximate a prescribed oscillatory behavior. This is significant because it reproduces the realistic ecological situation where the species and substrate concentrations oscillate. We show that the stability is maintained when the model is augmented by additional species that are being driven to extinction. We also give an input-to-state stability result for the chemostat-tracking equations for cases where there are small perturbations acting on the dilution rate and initial concentration. This means that the long-term species concentration and substrate behavior enjoys a highly desirable robustness property, since it continues to approximate the prescribed oscillation up to a small error when there are small unexpected changes in the dilution rate function.

    Citation: Frédéric Mazenc, Michael Malisoff, Patrick D. Leenheer. On the stability of periodic solutions in the perturbed chemostat[J]. Mathematical Biosciences and Engineering, 2007, 4(2): 319-338. doi: 10.3934/mbe.2007.4.319

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  • We study the chemostat model for one species competing for one nutrient using a Lyapunov-type analysis. We design the dilution rate function so that all solutions of the chemostat converge to a prescribed periodic solution. In terms of chemostat biology, this means that no matter what positive initial levels for the species concentration and nutrient are selected, the long-term species concentration and substrate levels closely approximate a prescribed oscillatory behavior. This is significant because it reproduces the realistic ecological situation where the species and substrate concentrations oscillate. We show that the stability is maintained when the model is augmented by additional species that are being driven to extinction. We also give an input-to-state stability result for the chemostat-tracking equations for cases where there are small perturbations acting on the dilution rate and initial concentration. This means that the long-term species concentration and substrate behavior enjoys a highly desirable robustness property, since it continues to approximate the prescribed oscillation up to a small error when there are small unexpected changes in the dilution rate function.


  • This article has been cited by:

    1. Frederic Mazenc, Michael Malisoff, Olivier Bernard, 2008, Lyapunov functions and robustness analysis under Matrosov conditions with an application to biological systems, 978-1-4244-2078-0, 2933, 10.1109/ACC.2008.4586941
    2. F. Mazenc, M. Malisoff, O. Bernard, A Simplified Design for Strict Lyapunov Functions Under Matrosov Conditions, 2009, 54, 0018-9286, 177, 10.1109/TAC.2008.2008353
    3. Frédéric Mazenc, Michael Malisoff, Stabilization of a chemostat model with Haldane growth functions and a delay in the measurements, 2010, 46, 00051098, 1428, 10.1016/j.automatica.2010.06.012
    4. Frederic Mazenc, Michael Malisoff, 2011, On stability and stabilization for chemostats with many limiting nutrients, 978-1-61284-801-3, 3700, 10.1109/CDC.2011.6160414
    5. Frederic Mazenc, Michael Malisoff, Jerome Harmand, 2007, Stabilization and robustness analysis for a chemostat model with two species and monod growth rates via a Lyapunov approach, 978-1-4244-1497-0, 3933, 10.1109/CDC.2007.4434316
    6. Frederic Mazenc, Zhong-Ping Jiang, 2009, Time-varying control laws with guaranteed persistence for a class of multi-species chemostats, 978-1-4244-3871-6, 7710, 10.1109/CDC.2009.5399908
    7. Iasson Karafyllis, Zhong-Ping Jiang, A new small-gain theorem with an application to the stabilization of the chemostat, 2012, 22, 10498923, 1602, 10.1002/rnc.1773
    8. Iasson Karafyllis, Zhong-Ping Jiang, Reduced order dead-beat observers for the chemostat, 2013, 14, 14681218, 340, 10.1016/j.nonrwa.2012.07.001
    9. Frederic Mazenc, Michael Malisoff, Jerome Harmand, 2007, Stabilization of a Periodic Trajectory for a Chemostat with Two Species, 1-4244-0988-8, 6128, 10.1109/ACC.2007.4282378
    10. Frédéric Mazenc, Michael Malisoff, Stability and stabilization for models of chemostats with multiple limiting substrates, 2012, 6, 1751-3758, 612, 10.1080/17513758.2012.663795
    11. FrÉdÉric Mazenc, Michael Malisoff, JÉrÔme Harmand, Further Results on Stabilization of Periodic Trajectories for a Chemostat With Two Species, 2008, 53, 0018-9286, 66, 10.1109/TAC.2007.911315
    12. Frédéric Mazenc, Michael Malisoff, Remarks on output feedback stabilization of two-species chemostat models, 2010, 46, 00051098, 1739, 10.1016/j.automatica.2010.06.035
    13. Global dynamics of the chemostat with different removal rates and variable yields, 2011, 8, 1551-0018, 827, 10.3934/mbe.2011.8.827
    14. Zhong Zhao, Baozhen Wang, Liuyong Pang, Ying Chen, Bifurcation Analysis of a Chemostat Model of Plasmid-Bearing and Plasmid-Free Competition with Pulsed Input, 2014, 2014, 1110-757X, 1, 10.1155/2014/343719
    15. Mary Ballyk, Ernest Barany, The role of resource types in the control of chemostats using feedback linearization, 2008, 211, 03043800, 25, 10.1016/j.ecolmodel.2007.08.019
    16. F. Mazenc, M. Malisoff, J. Harmand, Stabilization in a Two-Species Chemostat With Monod Growth Functions, 2009, 54, 0018-9286, 855, 10.1109/TAC.2008.2010964
    17. Frederic Mazenc, Michael Malisoff, 2010, Further results on robust output feedback control for the chemostat dynamics, 978-1-4244-7745-6, 822, 10.1109/CDC.2010.5717860
    18. Iasson Karafyllis, Costas Kravaris, Nicolas Kalogerakis, Relaxed Lyapunov criteria for robust global stabilisation of non-linear systems, 2009, 82, 0020-7179, 2077, 10.1080/00207170902912049
    19. Ningning Ye, Zengyun Hu, Zhidong Teng, Periodic solution and extinction in a periodic chemostat model with delay in microorganism growth, 2022, 21, 1534-0392, 1361, 10.3934/cpaa.2022022
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  • © 2007 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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