On the stability of periodic solutions in the perturbed chemostat
We study the chemostat model for one species competing for onenutrient using a Lyapunov-type analysis. We design the dilution ratefunction so that all solutions of the chemostat converge to aprescribed periodic solution. In terms of chemostat biology, this means that no matter whatpositive initial l...
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| Language: | English |
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AIMS Press
2007-01-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2007.4.319 |
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| author | Frédéric Mazenc Michael Malisoff Patrick D. Leenheer |
| author_facet | Frédéric Mazenc Michael Malisoff Patrick D. Leenheer |
| author_sort | Frédéric Mazenc |
| collection | DOAJ |
| description | We study the chemostat model for one species competing for onenutrient using a Lyapunov-type analysis. We design the dilution ratefunction so that all solutions of the chemostat converge to aprescribed periodic solution. In terms of chemostat biology, this means that no matter whatpositive initial levels for the species concentration and nutrientare selected, the long-term species concentration and substratelevels closely approximate a prescribed oscillatory behavior. Thisis significant because it reproduces the realistic ecologicalsituation where the species and substrate concentrations oscillate.We show that the stability is maintained when the model is augmentedby additional species that are being driven to extinction. We alsogive an input-to-state stability result for the chemostat-trackingequations for cases where there are small perturbations acting onthe dilution rate and initial concentration. This means that thelong-term species concentration and substrate behavior enjoys ahighly desirable robustness property, since it continues toapproximate the prescribed oscillation up to a small error whenthere are small unexpected changes in the dilution rate function. |
| format | Article |
| id | doaj-art-f7b2f1e4f4d64aee97b4c1c01bfd9070 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2007-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-f7b2f1e4f4d64aee97b4c1c01bfd90702025-01-24T01:53:27ZengAIMS PressMathematical Biosciences and Engineering1551-00182007-01-014231933810.3934/mbe.2007.4.319On the stability of periodic solutions in the perturbed chemostatFrédéric Mazenc0Michael Malisoff1Patrick D. Leenheer2Projet MERE INRIA-INRA, UMR Analyse des Systèmes et Biométrie INRA, 2, pl. Viala, 34060 MontpellierProjet MERE INRIA-INRA, UMR Analyse des Systèmes et Biométrie INRA, 2, pl. Viala, 34060 MontpellierProjet MERE INRIA-INRA, UMR Analyse des Systèmes et Biométrie INRA, 2, pl. Viala, 34060 MontpellierWe study the chemostat model for one species competing for onenutrient using a Lyapunov-type analysis. We design the dilution ratefunction so that all solutions of the chemostat converge to aprescribed periodic solution. In terms of chemostat biology, this means that no matter whatpositive initial levels for the species concentration and nutrientare selected, the long-term species concentration and substratelevels closely approximate a prescribed oscillatory behavior. Thisis significant because it reproduces the realistic ecologicalsituation where the species and substrate concentrations oscillate.We show that the stability is maintained when the model is augmentedby additional species that are being driven to extinction. We alsogive an input-to-state stability result for the chemostat-trackingequations for cases where there are small perturbations acting onthe dilution rate and initial concentration. This means that thelong-term species concentration and substrate behavior enjoys ahighly desirable robustness property, since it continues toapproximate the prescribed oscillation up to a small error whenthere are small unexpected changes in the dilution rate function.https://www.aimspress.com/article/doi/10.3934/mbe.2007.4.319robustness.species concentrationasymptotic stability analysischemostat |
| spellingShingle | Frédéric Mazenc Michael Malisoff Patrick D. Leenheer On the stability of periodic solutions in the perturbed chemostat Mathematical Biosciences and Engineering robustness. species concentration asymptotic stability analysis chemostat |
| title | On the stability of periodic solutions in the perturbed chemostat |
| title_full | On the stability of periodic solutions in the perturbed chemostat |
| title_fullStr | On the stability of periodic solutions in the perturbed chemostat |
| title_full_unstemmed | On the stability of periodic solutions in the perturbed chemostat |
| title_short | On the stability of periodic solutions in the perturbed chemostat |
| title_sort | on the stability of periodic solutions in the perturbed chemostat |
| topic | robustness. species concentration asymptotic stability analysis chemostat |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2007.4.319 |
| work_keys_str_mv | AT fredericmazenc onthestabilityofperiodicsolutionsintheperturbedchemostat AT michaelmalisoff onthestabilityofperiodicsolutionsintheperturbedchemostat AT patrickdleenheer onthestabilityofperiodicsolutionsintheperturbedchemostat |