Global stability of perturbed chemostat systems
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916566214377472 |
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| author | Alvarez-Latuz, Claudia Bayen, Terence Coville, Jerome |
| author_facet | Alvarez-Latuz, Claudia Bayen, Terence Coville, Jerome |
| contents | This paper is devoted to the analysis of global stability of the chemostat system with a perturbation term representing any type of exchange between species. This conversion term depends on species and substrate concentrations but also on a positive perturbation parameter. After having written the invariant manifold as a union of a family of compact subsets, our main result states that for each subset in this family, there is a positive threshold for the perturbation parameter below which, the system is globally asymptotically stable in the corresponding subset. Our approach relies on the Malkin-Gorshin Theorem and on a Theorem by Smith and Waltman about perturbations of a globally stable steady state. Properties of steady-states and numerical simulations of the system's asymptotic behavior complete this study for two types of perturbation term between species. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08011 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global stability of perturbed chemostat systems Alvarez-Latuz, Claudia Bayen, Terence Coville, Jerome Dynamical Systems 34A34, 34D23, 37C75, 93D99 This paper is devoted to the analysis of global stability of the chemostat system with a perturbation term representing any type of exchange between species. This conversion term depends on species and substrate concentrations but also on a positive perturbation parameter. After having written the invariant manifold as a union of a family of compact subsets, our main result states that for each subset in this family, there is a positive threshold for the perturbation parameter below which, the system is globally asymptotically stable in the corresponding subset. Our approach relies on the Malkin-Gorshin Theorem and on a Theorem by Smith and Waltman about perturbations of a globally stable steady state. Properties of steady-states and numerical simulations of the system's asymptotic behavior complete this study for two types of perturbation term between species. |
| title | Global stability of perturbed chemostat systems |
| topic | Dynamical Systems 34A34, 34D23, 37C75, 93D99 |
| url | https://arxiv.org/abs/2501.08011 |