Survey on chemostat models with bounded random input flow

Tomás Caraballo; Javier López-de-la-Cruz; Tomás Caraballo; Javier López-de-la-Cruz

doi:10.3934/mmc.2021005

Mathematical Modelling and Control

2021, Volume 1, Issue 1: 52-78. doi: 10.3934/mmc.2021005

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Survey

Survey on chemostat models with bounded random input flow

Tomás Caraballo ¹,
Javier López-de-la-Cruz ^{2
,
,}

1.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, C/Tarfia s/n, Facultad de Matemáticas, Universidad de Sevilla, 41012 Sevilla, Spain
2.
Departamento de Matemática Aplicada a las TIC, Escuela Técnica Superior de Ingenieros Informáticos, Campus de Montegancedo, Universidad Politécnica de Madrid, 28660 Boadilla del Monte, Madrid, Spain

Received: 27 February 2021 Accepted: 09 March 2021 Published: 19 March 2021

In this paper we study some chemostat models with random bounded fluctuations on the input flow. We start with the classical chemostat system and obtain new models incorporating, for instance, wall growth and different consumption functions, motivated by phenomena in real devices. In every case, we prove existence and uniqueness of positive global solution, existence of deterministic absorbing and attracting sets and we investigate the internal structure of the attracting sets to obtain detailed information about the long-time dynamics of the systems. This allows us to provide conditions under which either extinction or persistence of the species is ensured, the main goal for practitioners. In addition, we provide several numerical simulations to support the theoretical results.
- chemostat,
- bounded noise,
- Ornstein-Uhlenbeck,
- wall growth,
- non-monotonic consumption function
Citation: Tomás Caraballo, Javier López-de-la-Cruz. Survey on chemostat models with bounded random input flow[J]. Mathematical Modelling and Control, 2021, 1(1): 52-78. doi: 10.3934/mmc.2021005

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Abstract

In this paper we study some chemostat models with random bounded fluctuations on the input flow. We start with the classical chemostat system and obtain new models incorporating, for instance, wall growth and different consumption functions, motivated by phenomena in real devices. In every case, we prove existence and uniqueness of positive global solution, existence of deterministic absorbing and attracting sets and we investigate the internal structure of the attracting sets to obtain detailed information about the long-time dynamics of the systems. This allows us to provide conditions under which either extinction or persistence of the species is ensured, the main goal for practitioners. In addition, we provide several numerical simulations to support the theoretical results.

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