Bounded random fluctuations on the input flow in chemostat models with wall growth and non-monotonic kinetics

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

doi:10.3934/math.2021239

AIMS Mathematics

2021, Volume 6, Issue 4: 4025-4052. doi: 10.3934/math.2021239

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Bounded random fluctuations on the input flow in chemostat models with wall growth and non-monotonic kinetics

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
Dedicated to the memory of María José Garrido Atienza

Received: 17 November 2020 Accepted: 19 January 2021 Published: 04 February 2021
MSC : 34F05, 34K99

This paper investigates a chemostat model with wall growth and Haldane consumption kinetics. In addition, bounded random fluctuations on the input flow, which are modeled by means of the well-known Ornstein-Uhlenbeck process, are considered to obtain a much more realistic model fitting in a better way the phenomena observed by practitioners in real devices. Once the existence and uniqueness of global positive solution has been proved, as well as the existence of deterministic absorbing and attracting sets, the random dynamics inside the attracting set is studied in detail to provide conditions under which persistence of species is ensured, the main goal pursued from the practical point of view. Finally, we support the theoretical results with several numerical simulations.
- chemostat,
- wall growth,
- non-monotonic kinetics,
- absorbing set,
- real noise
Citation: Tomás Caraballo, Javier López-de-la-Cruz. Bounded random fluctuations on the input flow in chemostat models with wall growth and non-monotonic kinetics[J]. AIMS Mathematics, 2021, 6(4): 4025-4052. doi: 10.3934/math.2021239

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Abstract

This paper investigates a chemostat model with wall growth and Haldane consumption kinetics. In addition, bounded random fluctuations on the input flow, which are modeled by means of the well-known Ornstein-Uhlenbeck process, are considered to obtain a much more realistic model fitting in a better way the phenomena observed by practitioners in real devices. Once the existence and uniqueness of global positive solution has been proved, as well as the existence of deterministic absorbing and attracting sets, the random dynamics inside the attracting set is studied in detail to provide conditions under which persistence of species is ensured, the main goal pursued from the practical point of view. Finally, we support the theoretical results with several numerical simulations.

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