Study of the chemostat model with non-monotonic growth under random disturbances on the removal rate

Tomás Caraballo; Renato Colucci; Javier López-de-la-Cruz; Alain Rapaport; Tomás Caraballo; Renato Colucci; Javier López-de-la-Cruz; Alain Rapaport

doi:10.3934/mbe.2020382

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 7480-7501. doi: 10.3934/mbe.2020382

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Study of the chemostat model with non-monotonic growth under random disturbances on the removal rate

1.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, C/Tarfia s/n, Facultad de Matemáticas, Universidad de Sevilla, 41012 Sevilla, Spain
2.
Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche 12, 60131, Ancona, Italy
3.
MISTEA, Univ. Montpellier, INRAE, Montpellier SupAgro, 2, place Pierre Viala, 34060 Montpellier, France

Received: 21 May 2020 Accepted: 19 October 2020 Published: 29 October 2020

We revisit the chemostat model with Haldane growth function, here subject to bounded random disturbances on the input flow rate, as often met in biotechnological or waste-water industry. We prove existence and uniqueness of global positive solution of the random dynamics and existence of absorbing and attracting sets that are independent of the realizations of the noise. We study the long-time behavior of the random dynamics in terms of attracting sets, and provide first conditions under which biomass extinction cannot be avoided. We prove conditions for weak and strong persistence of the microbial species and provide lower bounds for the biomass concentration, as a relevant information for practitioners. The theoretical results are illustrated with numerical simulations.
- chemostat model,
- non-monotonic growth,
- bounded noise,
- Ornstein-Uhlenbeck,
- absorbing set
Citation: Tomás Caraballo, Renato Colucci, Javier López-de-la-Cruz, Alain Rapaport. Study of the chemostat model with non-monotonic growth under random disturbances on the removal rate[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7480-7501. doi: 10.3934/mbe.2020382

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Abstract

We revisit the chemostat model with Haldane growth function, here subject to bounded random disturbances on the input flow rate, as often met in biotechnological or waste-water industry. We prove existence and uniqueness of global positive solution of the random dynamics and existence of absorbing and attracting sets that are independent of the realizations of the noise. We study the long-time behavior of the random dynamics in terms of attracting sets, and provide first conditions under which biomass extinction cannot be avoided. We prove conditions for weak and strong persistence of the microbial species and provide lower bounds for the biomass concentration, as a relevant information for practitioners. The theoretical results are illustrated with numerical simulations.

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