Electron. J. Differential Equations, Vol. 2017 (2017), No. 194, pp. 1-26.

### Asymptotic stability of a coupled advection-diffusion-reaction system arising in bioreactor processes Maria Crespo, Benjamin Ivorra, Angel Manuel Ramos

Abstract:
In this work, we present an asymptotic analysis of a coupled system of two advection-diffusion-reaction equations with Danckwerts boundary conditions, which models the interaction between a microbial population (e.g., bacteria), called biomass, and a diluted organic contaminant (e.g., nitrates), called substrate, in a continuous flow bioreactor. This system exhibits, under suitable conditions, two stable equilibrium states: one steady state in which the biomass becomes extinct and no reaction is produced, called washout, and another steady state, which corresponds to the partial elimination of the substrate. We use the linearization method to give sufficient conditions for the linear asymptotic stability of the two stable equilibrium configurations. Finally, we compare our asymptotic analysis with the usual asymptotic analysis associated to the continuous bioreactor when it is modeled with ordinary differential equations.

Submitted March 1, 2017. Published August 7, 2017.
Math Subject Classifications: 35B30, 35B35, 35B40, 35K51, 35Q35.
Key Words: March 1, 2017. Published August 8, 2017.

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 María Crespo UMR MISTEA - Math., Informatique et Statistique pour lénvironnement et lÁgronomie (INRA/SupAgro). 2, Place P.Viala 34060 Montpellier, France email: maria.crespo-moya@umontpellier.fr Benjamin Ivorra Departamento de Matemáatica Aplicada & Instituto de Matemáatica Interisciplinar Universidad Complutense de Madrid Plaza de Ciencias, 3, 28040 Madrid, Spain email: ivorra@ucm.es Ángel Manuel Ramos Departamento de Matemáatica Aplicada & Instituto de Matemáatica Interisciplinar Universidad Complutense de Madrid Plaza de Ciencias, 3, 28040 Madrid, Spain email: angel@mat.ucm.es