Acta Biotheoretica

, Volume 65, Issue 3, pp 233–252 | Cite as

Asymptotic Behavior of a Chemostat Model with Constant Recycle Sludge Concentration

Regular Article
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Abstract

In this work, we study a several species aerobic chemostat model with constant recycle sludge concentration in continuous culture. We reduce the number of parameters by considering a dimensionless model. First, the existence of a global positive uniform attractor for the model with different removal rates is proved using the theory of dissipative dynamical systems. Hence, we investigate the asymptotic behavior of the model under small perturbations using methods of singular perturbation theory and we prove that, in the case of two species in competition, the unique equilibrium which is positive is globally asymptotically stable. Finally, we establish the link between the open problem of the chemostat with different removal rates and monotone functional responses, and our model when two species compete on the same nutrient. We give some numerical simulations to illustrate the results.

Keywords

Dissipative dynamical systems Chemostat with recycle sludge Local and global asymptotic stability Singular perturbation Competitive exclusion principle 

List of Symbols

Subscripts

i

The i-th species of microorganism.

Superscripts

r

Recycle stream.

0

Influent.

Variables and Parameters

V

Volume of aeration basin, volume.

Q

Flow rate of influent, volume/time.

D

Dilution rate; ratio of flow, Q, to volume, V, \(\text{ time }^{-1}\).

\(\mu\)

Specific biomass growth rate, \(\text{ time }^{-1}\).

\(\mu _{\max }\)

The maximum specific growth rate, \(\text{ time }^{-1}\).

Y

Yield coefficient, cell mass/substrate mass.

r

Recirculated flow ratio.

\(r\,Q\)

Flow rate of recycle, volume/time.

\(k_{\text{ S }}\)

Michaelis–Menten constant, mass/volume.

\(k_{d}\),\(k_{i}\)

Death rate, \(\text{ time }^{-1}\).

\(x_{i}\)

Concentration of i-th organism in reactor, mass/volume.

\(x_{i}^{0}\)

Concentration of i-th organism in influent, mass/volume.

\(x_{i}^{r}\)

Concentration of i-th organism in recycle flow to reactor, mass/volume.

S

Substrate concentration in reactor, mass/volume.

\(S_{0}\)

Influent substrate concentration, mass/volume.

\(\lambda _{i}\)

Break-even concentration of i-th organism, mass/volume.

Mathematics Subject Classification

92B05 37N25 34D23 34D10 34D15 

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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Laboratoire Systèmes Dynamiques et ApplicationsTlemcen UniversityTlemcenAlgeria

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