Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties

Frederic Mazenc; Gonzalo Robledo; Michael Malisoff

doi:10.3934/dcdsb.2018098

Article Contents

2018, Volume 23, Issue 4: 1851-1872. Doi: 10.3934/dcdsb.2018098

This issue Previous Article Determination of the area of exponential attraction in one-dimensional finite-time systems using meshless collocation Next Article

Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties

1.
EPI DISCO Inria-Saclay, CNRS, CentraleSupélec, Université Paris-Sud, 91192, Gif-sur-Yvette, France
2.
Departamento de Matemáticas, Universidad de Chile, Casilla 653, Santiago, Chile
3.
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA

^* Corresponding author: Michael Malisoff
^* Corresponding author: Michael Malisoff

Received: December 31, 2016

Revised: September 30, 2017

Early access: March 2018

Published: April 2018

The first and second authors were supported by the MATHAMSUD Cooperation Program (16 MATH-04 STADE). The third author was supported by NSF grant 1408295. A summary of some of this work that was confined to the case where the delays are zero was presented at the 2017 American Control Conference.

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Abstract

We study a chemostat model with an arbitrary number of competing species, one substrate, and constant dilution rates. We allow delays in the growth rates and additive uncertainties. Using constant inputs of certain species, we derive bounds on the sizes of the delays that ensure asymptotic stability of an equilibrium when the uncertainties are zero, which can allow persistence of multiple species. Under delays and uncertainties, we provide bounds on the delays and on the uncertainties that ensure input-to-state stability with respect to uncertainties.

Keywords:

Delay,
nonlinear,
bioreactors,
stability.

Mathematics Subject Classification: Primary: 92D40, 93D09; Secondary: 34D23.

Citation:

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Figure 1. Solution Components of (10) Plotted on Time Interval $[0, 25]$. Species $x_{1}(t)$ and $x_{2}(t)$ and Substrate $s(t)$. Initial State: $(s(0), x_1(0), x_2(0)) = (0.2, 0.1, 1)$.

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Figure 2. Solution Components of (10) Plotted on Time Interval $[0, 25]$. Species $x_{1}(t)$ and $x_{2}(t)$ and Substrate $s(t)$. Initial State: $(s(0), x_1(0), x_2(0)) = (1.3, 0.2, 0.1)$.

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