American Institute of Mathematical Sciences

Advanced
2007, 4(2): 319-338. doi: 10.3934/mbe.2007.4.319

On the stability of periodic solutions in the perturbed chemostat

Frédéric Mazenc 1, , Michael Malisoff 2, and  Patrick D. Leenheer 3,

1. 

Projet MERE INRIA-INRA, UMR Analyse des Systèmes et Biométrie INRA, 2, pl. Viala, 34060 Montpellier, France
2. 

Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, United States
3. 

Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, United States

Received  May 2006 Revised  December 2006 Published  February 2007

We study the chemostat model for one species competing for one nutrient using a Lyapunov-type analysis. We design the dilution rate function so that all solutions of the chemostat converge to a prescribed periodic solution. In terms of chemostat biology, this means that no matter what positive initial levels for the species concentration and nutrient are selected, the long-term species concentration and substrate levels closely approximate a prescribed oscillatory behavior. This is significant because it reproduces the realistic ecological situation where the species and substrate concentrations oscillate. We show that the stability is maintained when the model is augmented by additional species that are being driven to extinction. We also give an input-to-state stability result for the chemostat-tracking equations for cases where there are small perturbations acting on the dilution rate and initial concentration. This means that the long-term species concentration and substrate behavior enjoys a highly desirable robustness property, since it continues to approximate the prescribed oscillation up to a small error when there are small unexpected changes in the dilution rate function.
Keywords: robustness., species concentration, asymptotic stability analysis, chemostat.
Mathematics Subject Classification: 93D2.
Citation: Frédéric Mazenc, Michael Malisoff, Patrick D. Leenheer. On the stability of periodic solutions in the perturbed chemostat. Mathematical Biosciences & Engineering, 2007, 4 (2) : 319-338. doi: 10.3934/mbe.2007.4.319
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