On the stability of periodic solutions in the perturbed chemostat

Abstract

We study the chemostat model for one species competing for onenutrient using a Lyapunov-type analysis. We design the dilution ratefunction so that all solutions of the chemostat converge to aprescribed periodic solution. In terms of chemostat biology, this means that no matter whatpositive initial levels for the species concentration and nutrientare selected, the long-term species concentration and substratelevels closely approximate a prescribed oscillatory behavior. Thisis significant because it reproduces the realistic ecologicalsituation where the species and substrate concentrations oscillate.We show that the stability is maintained when the model is augmentedby additional species that are being driven to extinction. We alsogive an input-to-state stability result for the chemostat-trackingequations for cases where there are small perturbations acting onthe dilution rate and initial concentration. This means that thelong-term species concentration and substrate behavior enjoys ahighly desirable robustness property, since it continues toapproximate the prescribed oscillation up to a small error whenthere are small unexpected changes in the dilution rate function.

Keywords

• robustness.
• species concentration
• asymptotic stability analysis
• chemostat