Crowding effects promote coexistence in the chemostat

• Published in: Journal of mathematical analysis and applications Vol. 319; no. 1; pp. 48 - 60
• Main Authors: De Leenheer, Patrick, Angeli, David, Sontag, Eduardo D.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.07.2006; Elsevier
• Subjects: Chemostat; Coexistence; Crowding; Exact sciences and technology; Feedback systems; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Monotone systems; Ordinary differential equations; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Coexistence; Monotone systems; Chemostat; Feedback systems; Crowding; Input output; Monotone system; Sufficient condition; Feedback system; Mathematical analysis; Interconnection
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2006.02.036

This paper deals with an almost-global stability result for a particular chemostat model. It deviates from the classical chemostat because crowding effects are taken into consideration. This model can be rewritten as a negative feedback interconnection of two systems which are monotone (as input/output systems). Moreover, these subsystems behave nicely when subject to constant inputs. This allows the use of a particular small-gain theorem which has recently been developed for feedback interconnections of monotone systems. Application of this theorem requires—at least approximate—knowledge of two gain functions associated to the subsystems. It turns out that for the chemostat model proposed here, these approximations can be obtained explicitly and this leads to a sufficient condition for almost-global stability. In addition, we show that coexistence occurs in this model if the crowding effects are large enough.