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A new Liapunov function for the simple chemostat. (English) Zbl 1007.92034

From the introduction: The simple chemostat constitutes one of the most basic biochemical interactions, namely that of a consumer growing on a single limiting resource. The chemostat serves as a basic building block which underpins many multi-component population interaction models, and it also epitomizes an unfortunately rare situation where the mathematical model is completely validated by laboratory experimental data. Although the mathematical theory of the simple chemostat is well understood, related research on extensions and variations including multi-component population interactions continues to be an ongoing and active area of research, with many open questions.

We give what we think is a new Lyapunov function for the simple chemostat. This Lyapunov function directly verifies global asymptotic stability of the positive equilibrium of the simple chemostat whenever it exists. Global asymptotic stability of the simple chemostat is well known of course, and can be established in other ways, including an approach which uses another Lyapunov function, due to S.B. Hsu [SIAM J. Appl. Math. 34, 760-763 (1978; Zbl 0381.92014)]. Hsu’s Lyapunov function is only negative semi-definite along trajectories relative to the equilibrium. Thus the La Salle invariance principle must be applied to obtain the global asymptotic stability.

The function given in this note is sharper – its derivative along trajectories is negative definite relative to the equilibrium – and so no recourse to La Salle’s principle is needed. Although this does not seem to immediately lead to any new results about the simple chemostat, it provides another tool for the investigation of related models.

MSC:
92D40Ecology
34D20Stability of ODE