  In this paper, we study the global dynamics of a chemostat model with a
single nutrient and several competing species. Growth rates are not required to
be proportional to food uptakes. The model was studied by Fiedler and Hsu [J.
Math. Biol. (2009) 59:233-253]. These authors prove the nonexistence of
periodic orbits, by means of a multi-dimensional Bendixon-Dulac criterion. Our
approach is based on the construction of Lyapunov functions. The Lyapunov
functions extend those used by Hsu [SIAM J. Appl. Math. (1978) 34:760-763] and
by Wolkowicz and Lu [SIAM J. Appl. Math. (1997) 57:1019-1043] in the case when
growth rates are proportional to food uptakes.
