Volume 11, Number 2, Summer 2003
CONSIDERATIONS ON YIELD, NUTRIENT
UPTAKE, CELLULAR GROWTH, AND
COMPETITION IN CHEMOSTAT MODELS
JULIEN ARINO, SERGEI S. PILYUGIN
AND GAIL S. K. WOLKOWICZ
Abstract. We investigate some properties of a very general
model of growth in the chemostat. In the classical models
of the chemostat, the function describing cellular growth
is assumed to be a constant multiple of the function modeling
substrate uptake. The constant of proportionality is called the
growth yield constant. Here, this assumption of a constant describing
growth yield is relaxed. Instead, we assume that the
relationship between uptake and growth might depend on the
substrate concentration and hence that the yield is variable.
We obtain criteria for the stability of equilibria and for the
occurrence of a Hopf bifurcation. In particular, a Hopf bifurcation
can occur if the uptake function is unimodal. Then, in
this setting, we consider competition in the chemostat for a single
substrate, in order to challenge the principle of competitive
exclusion.
We consider two examples. In the first, the function describing
the growth process is monotone and in the second it
is unimodal. In both examples, in order to obtain a Hopf bifurcation,
one of the competitors is assumed to have a variable
yield, and its "uptake" is described by a unimodal function.
However, the interpretation is different in each case. We provide
a necessary condition for strong coexistence and a sufficient condition that guarantees the extinction of one or more
species. We show numerically by means of bifurcation diagrams
and simulations, that the competitive exclusion principle can be
breached resulting in oscillatory coexistence of more than one
species, that competitormediated coexistence is possible, and
that these simple systems can have very complicated dynamics.
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