A feedback perspective for chemostat models
with crowding eFects
Patrick De Leenheer
1
, David Angeli
2
, and Eduardo D. Sontag
3
1
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ
85287, USA
leenheer@math.la.asu.edu
2
Dip. di Sistemi e Informatica Universit´a di Firenze, Via di S. Marta 3, 50139
Firenze, Italy,
angeli@dsi.unifi.it
3
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903,
USA
http://www.math.rutgers.edu/
˜
sontag
,
sontag@control.rutgers.edu
Abstract.
This paper deals with an almost global stability result for a chemostat
model with including e±ects. The proof relies on a particular small-gain theorem
which has recently been developed for feedback interconnections of monotone sys-
tems.
1 Introduction
The chemostat is a well-known model used to describe the interaction between
microbial species which are competing for a single nutrient, see [12] for a
review. One of the prominent results in this area is the so-called ’competitive
exclusion principle’ which states roughly that in the long run only one of
the species survives. This is in contrast to what is observed in nature where
several species seem to coexist. This discrepancy has lead to modiFcations
of the model to try and bring theory and practice in better accordance; see
[14, 3, 9, 7]. Recently the chemostat has been
made
coexistent by means of
feedback control of the dilution rate [4].
In this paper we propose another modiFcation of the chemostat model:
˙
x
i
=
x
i
(
f
i
(
S
)
−
D
i
−
a
i
x
i
)
˙
S
=1
−
S
−
n
X
i
=1
x
i
f
i
(
S
)
(1)
where
i
,
2
, ..., n
,
x
i
is the concentration of species
i
and
S
is the nutrient
concentration. The positive parameters
D
i
are the sum of the (natural) death
rates of species
i
and the dilution rate, while the positive parameters
a
i
give
rise to death rates
a
i
x
i
which are due to crowding e±ects.
Throughout this paper we will assume the following: