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crowding-posta03 - A feedback perspective for chemostat...

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A feedback perspective for chemostat models with crowding effects 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 [email protected] 2 Dip. di Sistemi e Informatica Universit´a di Firenze, Via di S. Marta 3, 50139 Firenze, Italy, [email protected] 3 Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA http://www.math.rutgers.edu/ ˜ sontag , [email protected] Abstract. This paper deals with an almost global stability result for a chemostat model with including effects. 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 modifications 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 modification of the chemostat model: ˙ x i = x i ( f i ( S ) D i a i x i ) ˙ S = 1 S n i =1 x i f i ( S ) (1) where i = 1 , 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 effects. Throughout this paper we will assume the following:
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2 Patrick De Leenheer et al. f i : R + R + is continuously differentiable Moreover the functions f i are globally Lipschitz continuous on R + with Lipschitz constants L i . The classical Monod function f ( S ) = MS/ ( b + S ) with b, M > 0 satisfies these assumptions with global Lipschitz constant M/b . The only difference with the classical chemostat model [12] is that here crowding effects -modeled by the a i - are taken into consideration. Our main result is the following: Theorem 1. If n . max i ( L i a i ) . max i ( f i (1)) < 1 (2) then there exists an equilibrium point E of system (1) such that every solu- tion ξ ( t ) = ( x 1 ( t ) , x 2 ( t ) , ..., x n ( t ) , S ( t )) T starting in { ( x 1 , x 2 , ..., x n , S ) T R n +1 + | x i > 0 , i = 1 , ..., n } converges to E . Notice that our main result does not guarantee coexistence since the equi- librium point E could belong to the boundary of R n +1 + and correspond to the absence of one of the species. However, in the sequel we will provide conditions that do imply coexistence.
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