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

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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:
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2 Patrick De Leenheer et al. f i : R + R + is continuously diferentiable 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 satis±es these assumptions with global Lipschitz constant M/b . The only diference with the classical chemostat model [12] is that here crowding efects -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 2 , ..., x n ) 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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This note was uploaded on 12/27/2011 for the course MAS 3114 taught by Professor Olson during the Fall '08 term at University of Florida.

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

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