Dynamic behavior analysis of carboxymethylcellulose hydrolysis in a chemostat Rigel V. Gomez-Acata * , Pablo A. Lopez-Perez * , Rafael Maya-Yescas ** , Ricardo Aguilar-Lopez * *Department of Biotechnology & Bioengineering, CINVESTAV-IPN, No. 2508 D.F. Mexico (Tel: 55-5747-3800; e-mail: raguilar@cinvestav.mx). ** School of Chemical Engineering, Universidad Michoacana de San Nicolas de Hidalgo Michoacan, Mexico (e-mail: rmayay@umich.mx) Abstract: At the present time cellulosic waste substrates is a subject of global relevance because is an alternative source for biofuel production, the complex nature of this substrate and the presence of microorganisms for its biodegradation may result in complex dynamics which are important to understand why in many cases could have negative effects in the productivity process. In this paper dynamical behavior analysis of a chemostat model was carried out for two unstructured kinetic growth models (Luong and Han-Levenspiel), choosing as study case the carboxymethylcellulose hydrolysis by Cellulomonas cellulans, where differences related with multiplicity of steady states, hysteresis, periodicity and unstable operating regions were pointed out. Keywords: Dynamic behaviors, eigenvalues, hysteresis loop, limit cycle, bifurcation analysis, chemostat modeling, stability analysis, biological system. 1. INTRODUCTION Cellulose is the most abundant source of carbon on land; it is present in the wall of plant cells and is a polysaccharide consisting of a linear chain of hundreds to thousands β(1→4) linked D-glucose units. Cellulose is considered the most important reservoir of carbons to convert glucose, which can be used for production of ethanol, a promising alternative energy source for the limited crude oil, organic acids and other chemicals. (Bo-Kyung et al., 2009; Ye & Jiayang, 2002). Only a small percentage of all microorganisms in the earth can degrade cellulose principally bacteria and fungi (Wilson, 2011). The enzymes produced by these microorganisms have a lot of applications for example in the textile industry as a detergent in the pulp and textile industry, for wastewater treatment and for processing in coffee (Agarwal et al., 2009), reasons for which it is necessary to produce them in controlled environments such as bioreactors. Commonly, bioreactor mathematical models are proposed to describe the transformations of different substrates by microorganisms or their enzymes, through setting phenomenological balances in which unstructured kinetic growth models are employed to describe the velocity of the biochemical transformation of substrate to biomass and metabolites. These unstructured models present disadvantages like do not take into account the internal structure of the cell metabolism, biochemical reaction routes nor environmental effects. Bifurcation analysis is applied to linear and nonlinear (bio) mathematical models; it can be used to find optimal operating conditions and to avoid hazardous conditions for the process (Namjoshi, Kienle, & Ranmkrishna, 2003). This analysis is very useful and works revealing all possible equilibrium points for a given initial operation condition, maintenance free one or two system parameters in many cases (bifurcation parameter) such as the dilution rate and initial substrate concentration. With the above analysis is possible to predict if the system can present oscillations, hysteresis, steady state multiplicity and chaos (Alvarez-Ramirez et al. 2009) (Namjoshi et al. 2003). The local bifurcation is determined by calculated the system eigenvalues as in the situation of stability analysis (Gray et al. 1990), it occurs when one of the eigenvalues is close to the axis of imaginary numbers in the complex plane. The simplest bifurcations are associated when one of the eigenvalues takes the value zero (Fold bifurcation) as is the case of the branching point (BP) and limit point (LP) or when a pair of conjugate eigenvalues cross the imaginary axis (Hopf bifurcation, H). The Fold bifurcations are usually the cause of multiplicity of steady states and hysteresis. Hopf bifurcations are responsible for the appearance and disappearance of periodic solutions. (Zhang & Henson, 2001). Furthermore, this analysis can be used to find optimal operating conditions and to avoid hazardous conditions for the process. (Ajbar,2001) (Kuznetzov, 1998), (Namjoshi, Kienle, & Ranmkrishna, 2003). In this paper through a dynamic behaviour analysis of the carboxymethylcellulose hydrolysis by Cellulomonas cellulans 2012 IFAC Conference on Analysis and Control of Chaotic Systems The International Federation of Automatic Control June 20-22, 2012. Cancún, México 978-3-902823-02-1/12/$20.00 © 2012 IFAC 132 10.3182/20120620-3-MX-3012.00063