Dynamics of the chemostat : a bifurcation theory approach by Abdelhamid Ajbar

By Abdelhamid Ajbar

A ubiquitous device in mathematical biology and chemical engineering, the chemostat usually produces instabilities that pose safeguard risks and adversely impact the optimization of bioreactive platforms. Singularity thought and bifurcation diagrams jointly supply an invaluable framework for addressing those concerns. in accordance with the authors’ broad paintings during this box, Dynamics of the Chemostat: A Bifurcation idea Approach explores using bifurcation thought to investigate the static and dynamic habit of the chemostat.

The authors first survey the key paintings that has been performed at the balance of constant bioreactors. They subsequent current the modeling methods used for bioreactive platforms, the several kinetic expressions for progress premiums, and instruments, resembling multiplicity, bifurcation, and singularity conception, for examining nonlinear systems.

The textual content strikes directly to the static and dynamic habit of the fundamental unstructured version of the chemostat for consistent and variable yield coefficients in addition to within the presence of wall attachment. It then covers the dynamics of interacting species, together with natural and easy microbial pageant, biodegradation of combined substrates, dynamics of plasmid-bearing and plasmid-free recombinant cultures, and dynamics of predator–prey interactions. The authors additionally study dynamics of the chemostat with product formation for numerous progress types, supply examples of bifurcation idea for learning the operability and dynamics of continuing bioreactor types, and observe basic suggestions of bifurcation concept to investigate the dynamics of a periodically compelled bioreactor.

Using singularity concept and bifurcation ideas, this booklet provides a cohesive mathematical framework for studying and modeling the macro- and microscopic interactions taking place in chemostats. The textual content comprises types that describe the intracellular and working components of the bioreactive process. It additionally explains the mathematical concept at the back of the models.

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The singularity theory can therefore provide a useful tool for the classification of the different branching phenomena in the model. The second objective is to study the dynamic bifurcation of the bioreactor model. General conditions for the existence of Hopf points with respect to the selected growth kinetics are derived. The periodic behavior is examined for both constant and variable (substrate dependent) yield coefficient. 1: Schematic diagram of a bioreactor with cell recycle. 8) where µm is the maximum specific growth rate, ks is the saturation constant, and ki is the substrate inhibition constant.

11). 4: Normal forms and unfoldings for H10 and H20 singularities Type Normal form Universal unfolding Nondegeneracy conditions H10 x(x4 ± θ) x(x4 ± θ + αx2 ) a01 > 0(or < 0), a20 > 0 H20 x(x6 ± θ) x(x6 + ±θ + α1 x2 + α2 x4 ) a01 > 0(or < 0), a30 > 0 of H10 and H20 singularities. The definition of the coefficients describing the degeneracy conditions are also given in the appendix. 1 Introduction We start our study with the basic unstructured model of the ideal chemostat. 1. The purge fraction W containing the biomass is operated directly from the bioreactor.

A) stable node when the eigenvalues are unequal; (b), (c) proper and improper stable nodes when the eigenvalues are equal. = 0, and are called steady states. 1)) satisfy dx dt They are also called stationary points, equilibrium points, or critical points. 2) may have more than one solution xs . In this case we have a situation of multiple steady states, also called steady-state (static) multiplicity. 1) when it is linearized around xs . 2: (a) Saddle steady state; (b) Stable focal steady state; (c) Limit cycle.

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