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

By Abdelhamid Ajbar

A ubiquitous instrument in mathematical biology and chemical engineering, the chemostat frequently produces instabilities that pose safeguard dangers and adversely impact the optimization of bioreactive structures. Singularity thought and bifurcation diagrams jointly provide an invaluable framework for addressing those concerns. in keeping with the authors’ broad paintings during this box, Dynamics of the Chemostat: A Bifurcation concept Approach explores using bifurcation conception to investigate the static and dynamic habit of the chemostat.

Introduction
The authors first survey the main paintings that has been performed at the balance of continuing bioreactors. They subsequent current the modeling methods used for bioreactive structures, the various kinetic expressions for progress charges, and instruments, reminiscent of multiplicity, bifurcation, and singularity conception, for reading nonlinear systems.

Application
The textual content strikes directly to the static and dynamic habit of the elemental 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 straightforward microbial pageant, biodegradation of combined substrates, dynamics of plasmid-bearing and plasmid-free recombinant cultures, and dynamics of predator–prey interactions. The authors additionally research dynamics of the chemostat with product formation for numerous development types, supply examples of bifurcation idea for learning the operability and dynamics of constant bioreactor versions, and follow ordinary strategies of bifurcation conception to research the dynamics of a periodically pressured bioreactor.

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

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Extra info for Dynamics of the chemostat : a bifurcation theory approach

Example text

2b. Any abrupt change in the reactor operating parameters can cause the system to jump from a high conversion point to a low conversion operating point. 52358 and K = 4. 2d, for values of S¯f = 3 and K = 4. Isola and Mushroom The second possible qualitative change that can occur in the steady-state locus is the appearance of an isola and the growth of an isola into a mushroom. 2: (a) Branch sets for the hysteresis singularity; (b) Continuity diagram showing hysteresis in region H1 ; (c) Continuity diagram for the boundary H0 ; (d) Continuity diagram for region H2 ; solid line, stable branch; dashed line, unstable; circle, limit point.

But there exists a set of values of δ at which this function has the most degenerate singularities. Perturbing the system around these points results in forming several qualitatively different stable bifurcation diagrams. These singular points are characterized by the vanishing of a finite number of derivatives of f with respect to x and with respect to the bifurcation parameter θ. In addition, there always exists a normal form that is contact equivalent to f next to these singular points. We present in the following [33] a special case of a more general theorem proved by Golubitsky and Schaeffer [132].

52) [33, 366]. The periodic orbit is asymptotically stable when a is negative and unstable for positive values of a. 10). 10). 10: Examples of (a) Subcritical periodic branch; (b) supercritical; (c) at least one eigenvalue is positive. section, we discuss the degeneracies associated with the failure of H2 and H3 hypotheses. But first, we introduce the following definition for a general high order degeneracy of these two types. 57) where a1 = a is the stability coefficient. 59) where ν j denotes the derivative of order j of ν with respect to the bifurcation parameter θ.

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