Global Behavior of the Reduced System

Suppose that Ei and E2 exist and that (3.10) holds. Then 

In the next section, these local stability considerations will be shown to lead to corresponding global results. For this analysis, it will be important to approximate the one-dimensional unstable manifold of Ei when both El and 2 exist and (3.10) holds. To this end, we provide information on an eigenvector corresponding to the eigenvalue A] of f. Let x = (xi,Qi,X2,02) denote such an eigenvector. We find that 

The global asymptotic behavior of the reduced system, (4.3), is worked out in this section. The main result is stated immediately, for the convenience of the reader who may not wish to slog through the remainder 

Case (iv) is the interesting one, since both organisms can survive in the absence of competition in the chemostat. Recall that (3.10) is simply our convention of labeling as X2 the organism that can grow at the lowest nutrient concentration. 

The proof will be divided into various cases and presented as separate propositions. The key to the proof is the use of new variables defined by 

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In Section 5, the global behavior for the reduced system (4.3) was determined. It remains to show that the results obtained for this system carry over to the original model system (3.4). This will be done by making a change of variables in (3.4) and using the results of Appendix F. 

Global behavior of the system is determined by a single parameter, the reduced charge Q relative distance z is the internal variable, defined by the equilibrium condition... 

The chapter proceeds as follows. In the next section the variable-yield model of single-population growth is derived and analyzed. In Section 3, the competition model is formulated and its equilibrium solutions identified. The conservation principle is introduced in Section 4 in order to reduce the dimension of the system of equations by one local stability properties of the equilibrium solutions are also determined. The global behavior of solutions of the reduced system is treated in Section 5, and the global behavior of solutions of the original competitive system is discussed in Section 6. The chapter concludes with a discussion of the main results. 

The role of mixing has been studied in systems with more complex reaction schemes or considering more complex fluid-dynamical properties, and in the context of chemical engineering or microfluidic applications (for reviews on microfluidics see e.g. Squires (2005) or Ottino and Wiggins (2004)). Muzzio and Liu (1996) studied bi-molecular and so-called competitive-consecutive reactions with multiple timescales in chaotic flows. Reduced models that predict the global behavior of the competitive-consecutive reaction scheme were introduced by Cox (2004) and by Vikhansky and Cox (2006), and a method for statistical description of reactive flows based on a con-... 

The theorem shows clearly that plasmid loss is detrimental (or fatal) to the production of the chemostat. To compensate for this possibility, in commercial production a plasmid that codes for resistance to an antibiotic is added to the DNA that codes for the item to be produced. Thus, if the plasmid is lost then the wild type is susceptible to (inhibited by) the antibiotic. The antibiotic is introduced into the feed bottle along with the nutrient. The dynamics produced by adding an inhibitor to the chemostat was modeled in Chapter 4. A new direction for research on chemostat models would be to include the inhibitor, as in Chapter 4, and the plasmid model of this section (or one of the more general models) into the same model. This is a mathematically more difficult problem to analyze, since the reduced system will not be planar. Moreover, because the methods of monotone dynamical systems do not apply, other techniques would need to be found in order to obtain global results. The model also assumes extremely simple behavior for the plasmid more could be included in a model. 

Verification has also been performed directly on the DFG [17]. This is a highly interesting perspective, since data flow graphs are widely used as the starting point for synthesis. The verification here is a more global data-independent scope, allowing the verification of much larger systems at the cost of a reduced resolution detailed, data-dependent functional behavior is not modeled. 

Conduction and transport are two intimately related processes. Condnction is generally seen as a flow of entities (charges, molecnles, etc.) throngh a piece of material having an extension throngh space, so the movement of entities is considered as a transport under the influence of a difference of efforts (although it can be the converse). Condnctance is the system constitutive property that models the behavior of conduction at the global level. As transport requires space, it is not the conductance that is really able to model transport but the spatially reduced property called conductivity. 

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