Competitor analysis chapter

The inclusion of this chapter on SPM, in what otherwise would have been a book on surface analysis dealing with the traditional" techniques, will hopefully serve the additional purpose of demonstrating the merits of the two communities in being companions rather than competitors. The chapter will attempt to demonstrate the complementarities between the new" and the "old" techniques (see Table 1) as well as showing the common elements in the method-... 

Figure 1.1 shows the device used by Lovitt and Wimpenny, and Figure 1.2 is a schematic of the device to be analyzed in this chapter. The mathematical analysis is restricted to two vessels, two competitors, and one... 

There is an equilibrium point of the form (f, j, 0,0,0,0). This corresponds to both organisms washing out of the gradostat. In view of Lemma 3.1(a), there are potential equilibrium points of the form (5i,. S2, Wi, Ui, 0,0) and (5i, 52,0,0, Vi,V2), with all nonzero entries positive. These correspond to one of the competitors washing out of the gradostat. For coexistence to occur as a steady state, it must be shown that there also is an equilibrium point with all components strictly positive. The existence and the stability of these rest points are closely related, as the following sections show. To determine the stability of rest points it turns out that certain functions of the coordinates of the rest points must be evaluated, so it is important to be able to compute them explicitly. Our inability to make these computations when the number of vessels is more than two restricts the analysis, as the next chapter will show. 

Competition in a modified gradostat was considered in Smith and Tang [STa], There the rate E between the vessels (called the communication rate) was allowed to differ from the rate D (the dilution rate) from the feed bottle and to the overflow vessel. This, of course, still maintains the assumption that the volume in each vessel is constant. It was shown that the outcome of competition can be sensitive to the ratio E/D in the following sense As E/D is increased, first one competitor wins the competition, then coexistence occurs, and finally the second competitor wins. The analysis in the case E D k entirely similar to the case E = D discussed in this chapter. In [STa], a number of operating diagrams were determined numerically. For fixed population parameters fly, these oper-... 

The model in Chapter 4 had only one competitor taking up the inhibitor, but this is only a first approximation. One could justify this assumption only by showing that nothing changes if the system is modified to allow for uptake by both competitors. However, such modification complicates the analysis. Finally, could it be that a threshold amount needs to be consumed before the inhibitor is effective Adding thresholds to the model would be an interesting modeling task. 

Similarly, our analysis of the variable-yield model in Chapter 8 is limited to two competing populations because we rely on the techniques of monotone dynamical systems theory. One would expect the main result of Chapter 8 to remain valid regardless of the number of competitors, just as it did for the simpler constant-yield model treated in Chapters 1 and 2. Perhaps the LaSalle corollary of Chapter 2 can be used to carry out such an extension, using arguments similar to those used in [AM] (described in Chapter 2). As noted in [NG], a structured model in which... 

Section 3.2 of this chapter tackles the issues concerning the visibility of costing information. This form of analysis can be applied to benchmark an existing operation with a competitor, or it can be used to assess the implications on ROI against potential trade-offs (see section 1.4.3), such as comparing an in-house operation with a third party outsourcing alternative. 

Researchers fundamentally interested in C-C bond-forming methods for polyketide synthesis have at times viewed allylation methods as alternatives, and maybe even competitors, to aldol addition reactions. Both areas have dealt with similar stereochemical problems simple versus absolute stereocontrol, matched versus mismatched reactants. There are mechanistic similarities between both reaction classes open and closed transition states, and Lewis acid and base catalysis. Moreover, there is considerable overlap in the prominent players in each area boron, titanium, tin, silicon, to name but a few, and the evolution of advances in both areas have paralleled each other closely. However, this holds for an analysis that views the allylation products (C=C) merely as surrogates of or synthetic equivalents to aldol products (C=0). The recent advances in alkene chemistry, such as olefin metathesis and metal-catalyzed coupling reactions, underscore the synthetic utility and versatility of alkenes in their own right. In reality, allylation and aldol methods are complementary The examples included throughout the chapter highlight the versatility and rich opportunities that allylation chemistry has to offer in synthetic design. 

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