Competitive system

Entities involved in long-term contracts with electric utihties, such as fuel supphers and NUGs selling power to utihties, also have concerns that some utihties or industrial customers will not be able to honor their contracts under the new, more competitive system. Einahy, some utihties are concerned that they wih not be adequately reimbursed for opening up their transmission systems to competitors. The potential competitors in turn are concerned that utihties whl not provide unbiased access to their transmission systems if the utihties themselves are also in business of marketing power. There has also been some debate regarding which transmission facihties are eligible for open access. This is because some facihties are considered local distribution systems by utihties, which feel they should not be opened to competitors. 

As crude oil reserves dwindle, the marketplace will either transition to the electrifying of the transportation system (electric and fuel-cell vehicles and electric railways), with the electricity being produced by coal, natural gas, nuclear and renewables, or see the development of an industry to produce liquid fuel substitutes from coal, oil shale, and tar sands. It might also turn out to be a combination of both. The transition will vary by nation and will be dictated strongly by the fuels available, the economic and technological efficiencies of competitive systems, the relative environmental impacts of each technology, and the role government takes in the marketplace. 

This holds for noncatalytic reactions both isolated and in competitive system, as well as for isolated catalytic reactions. The rate of catalytic reaction in competitive (and generally in any coupled) system depends, however, on the concentrations of all the compounds present in the system, insofar as they are adsorbed on the same active centers on which the given reaction is taking place. 

This reaction was studied in the gaseous phase at 120°C with a sulfo-nated organic ion exchanger as catalyst (p. 27), under both the competitive system (Villa) (2 esters + 1 alcohol) and the system (VUIb) (1 ester + 2 alcohols)... 

Dimension of constants k1 is (mole X 102 hr-Ikg-Iatm-2) and of adsorption coefficients K, is (atm-1)-b From the study of competitive system (Villa). 

From the study of competitive system (Vlllb). d From the rate data of the transformation of the given substance. 

Since two mechanisms are possible for the competition between association and reaction, detailed ab initio calculations of the potential surface are even more necessary in theoretical determinations of the rates of association channels. More experimental work is also needed it is possible that as a larger number of competitive systems is studied, our understanding of the competition will increase. Critical systems for interstellar modeling include the association/reactive channels for C+ and bare carbon clusters, as well as for hydrocarbon ions and H2. 

In immunometric assays, unlike competitive systems, the amount of labelled antibody bound is directly proportional to the amount of unlabelled antigen present rather than inversely proportional. 

Enzyme labels are usually associated with solid-phase antibodies in the technique known as enzyme-linked immunosorbent assay (ELISA). There are several variants of this technique employing both competitive and non-competitive systems. However it is best used in combination with two monoclonal antibodies in the two-site format in which an excess of antibody is bound to a solid phase such as a test-tube or microtitre plate the test antigen is then added and is largely sequestered by the antibody (Figure 7.12). After washing... 

This type of approach is essentially non-competitive and usually requires the use of two monoclonal antibodies directed against two distinct epitopes on the analyte. Other devices have employed a two-stage competitive system in which analyte and labelled analyte compete for antibody in one part of the device. This is followed by transfer of the equilibrium mixture to a separate part of the device where membrane-immobilized antibody removes the unbound labelled material and allows the bound to go through the membrane into the absorbent pad. 

Batteries are likely to find an increasing application in this role. Although electrochemical storage may prove a less cost-effective alternative in some situations, it has many advantages. Batteries have a much shorter lead time in manufacture than most competitive systems, and being modular (unlike a hydroelectric dam or compressed air store), the energy storage facility can be added to, split into smaller units, or even transported... 

The power sector can and does pass through the bulk of marginal/opportunity Correlated costs to the wholesale power markets, as expected in a competitive system, resulting in substantial profits and downstream costs where retail markets are competitive. 

Process parameters for a commercial system will evolve as the technology matures. Conceptual design studies (References l.,.2>3) and experimental analyses have, however, led to the selection of ranges of interest and objectives for major process parameters that should result in a technically sound and economically competitive system. Table I summarizes these values. 

Since 1 is a local attractor, to prove the theorem it remains only to show that it is a global attractor. This is taken care of by the Poincare-Bendixson theorem. As noted previously, stability conditions preclude a trajectory with positive initial conditions from having 0 or 2 in its omega limit set. The system is dissipative and the omega limit set is not empty. Thus, by the Poincare-Bendixson theorem, the omega limit set of any such trajectory must be an interior periodic orbit or a rest point. However, if there were a periodic orbit then it would have to have a rest point in its interior, and there are no such rest points. Hence every orbit with positive initial conditions must tend to j. (Actually, two-dimensional competitive systems cannot have periodic orbits.) Figure 5.1 shows the X1-X2 plane. 

We interrupt the analysis of the inhibitor model in order to present some theorems on competitive systems which are needed in the analysis. Consider the system... 

In Remark 6.1 we have used < rather than because K is the usual positive cone. A consequence of these remarks is that planar cooperative or competitive systems do not have periodic orbits. For example, consider a planar cooperative system and let P be an arbitrary point on a periodic orbit. Impose the standard two-dimensional coordinate system at P. The orbit cannot be tangent to both the x and the y axes at P and so must have points in both quadrants II and IV (the sets unordered with respect to P), since points in quadrant I or III would be ordered. The orbit cannot pass through P again and cannot have points in quadrants I... 

A major difference between competitive and cooperative systems is that cycles may occur as attractors in competitive systems. However, three-dimensional systems behave like two-dimensional general autonomous equations in that the possible omega limit sets are similarly restricted. Two important results are given next. These allow the Poincare-Bendix-son conclusions to be used in determining asymptotic behavior of three-dimensional competitive systems in the same manner used previously for two-dimensional autonomous systems. The following theorem of Hirsch is our Theorem C.7 (see Appendix C, where it is stated for cooperative systems). 

Theorem 6.3 [Hi4]. Let L be a compact omega limit set of a competitive system in If L contains no equilibria, then L is a closed orbit. 

Theorem 7.1 guarantees the coexistence of both the Xi and X2 populations when Ec exists. However, it does not give the global asymptotic behavior. The further analysis of the system is complicated by the possibility of multiple limit cycles. Since this is a common difficulty in general two-dimensional systems, it is not surprising that such difficulties occur in the analysis of three-dimensional competitive systems. 

The requirement (4.2) means that (4.1) is a competitive system - an increase in X2 has a negative effect on the growth rate of x and vice versa. The system is said to be a cooperative system if the reverse inequalities hold in (4.2). Our interest here will be in the competitive case since (3.2) satisfies (4.2) in 2, as is easily checked. However, the cooperative case is... 

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. 

Theorem C.7 bears a strong resemblance to the Poincare-Bendixson theorem stated in Chapter 1. It will be used in Chapter 4 for the case where (C.l) is a competitive system, that is, for a system (C.l) where -/ is cooperative. Note that the omega (alpha) limit set of a competitive system is the alpha (omega) limit set of the time-reversed cooperative system, so Theorems C.5, C.6, and C.7 apply to competitive systems. Unlike cooperative systems, competitive systems can have attracting periodic orbits. For more on the Poincare-Bendixson theory of competitive and cooperative systems in see [S3], [SWl], and [ZS]. 

DS] P. deMottoni and A. Schiaffino (1982), Competition systems with periodic coefficients A geometric approach, Journal of Mathematical Biology 11 319-35,... 

SWl] H. Smith and P. Waltman (1987), A classification theorem for three dimensional competitive systems, Journal of Differential Equations 70 325-32. 

---