The Competition Model

Consider two populations, with densities Xx andA 2, competing for a single nutrient of concentration S in the chemostat. Competition occurs in the sense that each population consumes nutrient and so makes it unavailable for the competitior. The average amount of stored nutrient per individual of population Xx is denoted by Qx, and for population X2 by 02-Following the derivation of Section 2, we have the following equations  

The functions Hi(Qt) and p,(S, Q,) are, respectively, the per-capita growth rate and the per-capita uptake rate of population x,. We assume that p, is defined and continuously differentiable for Qj P, where P, 0, and satisfies  

Observe that (3.2) and (3.3) imply that 0/ 0 if 0, = P, and that therefore the interval of 0, values, [P, ), is positively invariant under the dynamics of (3.1). Biologically relevant initial values for (3.1) are 

Continuing as in Section 2, we scale the variables appearing in (3.1) as follows  

The Q are arbitrarily chosen representative values of the variables Q,. If we define 

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In its simplest form the competition model assumes the entire adsorbent surface is covered by a monolayer of solute and mobile phase molecules. Under normal chromatographic conditions, the concentration of sample molecules will be small and the adsorbed monolayer will consist mainly of mobile phase molecules. Retention of a solute molecule occurs by displacing a roughly equivalent volume of mobile phase molecules from the monolayer to make the surface accessible to the adsorbed solute aiolecule. For elution of the solute to occur -the above process must be reversible, and can be represented by the equilibrium depicted by equation (4.6)... 

The competition model and solvent interaction model were at one time heatedly debated but current thinking maintains that under defined r iitions the two theories are equivalent, however, it is impossible to distinguish between then on the basis of experimental retention data alone [231,249]. Based on the measurement of solute and solvent activity coefficients it was concluded that both models operate alternately. At higher solvent B concentrations, the competition effect diminishes, since under these conditions the solute molecule can enter the Interfacial layer without displacing solvent molecules. The competition model, in its expanded form, is more general, and can be used to derive the principal results of the solvent interaction model as a special case. In essence, it seems that the end result is the same, only the tenet that surface adsorption or solvent association are the dominant retention interactions remain at variance. 

The retention behavior of solutes in adsorption" chromatography can be described either by the "competition" model or by the "solute-solvent interaction" model depending on the eluent composition. It appears that both mechanisms are operative but their importance depends on the composition of the eluent mixture 84). 

The solute competes with eluent molecules for the ac ve adsorption sites on the surface of the stationary phase. Interactions tetween solute and solvent molecules in the liquid phase are cancelled by milar interactions in the adsorbed phase. This model has been introdu d by Snyder (2) and by Soczewinski (77, 78) and is called the competition model."... 

The competition model of Snyder assumes that the adso tion surface is completely covered by adsorbed eluent molecules forming a mono-layer. When solute molecules are adsorbed they displace olvent molecules. Due to the size differences one or more eluent mofecules are displaced by the solute molecules. The adsorbent surface is ijissumed to be homogeneous and each molecule tends to interact totally wRh the surface, i.e., it is adsorbed flatwise. Thus, the adsorbent surface area that the molecules require can be calculated from their molecular dimensions. Neglecting the interactions between solute and eluent molecules in the liquid and the adsorbed phase, the retention of an adsorbed molecule (expressed as net retention volume per unit weight of adsorbent K) can be related to the properties of the stationary phase, the eluent, and the sample by... 

Results from sensory evaluation of mixed solution are seen in Table IV. The data list the theoretical response for both the independent and competitive receptor hypothesis as well as the actual sensory score. The actual sensory scores were found to agree fairly well with the competitive model. The minor dissimilarity between the actual and theoretical is due to the inability of individual to taste bitterness in solutions that are extremely sweet, i.e., there is some masking of overall sensory perception which is concentration dependent. The data, therefore, clearly indicate that sweetness and bitterness act in a competitive manner and should be considered to compete for the binding sites at the same receptor. 

Due to the ongoing growth of Automobiltechnik Blau and the competitive model a further investment into the Chemical Leasing model is expected for the next future. 

Two models have been developed to describe the adsorption process. The first model, known as the competition model, assumes that the entire surface of the stationary phase is covered by mobile phase molecules and that adsorption occurs as a result of competition for the adsorption sites between the solute molecule and the mobile-phase molecules.1 The solvent interaction model, on the other hand, suggests that a bilayer of solvent molecules is formed around the stationary phase particles, which depends on the concentration of polar solvent in the mobile phase. In the latter model, retention results from interaction of the solute molecule with the secondary layer of adsorbed mobile phase molecules.2 Mechanisms of solute retention are illustrated in Figure 2.1.3... 

Snyder [350] has given an early description and interpretation of the behaviour of LSC systems. He explained retention on the basis of the so-called competition model . In this model it is assumed that the solid surface is covered with mobile phase molecules and that solute molecules will have to compete with the molecules in this adsorbed layer to (temporarily) occupy an adsorption site. Solvents which show a strong adsorption to the surface are hard to displace and hence are strong solvents , which give rise to low retention times. On the other hand, solvents that show weak interactions with the stationary surface can easily be replaced and act as weak solvents . Clearly, it is the difference between the affinity of the mobile phase and that of the solute for the stationary phase that determines retention in LSC according to the competition model. 

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 calculation of solvent (elution) strength parameters by the competition model is rather involved and a more empirical approach can be justified for routine purposes or for the separation of simple mixtures. For method development an estimate of solvent... 

The consequences of multisite attachment include the following. First, the competition model 18,45) can be construed to identify Z with the number of attachments of the solute molecule to the stationary phase in any case Z is expected to be proportional to this number. Consequently, large molecules will have larger values of 5 and Z as compared to small molecules. This is recognized in our version of conventional HPLC theory as applied to macromolecule samples. 

These considerations and results obtained from measurement of changes in macromolecules and their precursor pools throughout the growth cycle of B. amyloliquefaciens (Stormonth and Coleman, 1974) were used, together with the most reputable data available at the time, to develop the "competition" model of regulation based on the earlier proposal of Coleman (1967). Thus, Coleman et al. (1975) suggested that the rate of total exoprotein formation was coupled to growth by an inverse relationship. Two separate effects were considered to be involved, one of which was superimposable on the other. 

The "competition" model provided an explanation of the characteristics of secretion of those extracellular enzymes which are formed at the maximum rate after the end of exponential growth and which account for a significant part of the bacterium s total protein output. A mathematical model of exoprotein production in bacteria has been proposed, recently, which is consistent with the "competition" mechanism (Coleman and Fowler, 1984). It should be emphasized that the ideas implicit in such a mechanism may have been overlooked by those interested in only a single extracellular enzyme rather that the summation of all the individual extracellular proteins secreted by an organism. However, within its framework the opportunity for individual components to respond to other specific regulatory devices is not excluded. 

A variant of the competition model has been postulated by Osmond and Allaway (1974). These authors assume competition between PEP-C and RudP-C pathways for C3 carbon compounds rather than by CO2. 

This version of the competition model avoids the inconsistencies arising from the Km values and activity ratios of both carboxylases as outlined above. However, malic acid accumulation occurring in the light during the time of the regular dark period cannot be explained. 

Effective carbon flow from malic acid to carbohydrate with CO2 as an intermediate could be explained by the postulate that PEP-C might be temporarily suppressed as a competitor for CO2, thus interference with RuDP-C would be avoided. Both, the biological clock model and the competition model (see above) account for this postulate. However, considering the obvious limitations of these... 

Building competitive platforms that are grounded in this idea of value-based growth will require a much greater focus on managing the core processes that we referred to earlier. Whereas the competitive model of the past relied heavily on product innovation this will have to be increasingly supplemented by process innovation. The basis for competing in this new era will be ... 

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