Constant-yield model

In the classical model of the chemostat, discussed in Chapter 1, it is assumed (following Monod [Mol Mo2]) that the nutrient uptake rate is proportional to the reproductive rate. The constant of proportionality, which converts units of nutrient to units of organism, is called the yield constant. As a consequence of the assumed constant value of the yield, the classical model is sometimes referred to as the constant-yield model. 

The conclusions of Theorem 6.1 correspond precisely to those of Theorem 5.1 of Chapter 1 and Theorem 3.2 of Chapter 2. In fact, following Grover [G2], a constant-yield model can be associated with (3.1) in such a way that both models give the same predictions (this is not proved in [G2]). Consider the case where both Ei and 2 exist. Omit from (3.1) the equations for Qj and substitute... 

The predictions of the variable-yield model (3.1) and the corresponding constant-yield model (7.1) are identical. Typical solutions of each model approach the corresponding equilibrium in a monotone fashion (see Proposition 5.3). 

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... 

The effect on the normalized approach curves of allowing to take finite values is illustrated in Fig. 5, which shows simulated data for three rate constants, for redox couples characterized by y = 1. The rate parameters considered K = 100 (A), 10 (B), and 1 (C), are typical of the upper, medium, and lower constants that might be encountered in feedback measurements at liquid-liquid interfaces. In each case, values of = 1000 or 100 yield approach curves which are identical to the constant-composition model [44,47,48]. This behavior is expected, since the relatively high concentration of Red2 compared to Red] ensures that the concentration of Red2 adjacent to the liquid-liquid interface is maintained close to the bulk solution value, even when the interfacial redox process is driven at a fast rate. 

Astronomical Observatory, were used to carry out the calculations of theoretical equivalent widths of lines, synthetic spectra and a set of plane parallel, line-blanketed, flux constant LTE model atmospheres. The effective temperatures of the stars were determined from photometry, the infrared flux method and corrected, if needed, in order to achieve the LTE excitation balance in the iron abundance results. The gravities were found by forcing Fe I and Fe II to yield the same iron abundances. The microturbulent velocities were determined by forcing Fe I line abundances to be independent of the equivalent width. For more details on the method of analysis and atomic data see Tautvaisiene et al. (2001). 

An example of the use of this method with the constant capacitance model on the data for TiC>2 in 0.1 M KNO is illustrated in Figure 6. It appears from the figure that the problem is perfectly well determined, and that unique values of Ka and Ka2 can be determined. However, as is shown below, the values of Ka and Ka2 determined by this method are biased to fulfill the approximations made in processing the data (i) on the acidic branch, nx+, nx nx-, which yields a small value for Ka2, and (ii) on the basic branch, nx-, nx nx+, which yields a large value of Ka. Thus the approximation used to find values for Qa and Qa2 leads to values of Ka and Ka2 consistent with the approximation of a large domain of predominance of the XOH group. This constraint arose out of the need for mathematical simplicity, not out of any physical considerations. 

Figure 7. Covariability between values of C and Kd yielding best fit of diprotic surface hydrolysis model with constant capacitance model to titration data for TiC>2 in 0.1 M KNOj (Figure 5). The line is consistent with Equation 29. The crosses represent values of C and log found from a nonlinear least squares (NLLS) fit of the model to the data, with the value of capacitance imposed in all cases the fit was quite acceptable. The values of and C found by Method I (Figure 6) also fall near the line consistent with Equation 29. The agreement between these results supports the use of the linearized model (Equation 29) for developing an intuitive feel for surface reactions.

In these model calculations, the best fit to the observed spectrum was obtained with the butane carbon plane parallel to the surface with bonds to the bottom layer of hydrogens (Fig. 6(a)). The agreement worsened when carbon-substrate bonds were included in this orientation and became quite bad when the hydrogen-substrate bonds were neglected entirely (Fig. 6(b)). With the carbon plane perpendicular to the surface (Fig. 6(c)), no combination of force constants yielded as good a fit as in the plane-parallel configuration. 

For describing the elementary processes the transition state model (TSM) [20-23,57] is usually used. The TSM gives a clearly defined dependence of the preexponential factor on the type of the energy redistribution over the internal degrees of freedom of the reagents in the activated state. The reaction rate constant yields the following equation [20-23,57] ... 

One of the most attractive features of the CoMFA and CoMFA-like methods is that, because of the nature of molecular field descriptors, these approaches yield models that are relatively easy to interpret in chemical terms. Famous CoMFA contour plots, which are obtained as a result of any successful CoMFA study, tell chemists in rather plain terms how the change in the compounds size or charge distribution as a result of chemical modification correlate with the binding constant or activity. These observations may immediately suggest to a chemist possible ways to modify molecules to increase their potencies. However as demonstrated in the next section, these predictions should be taken with caution only after sufficient work has been done to prove the statistical significance and predictive ability of the models. 

G2] J. P. Grover (1992), Constant- and variable-yield models of population growth Responses to environmental variability and implications for competition, Journal of Theoretical Biology 158 409-28. 

Many attempts have been made to predict fracture toughness Kj(V ) of polymers as a function of material structure. Such calculations can start from statistical and molecular points of view, or from rheological points of view . In the context developed above, the fracture toughness may be expressed as a function of Xo in the following way if a simple constant stress model (i.e. the Dugdale model) is assumed along the craze boundary, then the fracture toughness yields ... 

In this work, a small-scale yielding model of a stationary crack is assumed, with the remote boundary being subjected to a mode I loading at a constant loading rate f i. The crack initially has a blunt tip with a radius n. Crack growth by crazing is allowed to occur only along the plane of the crack fe = 0). 

Nature of the Surface Complexes. The constant capacitance model assumes an inner-sphere molecular structure for surface complexes formed in reactions like equation 5a or 7. But this structure does not manifest itself explicitly in the composition dependence of Kc everything molecular is buried in which is an adjustable parameter. This encapsulating characteristic of the model was revealed dramatically by Westall and Hohl (13), who showed that five different surface speciation models, ranging from the Gouy-Chapman theory to the surface complex approach, could fit proton adsorption data on AL O., equally well, despite their mutually contradictory underlying molecular hypotheses [see also Hayes et al. (19)]. They concluded that "... no model will yield an unambiguous description of adsorption. .. . To this conclusion one may add that no model should provide such a description,... 

INS experiments evidence decoupling of the proton bending modes from carbonate entities [Fillaux 1988 Kashida 1994], Simulations of the spectral profile with valence-bond force-field models based on infrared and Raman spectra [Nakamoto 1965], yield spectacular differences between observation and calculations. Discrepancies arise from the force-field representation itself and cannot be eliminated by straightforward adjustment of the force constants. The model protons, bound to oxygen atoms by strong forces, ride displacements at low frequency of carbonate entities, mainly below 200 cm-1. Calculated intensities for these lattice modes are overestimated by at least one order of magnitude. 

In a critical evaluation of the results of anharmonic force constant calculations it is necessary to consider all the factors discussed above concerning the completeness, internal consistency, and accuracy of the data, the possible idiosyncrasies of the model potential used, the method of reducing the data, as well as the goodness of fit the force constants yield for the data points. It may be in order to mention that... 

Within the anisotropic cylindrical cell model the pressure is related to volume changes leaving the direction along the rod invariant. Simulations constantly yield a smaller osmotic coefficient than predicted by Poisson-Boltzmann theory. For multivalent systems it can even become negative. 

The crucial parameters are the concentration ks at which uptake is half of the maximum rate and the fixed ratio (or yield) q l at which nutrient is converted to biomass. The yield may be the Redfield ratio or some other optimum composition. Tilman etal. (1982) used the model to show how freshwater phytoplankters of different optimum composition or different half-saturation concentrations, might succeed to different extents depending on the ambient ratios of nutrient elements. Although the assumption of constant yield may be appropriate for pelagic heterotrophs, it is now seen to be too simple for accurate prediction of the growth of phytoplankters (Droop, 1983 Sommer, 1991 Ducobu etal., 1998). 

Fractionating effects of the escape process can now be calculated analytically if specific assumptions are made about the time dependence of the major (constituent 1) inventory Ni— that it is either replenished as fast as it escapes (constant inventory model), or is lost without replenishment along with the minor atmospheric species (Rayleigh fractionation model). In both cases the inventories N2 of minor species, here the noble gases, are assumed to be in the atmosphere at to and are lost without replenishment during the escape episode. For Rayleigh fractionation, adopted for this discussion. Equations (2), (3), and (7) and the definitions Fi = -dNi/dt and F2 = -dN2/dt may be combined and integrated, in the limit of Xi = 1, mc > m2 > mi, and mc > m2, to yield... 

In conventional fermentation models the central equation usually expresses cellular growth as a function of substrate, product and cellular concentration in the medium. Product formation is then related to cell growth or cellular concentration, whereas substrate uptake rate may include several contributions from cellular growth, maintenance and product formation. Moreover, these models most often are based on constant yields. 

---