Logistic equation/function

The biomass concentration A1 is a function of time so that the right hand side of equation 5.225 cannot be integrated directly, but the form of the integral will be that of the logistic equation (see equation 5.61). If, therefore, that integral is replaced by the logistic expression then, after rearranging, equation 5.225 becomes ... 

We assume that the population growth can be described by the logistic growth function F p) = p( - /o/Pmax) where p is the saturation density or carrying capacity. This growth function compares favorably with a wealth of experimental results [256]. Equation (7.1) leads to wavefronts with asymptotic velocity, see (5.60),... 

Fig. 15.4. The diagram or the fixed points and the limit cycles for the logistic equation as a function of the coupling constant K. From J. Gleick, Chaos , Viking, New York, 1988, reproduced with permission of the author.

Fig. 3.10 Decrease of concentration as a function of f/k (Fujii 1977). LE logistic equation model, KW Kitahara and Wollast model, MM Margenau and Murphy model (Margenau and Murphy 1956)...

FIGURE 11.14 Data set consisting of a control dose-response curve and curves obtained in the presence of three concentrations of antagonist. Panel a curves fit to individual logistic functions (Equation 11.29) each to its own maximum, K value, and slope. Panel b curves fit to the average maximum of the individual curves (common maximum) and average slope of the curves (common n) with only K fit individually. The F value for the comparison of the two models is 2.4, df = 12,18. This value is not significant at the 95% level. Therefore, there is no statistical support for the hypothesis that the more complex model of individual maxima and slopes is required to fit the data. In this case, a set of curves with common maximum and slope can be used to fit these data. 

FIGURE 5.9 Visualization of Equations 5.17 and 5.18 for modeling the posterior probabilities with LR. The left-hand plot pictures the logistic function P z) = ez/( 1 + ez) the right-hand plot shows P2(z) = 1 - Pi(z). 

Another method of detecting a dose-response relationship is to fit the data to various models for dose-response curves. This method statistically determines whether or not a dose-response model (such as a Logistic function) fits the data points more accurately than simply the mean of the values this method is described fully in Chapter 12. The most simple model would be to assume no dose-response relationship and calculate the mean of the ordinate data as the response for each concentration of ligand (horizontal straight line parallel to the abscissal axis). A more complex model would be to fit the data to a sigmoidal dose-response function (Equation 11.2). A sum of squares can be calculated for the simple model (response — mean of all response) and then for a fit of the data set refit to the four parameter Logistic shown... 

Figure 6.1 Average plankton population density as a function of the Damkohler number for logistic growth with non-uniform carrying capacity of the form K(x,y) = Kq + (5sin(27rx) sin(27ry) and chaotic mixing in the time-periodic sine-flow of Eq. (2.66). The continuous line represents results from the solution of the full partial differential equation with diffusion (Pe 104) and stars ( ) show the time-averaged plankton populations calculated from the non-diffusive Lagrangian representation.

Given that in this model the receptor and adenylate cyclase are separate entities, it is possible to take into account the independent variation of these two parameters. Moreover, the model based on receptor desensitization takes explicitly into account the activity of the two forms of phosphodiesterase. For each of these four parameters, namely the activity of adenylate cyclase, of intra- and extracellular phosphodiesterase, and the quantity of receptor present at the surface of the membrane, the variation observed in the hours that follow starvation takes the form of a sigmoidal increase as a function of time (fig. 7.3a-c). The evolution of each of these parameters can thus be described, to a first approximation, by an equation of the logistic type (Goldbeter Martiel, 1988 Martiel, 1988). 

The logistic map, referred to previously, is a now famous, but simple, quadratic function that displays the same period-doubling cascade into chaos exhibited by the Rossler system (and by quite a few experimental examples, as well). A map is an equation that gives an iterative rule for generating a sequence of points x, Xj, X3,... given an initial value Xq. The logistic map is given by... 

The asymptotic velocity depends explicitly on the shape of the initial conditions, if they do not have compact support. An adaptation of the Hamilton-Jacobi theory from classical mechanics is a usefiil technique to deal with this problem in a very general way, see below. The prototypical example of a concave reaction term is the KPP or logistic term F p) = rp — p). Equation (4.13) implies that v = 2 /rD. Examples for convex reaction functions typically occur in combustion theory, where F p) = — p) is referred to as the Arrhenius reaction term, or F p) =... 

This time-dependent solution (89) substitutes an elementary logarithinic dependency for the W-Lambert function. It is nevertheless remarkable that the solutirm of a generalized logistic kinetic version of the Michaelis-Menten instantaneous equation provides an analytically exact solution. It clearly reduces to the above Eq. (74) in the first order expansion of the chemical concentration time evolution with respect to the 50-effect concentration (EC50) observed. 

In preparative chromatography, the influence of salt, or, more generally speaking, the mobile phase modifier, on adsorption has to be taken into account [46]. The aforementioned equations do not reflect this situation. We have developed an empirical approach where we have combined the conventional isotherms and the dependence on salt using a logistic dose-response function [44]. Velayudhan and Horvath [47] have described a similar approach—the mobile phase modified adsorption isotherm. They assume that the maximum binding capacity is constant however, this assumption does not reflect real conditions in preparative chromatography. 

The logistics, weather constraints and the O M tasks are modelled and simulated using Generalized Stochastic Petri Nets with predicates and Monte Carlo simulation. The q and age dependent cost function for PM repairs (Equation 7) is compared with the cost function in terms of the effect on the turbine s O M costs and revenues. 

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