Logistical equation

The discussion of the randomness of the Bernoulli map iterates therefore applies equally well to the behavior of the a = 4 attractor set of the logistic equation. 

A simple two-dimensional analogue of the logistic equation, introduced by Henon in 1976 [henou76], is defined by... 

Fig. 4.13 Lyapunov exponent versus a for 2.9 < a < 4 for the logistic equation see text.

Figures 8.8-a through 8.8-d show a few snapshots illustrating intermittent behavior. We have used the logistic-equation driven CML (equation 8.34) and set D = 0.25 and a = 3.83. Each figure shows 16 times overlapped after a certain number of iterations have elapsed (100 iterations in figure 8.7-a, 200 in figure 8,7-b, 300 in 8.7-c, and 500 in 8.8-d). Notice how, depending on the time of the snapshot, some regions are periodic and others are turbulent. Islands cf order tend to come and go as time progresses i.e. the local order is intermittent.

Note that while a system s static complexity certainly influences its dynamical complexity, the two measures are clearly not equivalent. A system may be structurally rather simple (i.e. have a low static complexity), but have a complex dynamical behavior. (Think of the chaotic behavior of Feigenbaum s logistic equation, for example). 

Indeed, in those rare cases where a nonlinear system can be solved exactly -consider, for example, Feigenbaum s logistic equation when A = 4 = 4x (l —... 

There are statistical procedures available to determine whether the data can be fit to a model of dose-response curves that are parallel with respect to slope and all share a common maximal response (see Chapter 11). In general, dose-response data can be fit to a three-parameter logistic equation of the form... 

The responses are plotted on a semi-logarithmic axis, as shown in Figure 12.7a. The curves can be fit to a four-parameter logistic equation if there are appreciable effects on the basal response, or to a three-parameter logistic equation if basal effects are not observed. The curves for the data shown were fit to a three-parameter logistic equation (Equation 12.5). 

Section 11.4.5). The responses calculated by the logistic equations are subtracted from the actual data points and the result squared. The sum of these deviations becomes the sum of squares for the deviations. 

The logistic equation leads to a lag phase, an exponential initial growth rate and a stationary population of concentration (xm). In a population, it is often the case that the birth rate decreases as the population itself increases. The reasons may vary from increased scientific or cultural sophistication to a limited food supply. 

Appendix 1.2D Logits, the Logistic Equation, and their Relation to the Hill Plot and Equation... 

It is worth noting the distinction between the Hill equation and the logistic equation, which was first formulated in the 19th century as a means of describing the time-course of population increase. It is defined by the expression ... 

If an approximate Km value for the enzyme-substrate combination of interest is known, a full-scale kinetic assay may be done immediately. However, often an approximate value is not known and it is necessary first to do a range finding or suck and see preliminary assay. For such an assay, a concentrated substrate solution is prepared and tenfold serial dilutions of the substrate are made so that a range of substrate concentrations is available within which the experimenter is confident the Km value lies. Initial velocities are determined at each substrate concentration, and data may he plotted either hyperholically (as V versus [S]) or with [S] values expressed as logio values. In the latter case, a sigmoidal curve is fitted to data with a three parameter logistic equation (O Eq. 4) ... 

Reprint P (= [277]) is the last of three dialogue papers, the others being [145] = J, App. 1., and [204] = J, App. 2. There is some overlap in matter, and the description of the logistical equation here is much fuller than it was in the first dialogue simply because much more was known about it in 1993 than had been in 1977. A paper designed for a literary audience, having a... 

Very interesting, said C, without much conviction, for he was a bit miffed at the interruption of his exposition, but I hadn t quite finished with the logistical equation. ... 

The second term assumes that the inhibition is proportional to the square of the biomass concentration and this equation, on integration, yields the logistic equation ... 

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 ... 

Pearl, R. and Reed, L. J. Proc. Natl. Acad. Sci. 6 (1920) 275. Growth equation with inhibition factor leading to logistic equation. 

Dose-response curves for PCB standard solutions are carried out. The curves are estimated by nonlinear regression using the following logistic equation [35] by GraphPad Prism 4 program ... 

Fit the standard curve to a four-parameter logistic equation according to the formula y — (A-B/fl + fe/O l+B, where A is the maximal amperometric current, B is the minimum amperometric current, C is the concentration producing 50% of the maximal amperometric current, and D is the slope at the inflection point of the sigmoid curve (Fig. 33.1). 

The logistic equation was used as the basis for the nonlinear model. This model,... 

---