Logistical equation, evolution

Under these conditions, the evolution of the mechanism of synthesis of cAMP signals is governed by the three-variable system (5.12), to which are added four differential equations for the variation of parameters Rj, a, k, and these parameters will be taken as proportional to a variable X, whose time evolution is itself governed by the logistic equation ... 

The reader can check this behavior in May s numerical equation of the logistic model. If we change slightly the initial value in the domain of stability (for example, p = 2.7), the population converges to the same value of 0.6296. This point acts as an attractor. In the chaotic region, a similar weak variation gives way to conq>letely different succ sive evolutions. This is an indication of the "butterfly" effect. 

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). 

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. 

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