Example of combining positive and negative feedback loops in an evolving system

Take time increments of 10 units of time (u.t.). Suppose, in order to take account of the self-organized critical behaviour of the system, that the "bi" corresponding to the multiple negative feedback loops decrease as follows (see figure):

If we take u=1.5, n=0.01 , and "a" in Equation (2) such that the timescale of the process is 400 u.t. (a=0.0025) and zo=1, we can write:

The evolutions of y(t) and z(t)with time are shown on Figures 3and 4 respectively.

Such an evolution is equivalent to the case of a vessel whose filling rate is regulated by a float while the volume is increasing exponentially at a rate sufficiently low to allow the filling rate to decrease in a first stage. Note that above example is just one among other possible cases of combination of negative and positive feedback loops. It was chosen for the sake of simplicity. Other combinations can be made and/or other values of u, v, a and zochosen. In order to get a global curve of the kind shown on Figure 3, it is sufficient that : (1) reliability growth be taken into account, and (2) parameter "a" be not that high that it masks the effect of the negative feedback loops.

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