Power-law distributions and Levy-stable intermittent fluctuations in stochastic systems of many autocatalytic elements

A generic model of stochastic autocatalytic dynamics with many degrees of freedom wi, i=1,...,N, is studied using computer simulations. The time evolution of the wi's combines a random multiplicative dynamics w(i)(t + 1)= lambda w(i)(t) at the individual level with a global coupling through a constraint which does not allow the w(i)'s to fall below a lower cutoff given by <c(w)over bar>, where (w) over bar is their momentary average and 0<c<1 is a constant. The dynamic variables w(i) are found to exhibit a power-law distribution of the form p(w)similar to w(-1-alpha) The exponent alpha(c,N) is quite insensitive to the distribution nth of the random factor lambda, but it is nonuniversal, and increases monotonically as a function of c. The "thennodynamic" limit N-->infinity and the limit of decoupled free multiplicative random walks c-->0 do not commute: alpha(0,N)=0 for any finite N while alpha(c,infinity)greater than or equal to 1 which is the common range in empirical systems) for any positive c. The time evolution of (w) over bar(t) exhibits intermittent fluctuations parametrized by a (truncated) Livy-stable distribution L-alpha(r) with the same index alpha. This nontrivial relation between the distribution of the w(t)'s at a given time and the temporal fluctuations of their average is examined, and its relevance to empirical systems is discussed. [S1063-651X(99)09707-X].