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 $w_i$ $i=1,\mathrm{.}\mathrm{.}\mathrm{.},N$ is studied using computer simulations. The time evolution of the $w_i$’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 $c\cdot \bar{w}$, where $\bar{w}$ 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 $p(w)\sim w^{-1-\alpha }$. The exponent $\alpha (c,N)$ is quite insensitive to the distribution $\mathrm{\Pi }(\lambda )$ of the random factor $\lambda$, but it is non-universal, and increases monotonically as a function of $c$. The “thermodynamic” limit, N goes to infty and the limit of decoupled free multiplicative random walks c goes to 0, do not commute: $\alpha (0,N)=0$ for any finite $N$ while $\alpha (c,\mathrm{\infty })\ge 1$ (which is the common range in empirical systems) for any positive $c$. The time evolution of $\bar{w}(t)$ exhibits intermittent fluctuations parametrized by a (truncated) L'evy-stable distribution $L_{\alpha }(r)$ with the same index $\alpha$. This non-trivial relation between the distribution of the $w_i$’s at a given time and the temporal fluctuations of their average is examined and its relevance to empirical systems is discussed.