Generic Emergence of Power Law Distributions and Lévy-Stable Intermittent Fluctuations in Discrete Logistic Systems

Abstract

The dynamics of generic stochastic Lotka-Volterra (discrete logistic) systems of the form \cite{Solomon96a} $w_i(t+1)=\lambda (t)w_i(t)+a\bar{w}(t)-b{w}_i(t)\bar{w}(t)$ is studied by computer simulations. The variables $w_i$, $i=1,...N$, are the individual system components and $\bar{w}(t)=\frac{1}{N}{\sum }_i{w}_i(t)$ is their average. The parameters $a$ and $b$ are constants, while $\lambda (t)$ is randomly chosen at each time step from a given distribution. Models of this type describe the temporal evolution of a large variety of systems such as stock markets and city populations. These systems are characterized by a large number of interacting objects and the dynamics is dominated by multiplicative processes. The instantaneous probability distribution $P(w,t)$ of the system components $w_i$, turns out to fulfill a (truncated) Pareto power-law $P(w,t)\sim w^{-1-\alpha }$. The time evolution of $\bar{w}(t)$ presents intermittent fluctuations parametrized by a truncated Lévy distribution of index $\alpha$, showing a connection between the distribution of the $w_i$'s at a given time and the temporal fluctuations of their average.

18 pages and 5 figures (in one zipped file); [email protected], [email protected], [email protected], [email protected], http://shum.huji.ac.il/~sorin