Generic Emergence of Power Law Distributions and L\'evy-Stable Intermittent Fluctuations in Discrete Logistic Systems

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 ${\bar w (t)} = {1\over N} \sum_i w_i (t)$ 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 L\'evy 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.