Generic emergence of power law distributions and Lévy-Stable intermittent fluctuations in discrete logistic systems

The dynamics of generic stochastic Lotka-Volterra (discrete logistic) systems of the form ${w}_{i}(t+1)=\ensuremath{\lambda}{(t)w}_{i}(t)+a\overline{w}(t)\ensuremath{-}{\mathrm{bw}}_{i}(t)\overline{w}(t)$ is studied by computer simulations. The variables ${w}_{i}, i=1,\dots{},N,$ are the individual system components and $\overline{w}(t)=(1/N){\ensuremath{\sum}}_{i}{w}_{i}(t)$ is their average. The parameters $a$ and $b$ are constants, while $\ensuremath{\lambda}(t)$ 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 Pareto power law $P(w,t)\ensuremath{\sim}{w}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\alpha}}.$ The time evolution of $\overline{w}(t)$ presents intermittent fluctuations parametrized by a L\'evy-stable distribution with the same index $\ensuremath{\alpha},$ showing an intricate relation between the distribution of the ${w}_{i}'\mathrm{s}$ at a given time and the temporal fluctuations of their average.