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 $w_i\mathrm{}(t+1\mathrm{})=\lambda {\mathrm{}\left(t\right)w}_i\mathrm{}\left(t\right)+a\overline{w}\mathrm{}\left(t\right)\mathrm{}-{\mathrm{bw}}_i\mathrm{}\left(t\right)\mathrm{}\overline{w}\mathrm{}\left(t\right)\mathrm{}$ is studied by computer simulations. The variables $w_i,\hspace{0.5em}i=1,\dots ,N,$ are the individual system components and $\overline{w}\mathrm{}\left(t\right)=(1/N)\mathrm{}{\sum }_i{w}_i\mathrm{}\left(t\right)\mathrm{}$ is their average. The parameters $a$ and $b$ are constants, while $\lambda \mathrm{}\left(t\right)\mathrm{}$ 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 $P(w,t)\mathrm{}$ of the system components $w_i$ turns out to fulfill a Pareto power law $P(w,t)\mathrm{}\sim w^{-1-\alpha }.$ The time evolution of $\overline{w}\mathrm{}\left(t\right)\mathrm{}$ presents intermittent fluctuations parametrized by a Lévy-stable distribution with the same index $\alpha ,$ showing an intricate relation between the distribution of the $w_i’\mathrm{s}$ at a given time and the temporal fluctuations of their average.