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 [Formula Presented] is studied by computer simulations. The variables [Formula Presented] are the individual system components and [Formula Presented] is their average. The parameters [Formula Presented] and [Formula Presented] are constants, while [Formula Presented] 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 [Formula Presented] of the system components [Formula Presented] turns out to fulfill a Pareto power law [Formula Presented] The time evolution of [Formula Presented] presents intermittent fluctuations parametrized by a Lévy-stable distribution with the same index [Formula Presented] showing an intricate relation between the distribution of the [Formula Presented] at a given time and the temporal fluctuations of their average.