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

The dynamics of generic stochastic Lotka-Volterra (discrete logistic) systems of the form w(i)(t+l) = lambda(t)w(i)(t) + <a(w)over bar>(t) - bw(i)(r)(w) over bar(t) is studied by computer simulations. The variables w(i), i = 1, ..., N, are the individual system components and (w) over bar(t) = (1/N)Sigma(i)w(i)(t) is their average. The parameters a and b an 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 Pareto power law P(w, t)similar to w(-1-alpha). The time evolution of (w) over bar(t) presents intermittent fluctuations parametrized by a Levy-stable distribution with the same index alpha, showing an intricate relation between the distribution of the w(i)'s at a given time and the temporal fluctuations of their average.