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 \cite{Solomon96a}
w
i
(t+1)=λ(t)
w
i
(t)+a
w
¯
(t)−b
w
i
(t)
w
¯
(t)
is studied by computer simulations. The variables
w
i
,
i=1,...N
, are the individual system components and
w
¯
(t)=
1
N
∑
i
w
i
(t)
is their average. The parameters
a
and
b
are constants, while
λ(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)∼
w
−1−α
. The time evolution of
w
¯
(t)
presents intermittent fluctuations parametrized by a truncated Lévy distribution of index
α
, showing a connection between the distribution of the
w
i
's at a given time and the temporal fluctuations of their average.