Co-Evolutionist Stochastic Dynamics: Emergence of Power Laws

We propose a stochastic generalization of the Lotka-Volterra-Eigen-Schuster systems. In the generalized system, the set of dynamical variables is divided into several subsets, such that the variables within each subset interact with each other more strongly than with those in other subsets. It is shown that the size distribution of the dynamical variables within each subset approach a power-law form, while the exponents that characterize these distributions are different in different subsets. The exponent for each subset is found to depend only on the ratio between the stochastic term and the deterministic additive term. The model can be applied in the context of international trade, where the dynamical variables represent the revenues of firms or income/wealth of individuals.