Generalized Dynamical Mean Field Theory for Non-Gaussian Interactions

We present a generalized dynamical mean field theory for studying the effects of non-Gaussian quenched noise in a general set of dynamical systems. We apply the framework to the generalized Lotka-Volterra equations, a central model in theoretical ecology, where species interactions are fixed over time and heterogeneous. Our results show that the new mean field equations have solutions that depend on all cumulants of the distribution of species interactions. We obtain an analytic solution when the interaction couplings are α-stable distributed and find a relationship between the abundance distribution of species and the statistics of microscopic interactions. In the case of sparse interactions, which we investigate analytically, we establish a simple relationship between the distribution of interactions and the one of population densities.