Master equation and relative species abundance distribution for Lotka-Volterra models of interacting ecological...

Understanding the factors that control the dynamics of interacting species is a fundamental problem in ecology. The nature of the interactions among different species is usually not completely understood, but it is assumed that the species interaction plays an important role in the ecosystem properties as predicted by the niches models for an ecosystem. However, recent studies point out as the neutral hypothesis proposed by Hubbell of non-interacting species with an external source from the surrounding environment, allows to explain the relative species abundance distribution when the ecosystem has reached a stationary situation. In this paper we use the concept of fitness landscape to introduce a random dynamical model that describes the evolution of different communities near a stationary situation. The average dynamics can be associated to a system of Lotka-Volterra equations with mutualistic interactions. Then we derive a Master equation that satisfies the detailed balance condition of thermodynamical equilibria and allows to analytically compute the relative species abundance distribution near the stationary state as a multinomial negative distribution. These results suggest a possible approach to a synthetic theory that joins the niche theories and the Hubbell's neutral theory for RSA distribution.