Modelling Ecological Systems from a Niche Theory to Lotka-Volterra Equations

This paper is an attempt to analyze the notion of ecological niche as a community of different species and of ecosystem as a set of niches in order to formulate a dynamical model for an ecosystem. Our assumption is that the concept of fitness landscape allows to model the phenotype dynamics of an ensemble of species as a stochastic process. To take into account the interaction structure of different communities in the niches and the environment we introduce an ecological fitness potential to formulate a Lotka-Volterra system which describes the evolution of a mutual ecosystem in presence of finite resources. To explicitly consider the effect of fluctuations in the numerousness of the species, we associate a master equation to the average Lotka-Volterra system and we study the conditions of existence of a detailed balance equilibrium (i.e. a thermodynamic equilibrium) for the ecosystem. The explicit solution for the equilibrium probability distribution is a multinomial negative distribution and we discuss the relation between the detailed balance condition and relative species abundance distribution in the framework of Hubbell’s neutral theory. Moreover the theoretical distribution implies the existence of a correlation among the relative species distribution associated to the different communities. We use numerical simulations to illustrate the results on simple models