Stochastic Lotka-Volterra Equations

The Lotka-Volterra equations describe the population dynamics of species interacting with each other. It is of particular interest to study the number of coexisting species at stationarity, and their abundance distribution. Recently it has been proposed to use random matrix theory and tools from disordered system to study these systems in the case of quenched random interaction strengths. However, this approach is not able to solve the so called Complexity Stability Paradox, i.e. the more complex the community is, the more extinction we observe. With this thesis work we introduce the annealed disorder in the Lotka-Volterra equations, requiring that each entries of the interaction matrix is a Wiener process and studying the stability of the system. In particular we focus on the study of the phase diagram (mu, sigma^2), the average and the variance of the Wiener process. We find a transition between a phase in which a stationary state exists and a phase in which there is unbounded growth. We also provide an analytical prediction for species relative abundance distribution in the limit of large number of species using the Dean's method and upper bounds for the existence of a steady state solution and related numerical simulations of the model.