Complex systems in ecology: a guided tour with large Lotka-Volterra models and random matrices

Ecosystems represent archetypal complex dynamical systems, often modelled by coupled differential equations of the form dxidt=xi phi i(x1, horizontal ellipsis ,xN),where N represents the number of species and xi, the abundance of species i. Among these families of coupled differential equations, Lotka-Volterra (LV) equations, corresponding to phi i(x1, horizontal ellipsis ,xN)=ri-xi+(Gamma x)i,play a privileged role, as the LV model represents an acceptable trade-off between complexity and tractability. Here, ri is the intrinsic growth of species i and Gamma stands for the interaction matrix: Gamma ij represents the effect of species j over species i. For large N, estimating matrix Gamma is often an overwhelming task and an alternative is to draw Gamma at random, parameterizing its statistical distribution by a limited number of model features. Dealing with large random matrices, we naturally rely on random matrix theory (RMT). The aim of this review article is to present an overview of the work at the junction of theoretical ecology and large RMT. It is intended to an interdisciplinary audience spanning theoretical ecology, complex systems, statistical physics and mathematical biology.