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 $$\frac{d x_i}{d t} = x_i \varphi_i(x_1,\cdots, x_N)\ ,$$ where $N$ represents the number of species and $x_i$, the abundance of species $i$. Among these families of coupled diffential equations, Lotka-Volterra (LV) equations $$\frac{d x_i}{d t} = x_i ( r_i - x_i +(\Gamma \mathbf{x})_i)\ ,$$ play a privileged role, as the LV model represents an acceptable trade-off between complexity and tractability. Here, $r_i$ represents 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, parametrizing 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 random matrix theory. It is intended to an interdisciplinary audience spanning theoretical ecology, complex systems, statistical physics and mathematical biology.