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Ecosystems represent archetypal complex dynamical systems, often modelled bycoupled 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 diffentialequations, 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 modelrepresents an acceptable trade-off between complexity and tractability. Here,$r_i$ represents the intrinsic growth of species $i$ and $\Gamma$ stands forthe interaction matrix: $\Gamma_{ij}$ represents the effect of species $j$ overspecies $i$. For large $N$, estimating matrix $\Gamma$ is often an overwhelmingtask and an alternative is to draw $\Gamma$ at random, parametrizing itsstatistical distribution by a limited number of model features. Dealing withlarge 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 thejunction of theoretical ecology and large random matrix theory. It is intendedto an interdisciplinary audience spanning theoretical ecology, complex systems,statistical physics and mathematical biology.

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