Les Houches lectures on theoretical ecology: High-dimensional models and extreme events

The study of ecological systems is gaining momentum in modern scientific research, driven by an abundance of empirical data and advancements in bioengineering techniques. However, a full understanding of their dynamical and thermodynamical properties, also in light of the ongoing biodiversity crisis, remains a formidable endeavor. From a theoretical standpoint, modeling the interactions within these complex systems – such as bacteria in microbial communities, plant-pollinator networks in forests, or starling murmurations – presents a significant challenge. Given the intrinsically high dimensionality of these datasets, powerful approaches rely on random matrix theory and methods from disordered systems. In these lectures, we will explore two cornerstone models in theoretical ecology: the MacArthur/Resource-Consumer model, and the Generalized Lotka-Volterra model, with a special focus on systems composed of a large number of interacting species. In the second part, we will highlight timely research directions, particularly to bridge the gap with empirical observations and detect macroecological patterns.