Ecosystems represent archetypal complex dynamical systems, often modelled by coupled differential equations of the form $$ \frac{d x_i}{d t} = x_i φ_i(x_1,\cdots, x_N)\ , $$ 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 $$ \frac{d x_i}{d t} = x_i ( r_i - x_i +(Γ\mathbf{x})_i)\ , $$ 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 $Γ$ stands for the interaction matrix: $Γ_{ij}$ represents the effect of species $j$ over species $i$. For large $N$, estimating matrix $Γ$ is often an overwhelming task and an alternative is to draw $Γ$ 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.
Dimensionality-induced critical phase transition in stochastic Lotka-Volterra equation: From statistical averaging to systemic tipping point
State Key Laboratory of Animal Biodiversity Conservation and Integrated Pest Management, Institute of Zoology, Chinese Academy of Sciences（中国科学院动物研究所生物多样性保护与有害生物综合治理国家重点实验室）；School of Life Science and Technology, Northwestern Polytechnical University（西北工业大学生命科学与技术学院）；School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）；State Key Laboratory of Vegetation Structure, Function and Construction (VegLab), Institute of Ecology, College of Urban and Environmental Science, Peking University（北京大学城市与环境学院生态研究所植被结构、功能与构建国家重点实验室（VegLab））
Les Houches Lectures on Community Ecology: From Niche Theory to Statistical Mechanics
Kavli Institute for Theoretical Physics, University of California Santa Barbara（加州大学圣塔芭芭拉分校卡弗里理论物理研究所）；Pontifical University of the Holy Cross（十字神学院宗座大学）；Dept. of Physics and Faculty of Computing and Data Science, Boston University（波士顿大学物理系与计算数据科学学院）
Topologically robust zero-sum games and Pfaffian orientation -- How network topology determines the long-time dynamics of the antisymmetric Lotka-Volterra equation
Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）；Center for Systems Biology Dresden（德累斯顿系统生物学中心）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学数学研究所）
Impossible ecologies: Interaction networks and stability of coexistence in ecological communities
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Center for Systems Biology Dresden（德累斯顿系统生物学中心）；Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）；Cluster of Excellence Physics of Life, TU Dresden（德累斯顿工业大学生命物理卓越集群）
