  Ecosystems are among the most interesting and well-studied examples of
self-organized complex systems. Community ecology, the study of how species
interact with each other and the environment, has a rich tradition. Over the
last few years, there has been a growing theoretical and experimental interest
in these problems from the physics and quantitative biology communities. Here,
we give an overview of community ecology, highlighting the deep connections
between ecology and statistical physics. We start by introducing the two
classes of mathematical models that have served as the workhorses of community
ecology: Consumer Resource Models (CRM) and the generalized Lotka-Volterra
models (GLV). We place a special emphasis on graphical methods and general
principles. We then review recent works showing a deep and surprising
connection between ecological dynamics and constrained optimization. We then
shift our focus by analyzing these same models in "high-dimensions" (i.e. in
the limit where the number of species and resources in the ecosystem becomes
large) and discuss how such complex ecosystems can be analyzed using methods
from the statistical physics of disordered systems such as the cavity method
and Random Matrix Theory.
