Statistical mechanics of ecological systems: Neutral theory and beyond

The simplest theories often have much merit and many limitations, and, \nin this vein, the value of neutral theory (NT) of biodiversity has been \nthe subject of much debate over the past 15 years. NT was proposed at \nthe turn of the century by Stephen Hubbell to explain several patterns \nobserved in the organization of ecosystems. Among ecologists, it had a \npolarizing effect: There were a few ecologists who were enthusiastic, \nand there were a larger number who firmly opposed it. Physicists and \nmathematicians, instead, welcomed the theory with excitement. Indeed, NT \nspawned several theoretical studies that attempted to explain empirical \ndata and predicted trends of quantities that had not yet been studied. \nWhile there are a few reviews of NT oriented toward ecologists, the goal \nhere is to review the quantitative aspects of NT and its extensions for \nphysicists who are interested in learning what NT is, what its successes \nare, and what important problems remain unresolved. Furthermore, this \nreview could also be of interest to theoretical ecologists because many \npotentially interesting results are buried in the vast NT literature. It \nis proposed to make these more accessible by extracting them and \npresenting them in a logical fashion. The focus of this review is \nbroader than NT: new, more recent approaches for studying ecological \nsystems and how one might introduce realistic non-neutral models are \nalso discussed.