Self-organization of ecosystems to exclude half of all potential invaders

Species-rich Lotka-Volterra competition models of ecosystem dynamics transition with increasing species pool size from a phase with well-defined stable equilibrium to a dynamic phase that remains incompletely understood. We analytically describe the statistical mechanics of the steady state deep inside this dynamic phase, characterized by incessant turnover in species composition, and extract the distribution of invasion fitness of random invaders. We find that steady state invasion probability universally equals 1/2. This striking result agrees well with observations in plants and animals.