Multispecies ecosystems modeled by generalized Lotka-Volterra equations exhibit stationary population abundances, where a large number of species often coexist. Understanding the precise conditions under which this is at all feasible and what triggers species extinctions is a key, outstanding problem in theoretical ecology. Using standard methods of random matrix theory, I show that distributions of species abundances are Gaussian at equilibrium, in the weakly interacting regime. One consequence is that feasibility is generically broken before stability, for large enough number of species. I further derive an analytical expression for the probability that $N_e=0,1,2,...,$ species go extinct and conjecture that a single-parameter scaling law governs species extinctions. These results are corroborated by numerical simulations in a wide range of system parameters.

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• Complex systems
• Organismal, population, evolutionary & ecological systems
• Perturbation theory
• Population dynamics
• Collective dynamics
• Random matrix theory