Fig. 1.

(A) Species’ competitive abilities can be represented in a tournament in which we draw an arrow from the inferior to the superior competitor for all species pairs. A tournament is a directed graph composed by n nodes (the species) connected by n(n - 1)/2 edges (arrows). (B) Simulations of the dynamics for the tournament. The simulation begins with 25,000 individuals assigned to species at random (with equal probability per species). At each time step, we pick two individuals at random and allow the superior to replace the individual of the inferior. We repeat these competitions 10⁷ times, which generates relative species abundances that oscillate around a characteristic value (SI Text). (C) The average simulated density of each species from B (shown in lighter bars) almost exactly matches the analytic result obtained using linear programming (shown in darker bars).