Reaching steady state in simulations

We define the steady state of simulated communities as the community state in which neither the survival fraction nor the fluctuation fraction significantly changes as time goes on. In order to consistently analyze the steady state results for all the simulated communities, we analyzed the dependence of the phase diagrams on the simulated time. Fig. S5 shows that neither the survival fraction nor the fluctuation fraction significantly changes after t=5*10³. Accordingly, the phase diagrams in the paper show the state of communities at t=10⁴, unless otherwise stated.

To differentiate between stable and fluctuating communities, we computed the average coefficient of variation of N_i between t=5*10³ and t=10⁴. We define communities with this average coefficient of variation higher (lower) than 10^-3 as fluctuating (stable) communities (Fig. S1). Our simulation results show that all communities have reached steady states at t=5*10^3.