Extended Data Fig. 5. The aggregation-abundance relationship and the rare species advantage in individual-based simulations.

We used an individual-based implementation of the community model of eqs. 1 and 4 with initially 80 species on an area of 200 ha without immigration, simulated for 5000 time steps (25,000 years, ∆t = 5 years). A proportion p_d of recruits was placed with a Gaussian kernel around randomly distributed cluster centres, and the rest with the same kernel around their parents (see methods). The different simulations differ only in the parameter p_d = 0.1, 0.2, 0.35, 0.6, 0.8 (from top to bottom). a–e: temporal aggregation-abundance relationship for 2 or 3 selected species, taken every 50 years (black disks), and for all 80 species at year 25,000 (red discs). Fit of the temporal relationship with a power law with exponent e (blue line), f–j: time series showing the abundances of the first 25 (out of 80) species, where one species (red) invades at year 1000 (and 2 timesteps after extinction) starting with 50 individuals. The average abundance is shown as blue line. k–o: Total crowding index $\stackrel{\circledR }{C_f}+\stackrel{\circledR }{H_f}$ (i.e., distance-weighted number of neighbours) of the invading species in dependence on abundance [in k) we used a species that went extinct]. The red line shows the fit with a linear regression. p–t: same as k)-o), but only for conspecific crowding $\stackrel{\circledR }{C_f}$. The model parameters were the same for all species: r_f = 0.1/∆t, s_f = 0, B_f = 1, c = 0.000063, r = 20 m, β_ff = 0.02, σ = 10 m (see methods), and N_f^* = 1070 and J^* = 86,000 emerged. For comparison with Table 1, the normalized per capita population growth rates ${\tilde{\lambda }}_f(N_s=10)/r_f$ at the last timestep were 0.019, 0.012, 0.022, 0.031, and 0.038 for parameter values of p_d = 0.1, 0.2, 0.35, 0.6, and 0.8, respectively. The raw data with the output of the individual-based model can be found in the Supplementary Information as Supplementary Data Table 2.