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Study areasNine large forest dynamics plots of areas between 20 and 50 ha were used in the present study (Supplementary Table 1). The forest plots are part of the ForestGEO network4 and are situated in Asia and the Americas at locations ranging in latitude from 9.15° N to 45.55° N. Tree species richness among the plots ranges from 36 to 468. All free-standing individuals with diameter at breast height (dbh) ≥1 cm were mapped, size measured and identified. We focused our analysis here on individuals with dbh ≥ 10 cm (resulting in a sample size of 131,582 individuals) and focal species with more than 50 individuals (resulting in 289 species). The 10 cm size threshold excludes most of the saplings and enables comparisons with previous spatial analyses20,35,47,48. Shrub species were also excluded.Some of our analyses require estimation of the ratio βfi/βff that describes the relative individual-level competitive effect18 of individuals of species i on an individual of the focal species f. We used for this purpose phylogenetic distances49 based on molecular data, given in Myr, that assume that functional traits are phylogenetically conserved19,26,27. In this case, close relatives are predicted to compete more strongly or to share more pests than distant relatives26. To obtain consistent measures among forest plots, phylogenetic similarities were scaled between 0 and 1, with conspecifics set to 1, and a similarity of 0 was assumed for a phylogenetic distance of 1,200 Myr, which was somewhat larger than the maximal observed distance (1,059 Myr). This was necessary to avoid discounting crowding effects from the most distantly related neighbours26.Observed spatial patterns at species-rich forestsFigure 1 and Supplementary Data Table 1 show the intraspecific variation in our three crowding indices nkff, nkfh and nkfβ that can be approximated by gamma distributions. To assess how well the gamma distribution described the observed distribution, we used an error index defined as the sum of the absolute differences of the two cumulative distributions divided by the number of bins (spanning the two distributions). The maximal value of the error index is one, and a smaller value indicates a better fit.Equations (6, 8 and 9) relate the measures of the emerging spatial patterns (that is, kff, kfh and Bf) to macroscale properties and conditions for species coexistence. Even though our multiscale model (equation (7)) is simplified, it allows for a direct comparison with the emerging patterns in our nine fully stem-mapped forest plots. We estimate the key quantities of equations (8) and (9) directly from the forest plot data (Fig. 4), with the exception of the carrying capacities Kf, which were indirectly estimated from the observed species abundances (assuming approximate equilibrium; equation (8) and Supplementary Data Table 1). This allowed us to estimate the feasibility index µf (equation (9)). Because statistical analyses with individual-based neighbourhood models19,26 based on neighbourhood crowding indices have shown that the performance of trees depends on their neighbours for R between 10 and 15 m, we estimate all measures of spatial neighbourhood patterns with a neighbourhood radius of R = 10 m. Analyses with R = 15 or R = 20 gave similar results.The spatial multispecies model and equilibriumWe use a general macroscale model to describe the dynamics of a community of S species:$$frac{{N_fleft( {t + {Delta}t} right) – N_fleft( t right)}}{{{Delta}t}} = N_fleft( t right)left[ {left( {r_f – 1} right) + s_fexp left( { – alpha _{ff}N_fleft( t right) – mathop {sum }limits_{i ne f} alpha _{fi}N_i(t)} right)} right]$$
(11)
where rf is the mean number of recruits per adult of species f within time step Δt, sf is a density-independent background survival rate of species f and the αfi are the population-level interaction coefficients, yielding αff = c γff kff βff and αfi = c γfβ kfh βff Bf (equation (6)). The βfi are the assumed individual-level interaction coefficients between individuals of species i and f; kff = Kff(R) / π R2 and kfh = Kfh(R) / π R2 measure intraspecific clustering and interspecific segregation, respectively, with Kff(R) being the univariate K function for species f and Kfh(R) the bivariate K function describing the pattern of all heterospecifics ‘h’ around individuals of species f. A is the area of the observation window.Following equation (5), Bf can be estimated as$$B_f = frac{{bar n_{fbeta }}}{{bar n_{f{mathrm{h}}}}} = frac{{mathop {sum }nolimits_{i ne f} left[ {ck_{fi}N_ileft( t right)} right]frac{{beta _{fi}}}{{beta _{ff}}}}}{{mathop {sum }nolimits_{i ne f} left[ {ck_{fi}N_ileft( t right)} right]}} = frac{{mathop {sum }nolimits_{i ne f} k_{fi}N_ileft( t right)frac{{beta _{fi}}}{{beta _{ff}}}}}{{k_{f{mathrm{h}}}mathop {sum }nolimits_{i ne f} N_ileft( t right)}},$$
(12)
and is the weighted average of the relative individual-level interaction coefficients βfi/βff between species i and the focal species f, weighted by the mean number of individuals of species i in the neighbourhoods of the individuals of the focal species (that is, c kfi Ni(t)). For competitive interactions, Bf ranges between zero and one; Bf = 1 indicates that heterospecific and conspecific neighbours compete equally, and smaller values of Bf indicate reduced competition with heterospecific neighbours. The denominator can be rewritten in terms of segregation kfh to all heterospecifics and the total number of heterospecifics ∑i≠f Ni(t).The analytical expression of the equilibrium (equation (8)) relies on the assumption that the values of Bf are approximately constant in time. This assumption may not apply in our model during the initial burn-in phase of the simulations if the βfi/βff show large intraspecific variability (Supplementary Text and Figs. 1–5). The underlying mechanism is the central niche effect introduced by Stump45 where a species has reduced average fitness if it has high niche overlap with competitors.Finally, the factors γff = ln(1 + bff βff) (bff βff)−1 and γfβ = ln(1+ bfβ βff) (bfβ βff)−1 describe the influence of the variance-to-mean ratios bff and bfβ of the gamma distribution of the crowding indices nkff and nkfβ, respectively. For high survival rates during one time step (for example, >85%), the values of γff and γfβ are close to one; in this case the exponential function in equation (1a) can be approximated by its linear expansion and γff = γfβ = 1.In equilibrium we have (Nf(t + Δt) ‒ Nf(t))/Δt = 0, which leads, with equation (7), to:$$N_f^{ast} = left( {K_f – frac{{alpha _{f{mathrm{h}}}}}{{alpha _{ff}}}J^{ast} } right) left( {1 – frac{{alpha _{f{mathrm{h}}}}}{{alpha _{ff}}}} right)^{-1}$$
(13)
with (K_f = – {mathrm{ln}}left( {frac{{1 – r_f}}{{s_f}}} right) left( alpha _{ff} right)^{-1}) and the total number of individuals being (J^{ast} = sum _iN_i^{ast}). Rewriting equation (13) yields (frac{K_f}{J^{ast}} = left( frac{N_f^{ast}}{J^{ast}} right) left(1- frac{alpha_{f{mathrm{h}}}}{alpha_{ff}}right) + frac{alpha_{f{mathrm{h}}}}{alpha_{ff}}). For αfh/αff  More

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