Consequences of spatial patterns for coexistence in species-rich plant communities

Abstract

Ecology cannot yet fully explain why so many tree species coexist in natural communities such as tropical forests. A major difficulty is linking individual-level processes to community dynamics. We propose a combination of tree spatial data, spatial statistics and dynamical theory to reveal the relationship between spatial patterns and population-level interaction coefficients and their consequences for multispecies dynamics and coexistence. Here we show that the emerging population-level interaction coefficients have, for a broad range of circumstances, a simpler structure than their individual-level counterparts, which allows for an analytical treatment of equilibrium and stability conditions. Mechanisms such as animal seed dispersal, which result in clustering of recruits that is decoupled from parent locations, lead to a rare-species advantage and coexistence of otherwise neutral competitors. Linking spatial statistics with theories of community dynamics offers new avenues for explaining species coexistence and calls for rethinking community ecology through a spatial lens.

Main

Understanding the mechanisms that maintain high species diversity in plant communities such as tropical forests has long challenged ecologists¹ and has stimulated major efforts in field and theoretical ecology^2,3,4,5. However, despite a multitude of coexistence mechanisms that have been proposed⁶ and recent advances in coexistence theories^{7,8,9,10,11,12,13,14,15,16,17,18}, this fundamental question has not been fully resolved^8,9,14. For example, theoretical models indicate that stable coexistence is difficult to reach in large communities^11,13. We argue that consideration of spatial patterns of plant individuals, such as intraspecific clustering and interspecific segregation, may allow for a better understanding of mechanisms of coexistence in species-rich communities¹⁷.

Although many studies suggest that spatial patterns and neighbourhood effects may play an important role in diversity maintenance^{17,18,19,20,21}, the integration of spatial patterns into coexistence theories of species-rich communities is difficult. A major difficulty is linking spatial processes at the individual level to community dynamics. One reason for this is a scale mismatch. The analytical models that form the basis of most coexistence theories^{7,8,11,12,13,14,15,16,22} have state variables that operate at the macroscale (that is, the population or community-level abundances), use parameters that describe average ‘mesoscale’ properties of the individuals (such as population-level interaction coefficients and demographic rates) and often rely on ‘mean-field’ approximations^18,23 where spatial patterns are neglected. However, spatial patterns and population-level interaction coefficients emerge at the mesoscale from the microscale behaviour of individuals and their interactions with other individuals and the environment. Therefore, studying the impact of spatial patterns on species coexistence requires multiscale approaches such as spatial moment equations^18,23,24 that incorporate pattern-forming processes operating at the level of individuals and translate these into population and community dynamics.

We propose here such a multiscale approach. To this end, we first derive population-level interaction coefficients α_fi from individual-level interaction coefficients β_fi and neighbourhood crowding indices^19,21 that are commonly used to describe interactions among tree individuals at the microscale, and then incorporate the emerging coefficients α_fi into analytical macroscale models. Our approach is based on separation of timescales (adiabatic approximation²⁵), given that mesoscale spatial patterns usually build up quickly and approach a quasi-steady state whereas the macroscale state variables (for example, abundances) change slowly²³. Therefore, we do not need to describe the dynamics of the quick mesoscale patterns explicitly (as, for example, is done in approaches based on moment equations^18,23,24) but concentrate instead on spatial patterns that transport the critical information from the microscale into macroscale models. This approach requires information on mesoscale spatial patterns that can be obtained from fully mapped forest plots such as those of the Forest Global Earth Observatory (ForestGEO) network⁴.

More specifically, we (1) derive species-level interaction coefficients from individual-level interactions using empirical information on spatial patterns in nine ForestGEO megaplots⁴, (2) integrate the resulting species-level interaction coefficients into analytical macroscale multispecies models and (3) study their consequences for multispecies dynamics and coexistence.

Results and discussion

Species-level interaction coefficients

We first derive species-level interaction coefficients from individual-level neighbourhood crowding indices^19,21,26 that quantify how the performance of a focal individual depends on interactions with its neighbours. To this end, we describe the survival rate of a focal individual k of species f in dependence on the local number of neighbours as

$$s_{kf} = s_f \exp \left( - \beta _{ff} \left(n_{kff} + \underbrace {\mathop {\sum}\nolimits_{i \ne f} {(\beta _{fi}/\beta _{ff})n_{kfi}} }_{n_{kf\beta }} \right) \right),$$

(1a)

where s_f is a density-independent background survival rate of species f; the crowding indices n_kff, n_kfi and n_kfh are the number of conspecifics, neighbours of species i and heterospecific neighbours within distance R of a focal individual k, respectively; the subscript ‘h’ indicates all heterospecifics together (that is, n_kfh = ∑_i≠f n_kfi) and the crowding index n_kfβ weights each heterospecific neighbour by its relative competition strength β_fi/β_ff (equation (1a)), with β_fi being the individual-level interaction coefficients between species f and i (Fig. 1a–c). The corresponding population-level survival rate is given by

$${\mathrm{surv}}_f = s_f\exp \left( - \alpha _{ff} \left(N_f(t) + \mathop {\sum}\limits_{i \ne f} {(a_{fi}/ \alpha _{ff})} N_i(t) \right) \right),$$

(1b)

where N_i(t) is the abundance of species i at time step t and α_fi is the population-level interaction coefficient between species f and i. To estimate surv_f we average the survival rates s_kf (equation (1a)) of all individuals k of species f:

$${\mathrm{surv}}_f = \frac{1}{{N_f\left( t \right)}}\mathop {\sum}\limits_{k = 1}^{N_f\left( t \right)} {s_{kf} = s_f} \mathop {\int}\limits_x {\mathop {\int}\limits_y {\exp } } \left( { - \beta _{ff}\left( {x + y} \right)} \right)p_x p_y {\mathrm{d}}x {\mathrm{d}}y$$

(2)

where p_x and p_y are the distributions of the crowding indices x = n_kff and y = n_kfβ for individuals of species f, respectively.

Fig. 1: Neighbourhood crowding indices describe individual-level interactions and their intraspecific variability.

a, The conspecific crowding index n_kff is the number of conspecific neighbours (filled red circles) within distance R (black circle) of the focal individual k (filled red square). b, The heterospecific crowding index n_kfh is the number of heterospecific neighbours (filled grey circles) within distance R of the focal individual k. c, The heterospecific interaction crowding index n_kfβ additionally weights heterospecifics by their relative competitive effect β_fi/β_ff, symbolized by the arrows. Different colours indicate different species. d, Distribution of the number n_kff of conspecific neighbours with diameter at breast height (dbh) ≥ 10 cm of the species Castanopsis cuspidata of the 25 ha Fushan plot. e, Corresponding distribution of heterospecific neighbours n_kfh. f, Corresponding distribution of the crowding index n_kfβ. In d–f, blue lines show gamma distributions with the same means and variance-to-mean ratios as the observed distributions, and the vertical black line indicates the mean value. See Supplementary Data Table1 for additional examples.

To determine the distributions p_x and p_y, we analysed forest inventory data from nine 20–50 ha forest dynamics plots (Supplementary Table 1) in the ForestGEO network⁴. We used phylogenetic similarity between tree species as a surrogate for the relative competition strength β_fi/β_ff because it is available for the species in the nine plots (Methods). This is an established approach in species-rich communities^19,26,27 to approximate niche differences in the absence of other data.

The number of con- and heterospecific neighbours and the heterospecific interaction index n_kfβ vary widely among conspecifics and can be described by gamma distributions (Fig. 1 and Extended Data Figs. 1 and 2). Detailed analysis of the empirical crowding indices reveals additional relationships that are relevant for our subsequent analysis. First, we find that the crowding indices n_kff and n_kfβ are not, or are only weakly, correlated for a given species f (Extended Data Fig. 3a). Second, we find for trees of a given species f high correlations between the two crowding indices n_kfh and n_kfβ (Extended Data Fig. 3b) with a common factor B_f (that is, n_kfβ ≈ B_fn_kfh). This result suggests operation of diffuse neighbourhood competition, in which the competition strength of heterospecifics is on average a factor B_f lower than that of conspecifics.

The integral of equation (2) can be solved analytically for independent gamma distributions p_x and p_y and yields

$${\mathrm{surv}}_f = s_f \exp \left( { - \beta _{ff}\left( {\gamma _{ff}\bar n_{ff} + \gamma _{f\beta }\bar n_{f\beta }} \right)} \right)$$

(3)

where $\bar n_{ff}$ and $\bar n_{f\beta }$ are the average values of the crowding indices n_kff and n_kfβ, respectively, and γ_ff and γ_fβ contain the variance-to-mean ratios of the gamma distributions p_x and p_y, respectively, but in our case have values close to one (Methods).

The last step in deriving pairwise population-level interaction coefficients is to relate the averages of the different crowding indices to the macroscale population abundances N_f(t). We accomplish this by taking advantage of connections between crowding indices and the summary functions of spatial point process theory²¹. The mean of the crowding index n_kff (that is, the mean number of further conspecific neighbours within distance R) is proportional to Ripley’s K, a well-known quantity in point process theory^28,29:

$$\bar n_{ff} = K_{ff}\left( R \right)N_f(t)/A$$

(4a)

where K_ff(R) is the univariate K function for species f and A is the area of the observation window. The K function describes the spatial pattern of conspecifics within a neighbourhood distance R, indicating clustering if K_ff(R) > πR², a random pattern if K_ff(R) = πR² and regularity if K_ff(R) < πR². In the following, we use the normalized K function $k_{ff}\left( R \right) = K_{ff}\left( R \right)/\uppi R^2$ to quantify the spatial neighbourhood patterns, and therefore $\bar n_{ff} = k_{ff}\left( R \right)\frac{{\uppi R^2}}{A}N_f(t)$.

Analogously, the mean number of heterospecific neighbours is given by

$$\bar n_{f{\mathrm{h}}} = k_{f{\mathrm{h}}}\left( R \right)\frac{{\uppi R^2}}{A}\mathop {\sum }\limits_{i \ne f} N_i(t),$$

(4b)

where the bivariate normalized K function k_fh(R) indicates segregation to heterospecifics (subscript ‘h’) within distance R if k_fh(R) < 1. Independent placement occurs if k_fh(R) = 1, and attraction if k_fh(R) > 1.

Motivated by the finding n_kfß ≈ B_f n_kfh (Extended Data Fig. 3b) we rewrite the mean crowding index $\bar n_{f\beta }$ as

$$\bar n_{f\beta } = \underbrace {B_f}_{\frac{{\bar n_{f\beta }}}{{\bar n_{f{\mathrm{h}}}}}}\bar n_{f{\mathrm{h}}},$$

(5)

where the point process summary function B_f indicates how much the competition strength of one heterospecific neighbour differs on average from that of one conspecific neighbour. The values of B_f depend mainly on the individual-level interaction coefficients β_fi but also on the spatial pattern of the different species and their relative abundances (equation (12)).

Inserting the expressions for $\bar n_{ff}$ and $\bar n_{f\beta }$ into equation (3) and comparing with equation (1b) leads to our first main result, the analytical expressions of the population-level interaction coefficients:

$$\alpha _{ff} = c\,\gamma _{ff}\,k_{ff}\beta _{ff} \qquad {\mathrm{intraspecific}}\,{\mathrm{interactions}}\,{\mathrm{of}}\,{\mathrm{species}}\,f$$

(6a)

$$\alpha _{fi} = c\gamma _{f\beta }k_{f{\mathrm{h}}}\beta _{ff}B_f \, \quad {\mathrm{interspecific}}\,{\mathrm{interactions}}\,{\mathrm{with}}\,{\mathrm{species}}\,i$$

(6b)

with scaling constant c = πR²/A. Notably, equation (6b) indicates that the emerging population-level interaction coefficients α_fi are the same for all heterospecifics (that is, α_fi = α_fh for i ≠ f). Thus, even if the individual-level interactions coefficients β_fi differ among species pairs, the emerging population-level interaction coefficients α_fi have a substantially simpler structure. This phenomenon is an example of simplicity emerging from complex species interactions³⁰ and is likely to occur only in species-rich communities^9,12,31.

The population-level interaction coefficients α_fi depend on several factors that can influence the macroscale balance between intra- and interspecific competition: (1) intraspecific clustering k_ff and interspecific segregation k_fh (Fig. 2a,b), (2) the relative competition strength B_f of one heterospecific neighbour (Fig. 2c) and (3) the shape of the response of survival to crowding (γ_ff and γ_fβ, which contain the variance-to-mean ratios of the distribution of the crowding indices). Note that absence of spatial patterns (that is, k_ff = 1 and k_fh = 1) and a linear approximation of equation (3) lead to γ_ff = γ_fβ = 1 and direct proportionality α_fi = cβ_fi of the individual- and population-level interaction coefficients, as assumed by Lotka–Volterra models.

Fig. 2: Emergent spatial patterns in the nine forest dynamics plots.

a–c, Distribution of the values of the different measures of spatial patterns, taken over the focal species of the different forest plots. Boxplots show the 10th, 25th, 50th, 75th and 90th percentiles; outliers are indicated by filled black points. Intraspecific clustering is indicated by k_ff > 1 and interspecific segregation by k_fh < 1, and the less a heterospecific neighbour competes on average relative to a conspecific neighbour, the more B_f decreases. The neighbourhood radius used was R = 10 m. For the analysis, we used all individuals with dbh ≥ 10 cm and included focal species with more than 50 individuals. For forest plot names, see Supplementary Data Table 1.

The information on spatial patterns extracted from the inventory data of our nine forests allows us to estimate the relative population-level interaction coefficients α_fi/α_ff for all pairs of species i and f (Fig. 3 and Extended Data Fig. 4). For example, at BCI, the values of α_fi/α_ff differ substantially from the corresponding individual-level coefficients β_fi/β_ff (Fig. 3a,c), and for 83% of all species pairs, we find α_fi/α_ff < β_fi/β_ff. Thus, the mesoscale spatial patterns can reduce, at the population level, the strength of heterospecific interactions relative to conspecific interactions by ‘diluting’ encounters with heterospecific neighbours relative to conspecific neighbours. Spatial patterns therefore have a strong potential to alter the outcome of deterministic individual-level interactions.

Fig. 3: The emergent population-level interaction coefficients.

a, Relationship between the relative population-level interaction coefficients α_fi/α_ff and the corresponding relative individual-level interaction coefficients β_fi/β_ff, (equation (6)) for the 75 focal species of the BCI plot. The β_fi/β_ff were based on phylogenetic dissimilarity. b, Distribution of the values of α_fi/α_ff, taken over the 289 focal species of the different forest plot. Boxplots show the 10th, 25th, 50th, 75th and 90th percentiles; outliers are indicated by filled black points. For full, separate distributions for tropical, subtropical and temperate forests, see Extended Data Fig. 4. c, Example for the distribution of the relative individual- and population-level interaction coefficients for the BCI plot. The neighbourhood radius used was R = 10 m. For the analysis, we used all individuals with dbh ≥ 10 cm and included focal species with more than 50 individuals. For forest plot names, see Supplementary Data Table 1.

Conditions for coexistence in the multiscale model

To study the consequences of the emerging spatial patterns for community dynamics and coexistence we insert the population-level interaction coefficients α_fi (equation (6)) into a simple macroscale model

$$\begin{array}{l}\frac{{N_f\left( {t + {\Delta}t} \right) - N_f\left( t \right)}}{{{\Delta}t}} = N_f\left( t \right)\\\left[ {\left( {r_f - 1} \right) + s_f \exp \left( - \alpha _{ff}N_f\left( t \right) - \alpha _{f{\mathrm{h}}}\mathop {\sum }\limits_{i \ne f} N_i(t)\right)} \right]\end{array}$$

(7)

In this model we assume that survival is governed by neighbourhood competition with α_fi = α_fh, and the number of recruits of species f during a time step Δt is given by r_fN_f(t), where r_f is the per capita reproduction rate of species f.

The carrying capacity of species f (that is, the equilibrium of equation (7) with N_i(t) = 0 for all i ≠ f) is given by $K_f = - \ln \left( {\frac{{1 - r_f}}{{s_f}}} \right) {\alpha _{ff}}^{-1}$. Note that our theory also applies, after redefinition of the carrying capacity, to alternative macroscale models (Supplementary Table 3). From equation (7) we find $K_f = N_f^{\ast} + \frac{{\alpha _{f{\mathrm{h}}}}}{{\alpha _{ff}}}\mathop {\sum }\limits_{i \ne f} N_i^{\ast} = N_f^{\ast} \left( {1 - \frac{{\alpha _{f{\mathrm{h}}}}}{{\alpha _{ff}}}} \right) + \frac{{\alpha _{f{\mathrm{h}}}}}{{\alpha _{ff}}}J^{\ast}$, where $N_f^{\ast}$ is the abundance of species f in equilibrium and J^* the equilibrium community size (that is, $J^{\ast} = \sum _i N_i^{\ast}$; see also equation (14)). This leads, under the assumption that the population-level interaction coefficients α_fh are constant (Supplementary text), to a single equilibrium of the macroscale model for species f

$$N_f^{\ast} = \frac{{\overbrace {\alpha _{ff}K_f}^{ - \ln \left( {\frac{{1 - r_f}}{{s_f}}} \right)} - \alpha _{f{\mathrm{h}}}J^{\ast}}}{{\alpha _{ff} - \alpha _{f{\mathrm{h}}}}}$$

(8)

that is positive if denominator and numerator are both positive or both negative. However, the invasion criterion (equation (18)) is only fulfilled if both are positive. In this case, equation (8) suggests two different ways a species can go extinct. First, the denominator indicates that a species with strong clustering k_ff will show a small equilibrium abundance since in this case α_ff ≫ α_fh (equation (6)). Large values of k_ff can be expected for species of low abundance under dispersal limitation, where recruitment happens close to conspecific adults.

Second, the numerator of equation (8) indicates a positive abundance of species f if α_ffK_f > α_fhJ^* and α_fh/α_ff < 1. Therefore, we introduce a new feasibility index

$$\mu _f = \frac{{\alpha _{f{\mathrm{h}}}J^{\ast}}}{{\alpha _{ff}K_f}}$$

(9)

that indicates a positive abundance if µ_f < 1 given that heterospecific interactions at the population level are weaker than conspecific interactions (that is, α_fh/α_ff < 1). The invasion criterion^7,8 that tests whether a species with low abundance can invade the equilibrium community of all other species turned out to be basically the same as the feasibility condition (equation (9)) if the invading species does not show strong clustering (Methods and equation (18)). Note that we did not assume Allee effects⁸.

Further analysis that considers the dependency of J^* on the values of K_f and α_fh/α_ff shows that the values of µ_f must be similar for all species f to fulfil the condition µ_f < 1, and that µ_f can show larger interspecific variability if the species richness S is smaller and/or if the mean of $\frac{{\alpha _{f{\mathrm{h}}}}}{{\alpha _{ff}}} \left( {1 - \frac{{\alpha _{f{\mathrm{h}}}}}{{\alpha _{ff}}}} \right)^{-1}$ is smaller (equations (14) and (15)). The feasibility index µ_f therefore governs species assembly by determining the subset of species of a larger species pool that can persist¹³, but any addition of a new species changes µ_f and may lead to reassembly of the community.

Using the observed abundances in the forest plots (and assuming equilibrium) allows us to test our theory. We can estimate from the observed abundances the carrying capacities K_f and therefore also the indices µ_f for all focal species of our nine plots (Fig. 4). For 282 of our 289 focal species, we found µ_f < 1 and α_fh/α_ff < 1, which means that the two conditions for stable coexistence are indeed satisfied for nearly all species. However, this is not a given, as shown by the seven species from BCI with µ_f > 1 and α_fh/α_ff > 1. Thus, our theory is compatible with the observed coexistence of most species at our nine forest plots if the assumption of approximately constant population-level interaction coefficients holds.

Fig. 4: The feasibility index µ_f for the 289 focal species of different forest plots.

a, The distribution of the index µ_f (equation (9)) for the 289 analysed tree species. The vertical dashed line indicates the median of the distribution. b, Distribution of µ_f, taken over the 289 analysed species of the different forest plot. Boxplots show the 10th, 25th, 50th, 75th and 90th percentiles; outliers are represented by black points. c, Relationship between the feasibility index µ_f and the ratio α_fh/α_ff of heterospecific to conspecific population-level interaction coefficients. Values of µ_f < 1 indicate a positive abundance. The red lines show the dependence of µ_f on α_fh/α_ff expected for communities without interspecific variability in µ_f (equation (10a)) with 18 focal species (CBS plot, lower line) and 75 focal species (BCI plot, upper line). The neighbourhood radius used was R = 10 m. For the analysis, we used all individuals with dbh ≥ 10 cm and included tree species with more than 50 individuals. For plot names, see Supplementary Data Table 1.

In addition, we get information on the typical values of µ_f and α_fh/α_ff that allows for insight into the stability of the communities. In agreement with the predictions of our theory, we find that µ_f tends to be smaller if α_fh/α_ff is smaller (Fig. 4c). Furthermore, the values of µ_f were, for most species, larger than the expectation of µ_f for the corresponding communities without interspecific variability in µ_f (equation (10a)) but with the same number of species and the same mean values of α_fh/α_ff (Fig. 4c), but all of the values were relatively close to the critical value of 1 (the median of all 289 species was 0.938; Fig. 4a).

Consequences of spatial patterns for coexistence

Our theory predicts that coexistence requires, in the limit of high species richness, that species approach functional identity with respect to the feasibility index µ_f (Methods). This resembles neutral theory^2,32, but our theory allows for trade-offs among demographic parameters and emerging spatial patterns to reach this equivalence (equation (9)). To study the consequences of spatial patterns for coexistence, we analysed a symmetric³³ version of our model where all species have the same parameters and follow the same stochastic rules and where all individuals compete identically (leading to B_f = 1). Thus, we eliminate any potential coexistence mechanism other than that resulting from spatial patterns.

If the mesoscale patterns k_ff and k_fh converge to a stochastic equilibrium, we find that the feasibility and invasion criteria are always fulfilled if α_fh/α_ff < 1 since

$$\mu _f = \frac{{S\frac{{\alpha _{f{\mathrm{h}}}}}{{\alpha _{ff}}}}}{{1 + \frac{{\alpha _{f{\mathrm{h}}}}}{{\alpha _{ff}}}\left( {S - 1} \right)}} < 1$$

(10a)

$$\mu _f^i = \frac{{(S - 1)\frac{{\alpha _{f{\mathrm{h}}}}}{{\alpha _{ff}}}}}{{1 + \frac{{\alpha _{f{\mathrm{h}}}}}{{\alpha _{ff}}}\left( {S - 2} \right)}} < 1$$

(10b)

Equations (10a) and (10b) follow from equations (16) and (18), respectively, if α_fh/α_ff is the same for all species f. Thus, the spatial patterns that emerge at the mesoscale from the individual-level interactions can stabilize if α_fh/α_ff < 1. The underlying mechanism is a positive fitness–density covariance³⁴ (Methods).

To reveal the conditions that can lead to coexistence in a spatially explicit context, we use a spatially explicit and individual-based³⁵ implementation of the symmetric version of the multiscale model (equation (7) and Methods). Models of this type are able to produce realistic spatial patterns consistent with mapped species distributions of large forest plots^17,35. While our analytical approach in equation (7) only allows us to make simplified assumptions about the spatial component of the recruitment process, the simulation model allows us to explore the role of the spatial component of recruitment in more detail.

Indeed, the way recruits were placed was critical for coexistence. Randomly placed recruits produced unstable dynamics (Extended Data Fig. 5a) characterized by regularity (mean of k_ff = 0.92) and segregation (mean of k_fh = 0.92), both caused by competition¹⁸, and the instability was caused by con- and heterospecifics competing equally at the population level (that is, α_fh/α_ff ≈ 1) (Extended Data Fig. 5d,g). When we followed the common approach of placing recruits with a kernel around conspecific adults^{17,18,35,36,37,38,39} to mimic dispersal limitation, we again found unstable dynamics (Extended Data Fig. 5b), despite intraspecific clustering and interspecific segregation (that is, α_fh/α_ff < 1; Extended Data Fig. 5n). The reason for the instability was high clustering of rare species^24,35 (Extended Data Fig. 6) that completely negated the potentially positive effects of α_fh/α_ff < 1.

In contrast, community dynamics can be stabilized if recruits are placed in small clusters but independent of the location of conspecific adults (Extended Data Fig. 5c). With this mechanism we mimic canopy gaps⁴⁰, animal seed dispersal⁴¹ or other mechanisms that can generate clustering independent of parent locations, as found at BCI⁴². Decoupling clustering from the parent locations does not lead to the negative relationship between clustering and abundance, and all measures of spatial patterns converged quickly into quasi-equilibrium (Extended Data Fig. 5f,i,o).

The simulation data reveal the spatial coexistence mechanism underlying the positive fitness–density covariance³⁴ (Extended Data Figs. 7 and 8). We find that the emerging spatial patterns lead to a situation where individuals of a common species are more likely to be near more neighbours and tend therefore to experience stronger competition (Extended Data Fig. 7). While the number of heterospecific neighbours remains approximately constant, the number of conspecific neighbours decreases with decreasing abundance if clustering does not change with abundance (equation (4a)). However, if clustering increases with decreasing abundance, the rare-species advantage is weakened and the dynamics become unstable²⁴.

The data of several ForestGEO forest plots were compatible with a positive fitness–density covariance (Extended Data Fig. 8f–i) as they show that, when a species becomes rare, areas of higher conspecific crowding tend to have fewer total competitors. Comparison of the results of the stable versus unstable simulation showed that even relatively weak tendencies in this relationship are sufficient to stabilize the dynamics (Extended Data Figs. 8a,b). However, this was not the case for three temperate forest plots where the power-law clustering–abundance relationship showed exponents of b < −0.5 (that is, species tended to have high clustering at low abundances), but the other plots showed b > −0.5 (Extended Data Fig. 8n–p).

The apparent contradiction with previous theoretical studies^18,24,35,37 where intraspecific clustering and interspecific segregation could not stabilize community dynamics thus arises as a consequence of the assumption of placing recruits close to their parents. This finding has important consequences for ecological theory because it shows, in contrast to the prevalent view^36,43,44, that spatial patterns alone can lead to coexistence of multiple species. This is even more important since the specific spatial patterns required for this coexistence mechanism also exist in real forests.

Conclusions

Understanding the mechanisms that maintain high species diversity in communities such as tropical forests is at the core of ecological theory, but these mechanisms are not yet fully resolved. Here, we argue that spatial patterns may play an important role in species coexistence of high diversity plant communities¹⁷. To test this hypothesis, we introduced a multiscale framework that reveals how pattern-forming processes operating at the level of individuals translate into mesoscale spatial patterns and how those patterns influence macroscale community dynamics.

We showed that the population-level interaction coefficients α_fi can have, for a broad range of common circumstances, a simpler structure than the underlying individual-level interaction coefficients β_fi. This simplicity, which emerged from spatially explicit species interactions³⁰, allowed for an analytical treatment of equilibrium, feasibility and invasion conditions of the corresponding macroscale models (equations (8–10)). Inserting the emerging α_fi coefficients into macroscale community models (for example, equation (7); Supplementary Table 3) should, in principle, allow us to take advantage of macroscale theory^{9,11,12,13,45}. However, our results also indicate that the population-level interaction coefficients may not be temporally constant as commonly assumed but depend on spatial patterns that may change with abundance. This is especially likely if recruitment is mainly located close to the parents.

It is also possible to expand our framework to take into account more detailed neighbourhood crowding indices that consider not only the number of neighboured trees of a given species but also their distance and size^19,21,26. This requires redefinition of the quantities k_ff and k_fh that describe intraspecific clustering and interspecific segregation, respectively, but does not change the overall structure of our equations. A special strength of our approach is that the population-level interaction coefficients contain measures of spatial neighbourhood patterns that can be directly estimated from fully mapped forest plots⁴. Together with additional information, this may allow for estimating network structures as well as stability of the whole community.

Our analysis revealed that communities of competing species can show a stable mode where the mesoscale patterns converge quickly into quasi-equilibrium and an unstable mode where negative relationships between species clustering and abundance emerge (Extended Data Figs. 5 and 6). The two modes are governed by the way species clustering is generated: the well-known unstable mode is related to clustering of recruits around their parents^{17,18,35,36,37,38,39} whereas the stable mode is related to clustering in locations that are independent from the parent locations, due, for example, to animal seed dispersal⁴¹ or canopy gaps⁴⁰. This result calls for a closer examination of the spatial relationship between the recruits and adults. Indeed, independent placement of recruits from conspecific large trees may not be unusual. For example, Getzin et al.⁴² found in detailed analyses of the BCI forest that recruits were for most species spatially independent of large conspecific trees. For the stable mode we could identify conditions for coexistence, and forthcoming work may extend to quantifying the ability of additional mechanisms such as niche differences^7,8, habitat associations⁴⁶, spatial and temporal relative nonlinearity^7,8 and storage effects^7,8 to alleviate the destabilizing increase of clustering if species become rare.

This study explicitly incorporates spatial patterns in theoretical models of plant communities and combines analytical theory with spatial simulations and field data analysis. Our finding that species with similar attributes may show stable coexistence has profound implications for ecological theory. Furthermore, the multiscale framework we propose here opens exciting new avenues to explain species coexistence through a spatial lens.

Methods

Study areas

Nine 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 analyses^20,35,47,48. Shrub species were also excluded.

Some of our analyses require estimation of the ratio β_fi/β_ff 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 conserved^19,26,27. 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 forests

Figure 1 and Supplementary Data Table 1 show the intraspecific variation in our three crowding indices n_kff, n_kfh and n_kfβ 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, k_ff, k_fh and B_f) 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 K_f, 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 µ_f (equation (9)). Because statistical analyses with individual-based neighbourhood models^19,26 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 equilibrium

We use a general macroscale model to describe the dynamics of a community of S species:

$$\frac{{N_f\left( {t + {\Delta}t} \right) - N_f\left( t \right)}}{{{\Delta}t}} = N_f\left( t \right)\left[ {\left( {r_f - 1} \right) + s_f\exp \left( { - \alpha _{ff}N_f\left( t \right) - \mathop {\sum }\limits_{i \ne f} \alpha _{fi}N_i(t)} \right)} \right]$$

(11)

where r_f is the mean number of recruits per adult of species f within time step Δt, s_f is a density-independent background survival rate of species f and the α_fi are the population-level interaction coefficients, yielding α_ff = c γ_ff k_ff β_ff and α_fi = c γ_fβ k_fh β_ff B_f (equation (6)). The β_fi are the assumed individual-level interaction coefficients between individuals of species i and f; k_ff = K_ff(R) / π R² and k_fh = K_fh(R) / π R² measure intraspecific clustering and interspecific segregation, respectively, with K_ff(R) being the univariate K function for species f and K_fh(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), B_f can be estimated as

$$B_f = \frac{{\bar n_{f\beta }}}{{\bar n_{f{\mathrm{h}}}}} = \frac{{\mathop {\sum }\nolimits_{i \ne f} \left[ {ck_{fi}N_i\left( t \right)} \right]\frac{{\beta _{fi}}}{{\beta _{ff}}}}}{{\mathop {\sum }\nolimits_{i \ne f} \left[ {ck_{fi}N_i\left( t \right)} \right]}} = \frac{{\mathop {\sum }\nolimits_{i \ne f} k_{fi}N_i\left( t \right)\frac{{\beta _{fi}}}{{\beta _{ff}}}}}{{k_{f{\mathrm{h}}}\mathop {\sum }\nolimits_{i \ne f} N_i\left( t \right)}},$$

(12)

and is the weighted average of the relative individual-level interaction coefficients β_fi/β_ff 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 k_fi N_i(t)). For competitive interactions, B_f ranges between zero and one; B_f = 1 indicates that heterospecific and conspecific neighbours compete equally, and smaller values of B_f indicate reduced competition with heterospecific neighbours. The denominator can be rewritten in terms of segregation k_fh to all heterospecifics and the total number of heterospecifics ∑_i≠f N_i(t).

The analytical expression of the equilibrium (equation (8)) relies on the assumption that the values of B_f are approximately constant in time. This assumption may not apply in our model during the initial burn-in phase of the simulations if the β_fi/β_ff 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 γ_ff = ln(1 + b_ff β_ff) (b_ff β_ff)⁻¹ and γ_fβ = ln(1+ b_fβ β_ff) (b_fβ β_ff)⁻¹ describe the influence of the variance-to-mean ratios b_ff and b_fβ of the gamma distribution of the crowding indices n_kff and n_kfβ, respectively. For high survival rates during one time step (for example, >85%), the values of γ_ff and γ_fβ are close to one; in this case the exponential function in equation (1a) can be approximated by its linear expansion and γ_ff = γ_fβ = 1.

In equilibrium we have (N_f(t + Δt) ‒ N_f(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 α_fh/α_ff < 1 we therefore find K_f < J^*, which indicates that a multispecies forest would host more individuals than a monoculture. To estimate J^* we sum equation (13) over all species i and find $J^{\ast} = \mathop {\sum }\limits_i \frac{{K_i}}{{1 - \alpha _{i{\mathrm{h}}}/\alpha _{ii}}} - J^{\ast} \mathop {\sum }\limits_i \frac{{\alpha _{i{\mathrm{h}}}/\alpha _{ii}}}{{1 - \alpha _{i{\mathrm{h}}}/\alpha _{ii}}}$. Therefore, we obtain

$$J^{\ast} = \mathop {\sum }\limits_{i = 1}^S \frac{{K_i}}{{1 - \alpha _{i{\mathrm{h}}}/\alpha _{ii}}} \left( {1 + \mathop {\sum }\limits_{i = 1}^S \frac{{\alpha _{ih}/\alpha _{ii}}}{{1 - \alpha _{ih}/\alpha _{ii}}}} \right)^{-1} = \frac{{Sm_K}}{{1 + Sm_\alpha }}$$

(14)

with $m_K = \frac{1}{S}\mathop {\sum }\limits_i^{\,} \frac{{K_i}}{{1 - \alpha _{i{\mathrm{h}}}/\alpha _{ii}}}$ and $m_\alpha = \frac{1}{S}\mathop {\sum }\limits_i^{\,} \frac{{\alpha _{i{\mathrm{h}}}/\alpha _{ii}}}{{1 - \alpha _{i{\mathrm{h}}}/\alpha _{ii}}}$ being averages over the S species of the community.

All species have positive abundances at equilibrium if the two conditions µ_f = α_fhJ^*/α_ffK_f < 1 and α_fh/α_ff < 1 are met (see equation (9)). We now show that the chance that these conditions are satisfied for all species is larger if the values of µ_f show little intraspecific variability. To understand this, we assume that the µ_f can be approximated by their mean $\bar \mu$. In this case $J^{\ast} /\bar \mu$ is also approximately constant and we can replace K_i in equation (14) by $(J^{\ast} /\bar \mu ) (\alpha _{i{\mathrm{h}}}/\alpha _{ii})^{-1}$ and obtain

$$J^{\ast} = \mathop {\sum }\limits_{i = 1}^S \frac{{\left( {\frac{{J^{\ast} }}{{\bar \mu }}} \right)\frac{{\alpha _{i{\mathrm{h}}}}}{{\alpha _{ii}}}}}{{1 - \frac{{\alpha _{i{\mathrm{h}}}}}{{\alpha _{ii}}}}} \left( {1 + \mathop {\sum }\limits_{i = 1}^S \frac{{\frac{{\alpha _{i{\mathrm{h}}}}}{{\alpha _{ii}}}}}{{1 - \frac{{\alpha _{i{\mathrm{h}}}}}{{\alpha _{ii}}}}}} \right)^{-1} = \frac{{J^{\ast} }}{{\bar \mu }}\frac{{Sm_\alpha }}{{1 + Sm_\alpha }}$$

(15)

where S is the number of species in the community, and therefore

$$\bar \mu = \frac{{Sm_\alpha }}{{1 + Sm_\alpha }} < 1$$

(16)

Thus, in the case of a perfect interspecific balance in µ_f we always have a feasible equilibrium if α_fh/α_fh < 1, and species can go extinct only if the intraspecific variability in µ_f becomes too large. The smaller the mean value of µ_f, the more variability in µ_f is allowed. Equation (16) shows that $\bar \mu$ is smaller if the number S of species in the community is smaller and/or if the mean value of m_α is smaller.

Equation (16) also suggests that communities with more species need to show stronger species equivalence in µ_f because the term S m_α(1 + S m_α)⁻¹ approaches a value of one for a large number of species S. This finding mirrors the results of analyses of Lotka–Volterra models with random interaction matrices¹¹ that showed that the larger the number of species S, the more difficult it becomes to generate a feasible community.

Invasion criterion

Using the population-level interaction coefficients (equation (6)) in the macroscale model, we now derive conditions for coexistence based on the invasion criterion^7,8 for a species m. The growth rate of an invading species m with low density M(t) into the equilibrium community of all other S – 1 species N_i(t) should be positive; thus, with equation (7), we have

$$\left( {r_m - 1} \right) + s_m \exp\left( { - \alpha _{mm}M\left( t \right) - \alpha _{m{\mathrm{h}}}\mathop {\sum }\limits_{i = 1}^{S - 1} N_i^{\ast} } \right) > 0.$$

(17)

Considering that $J_m^{\ast} = \mathop {\sum}\nolimits_{i = 1}^{S - 1} {N_i^{\ast}}$ and $\alpha_{mm} M(t) \ll \alpha_{f{\mathrm{h}}} J_m^{\ast}$ (that is, the invading species m is at low abundance and does not show strong clustering) we find $- \ln \left( \frac{1-r_m}{s_m} \right) > \alpha_{m{\mathrm{h}}} J_m^ \ast$, and by dividing by α_mm we obtain the invasion condition

$$\mu _m^i = \frac{{\alpha _{m{\mathrm{h}}}J_m^{\ast} }}{{\alpha _{mm}K_m}} < 1$$

(18)

which is basically identical to the condition for feasibility (equation (9)), but here the community size $J_m^ \ast$ of the reduced community appears instead of the equilibrium community size J^* of all species, including species m. Thus, a new species m is more likely to invade if it has a high value of the carrying capacity K_m and if it more strongly reduces heterospecific interactions relative to conspecific interaction (that is, α_mh/α_mm is smaller). However, if the species is too efficient (that is, has too large a capacity K_m and/or too low an α_mh/α_mm) it may increase the value of J^* too much (equation (14)), thereby causing the extinction of the weakest species with the highest values of µ_f (that is, a too-low value of K_m and a too-high value of α_fh/α_ff). Equation (18) also suggest that an equilibrium with µ_f > 1 and α_mh/α_mm > 1 will be unstable.

Fitness–density covariance

To place our new spatial coexistence mechanism in the context of existing coexistence theory, we apply scale transition theory³⁴ to our model version where spatial effects are the only potential coexistence mechanism (that is, all species have the same parameters and all individuals compete equally; β_fi/β_ff = 1, B_f = 1).

Following equation (1a), the expected fitness of an individual k of a focal species f (that is, its expected contribution to the population after some defined interval of time Δt) in the macroscale model (equation (7)) is

$$\lambda _{kf} = r_f + s_{f}f\left( {W_k} \right) = r_f + s_f \exp\left( { - \beta _{ff}\left( {n_{kff} + \mathop {\sum }\limits_{i \ne f} n_{kfi}} \right)} \right)$$

(19)

where W_k = n_kff + ∑_i≠fn_kfi is the fitness factor of individual k, f(W_k) = exp(−β_ff  W_k) is the fitness function and n_kff and ∑_i≠f n_kfi are the number of conspecific and heterospecific neighbours, respectively, of individual k within distance R. The spatial average of the fitness factor over the entire plot is

$$\bar W_k = cN_f\left( t \right) + c\mathop {\sum }\limits_{i \ne f} N_i(t) = cJ(t)$$

(20)

where c = πR² / A, and J(t) = ∑_iN_i(t) is the total number of individuals in the plot. Given that J(t) converges very quickly into equilibrium J^* (Extended Data Fig. 5 and Supplementary Figs. 1 and 2), we find for the spatial average fitness $\bar \lambda _f = 1$.

The average individual fitness $\tilde \lambda _f(t)$ of a focal species f is the average of λ_k,f over all individuals k of species f and can be estimated for the macroscale model (equations (6 and 7)) as $\tilde \lambda _f\left( t \right) = N_f\left( {t + {\Delta}t} \right)/N_f\left( t \right)$. A key ingredient of scale transition theory³⁴ is that the fitness–density covariance is given by ${\mathrm{cov}} = \tilde \lambda _f\left( t \right) - \bar \lambda _f$. With equation (3) and γ_ff ≈ γ_fβ ≈ 1 and $\bar n_{f\beta } = \bar n_{f{\mathrm{h}}}$ we find

$$\tilde \lambda _f\left( t \right) - \bar \lambda _f = (r_f - 1) + s_f \exp( - \beta _{ff}(\bar n_{ff} + \bar n_{f{\mathrm{h}}}))$$

(21)

where the mean of the crowding indices is given by $\bar n_{ff}(N_f) = ck_{ff}(N_f)N_f$ and $\bar n_{f{\mathrm{h}}}(N_f) = ck_{f{\mathrm{h}}}(J^{\ast} - N_f)$ (equation (4)). Therefore, if clustering k_ff and segregation k_fh are independent from abundance N_f, more abundant species have more neighbours, since

$$\bar n_{ff} + \bar n_{f{\mathrm{h}}} = c(k_{ff} - k_{f{\mathrm{h}}})N_f + ck_{f{\mathrm{h}}}J^{\ast}$$

(22)

Thus, a positive fitness–density covariance in our model means that individuals of a common species are more likely to be near more trees in total.

Extended Data Fig. 7 shows the quantities $\bar n_{ff} + \bar n_{f{\mathrm{h}}}$, $\bar n_{ff}$, $\bar n_{f{\mathrm{h}}}$ and $\tilde \lambda _f - \bar \lambda _f$ plotted over abundance N_t for data generated by our spatially explicit simulation model for the scenarios of stable and unstable dynamics (Extended Data Fig. 5b,c). Indeed, the stable simulations show a positive fitness–density covariance, however, there is no such trend for the dynamics of the unstable community (Extended Data Fig. 7g,h).

Spatial patterns will act as positive fitness–density covariance if, when a species becomes rare, areas of high conspecific crowding have fewer competitors. We tested this for the data generated by our simulation model and for the nine forest plots (Extended Data Fig. 8). We could estimate for each focal species f the covariance between the number of conspecific neighbours (that is, n_kff) and the total number of neighbours (that is, n_kff + n_kfh) and demand that the covariance should be mostly positive and larger for more abundant species. However, since the quantity n_kff appears in this test on both sides, a positive covariance can be expected. To compensate for this artefact, we instead used the covariance between the local dominance of conspecifics in the neighbourhood of individuals k (that is, d_kff = n_kff (n_kff + n_kfh)⁻¹) and total number of neighbours (that is, n_kff + n_kfh) (Extended Data Fig. 8).

Spatially explicit simulation model

The model is a spatially explicit and stochastic implementation of the spatial multispecies model (equation (7)), similar to that of May et al.^35,37 and Detto and Muller-Landau¹⁷, and simulates the dynamics of a community of S tree species in a given plot of a homogeneous environment (for example, 50 ha) in 5 yr time steps adapted to the ForestGEO census interval (Extended Data Fig. 5 and Supplementary Figs. 1 and 2). Only reproductive (adult) trees are considered, but size differences between them are not considered. During a given time step the model first simulates stochastic recruitment of reproductive trees and placement of recruits, and second, stochastic survival of adults that depends on the neighbourhood crowding indices for conspecifics (n_kff) and heterospecifics (n_kfβ) (but excluding recruits). In the next time step, the recruits count as reproductive adults and are subject to mortality. No immigration from a metacommunity is considered. To avoid edge effects, torus geometry is assumed.

The survival probability of an adult k of species f is given by $s_f \exp \left(-\beta_{ff} \left( n_{kff}+n_{kf\beta}\right)\right)$ (equation (1a)). The two neighbourhood indices n_kff and n_kfβ describe the competitive neighbourhood of the focal individual k and sum up all conspecific and heterospecific neighbours, respectively, within distance R, but weight them with the relative individual-level interaction coefficients β_fi/β_ff (refs. ^19,21,26).

Each individual produces on average r_f recruits, and their locations are determined by a type of Thomas process²⁸ to obtain clustering. To this end, the spatial position of the recruits is determined by two independent mechanisms. First, a proportion 1 – p_d of recruits is placed stochastically around randomly selected conspecific adults by using a two-dimensional kernel function (here a Gaussian with variance σ²). This is the most common way to generate species clustering in spatially explicit models^{17,18,35,36,37,38,39}. Specifically, we first randomly select one parent for each of these recruits among the conspecific adults and then determine the position of the recruit by sampling from the kernel. Second, the remaining proportion p_d of recruits is distributed in the same way around randomly placed cluster centres that are located independently of conspecific adults. This mode mimics spatial clustering of recruits independent of the parent locations⁴² in a simple way, such as contagious seed dispersal by animals⁵⁰ or forest gaps that may imprint clumped distributions of recruits of pioneer species⁴⁰. For each species we assume a density λ_fc of randomly distributed cluster centres, which have, at each time step, a probability p_fp of changing location. For each of these recruits, we first randomly select one cluster centre among the cluster centres of the corresponding species and then determine the position of the recruit by sampling from the kernel. For the simulation shown in Extended Data Fig. 5a, the recruits were located at random positions within the plot.

Parameterization of the simulation model

Extended Data Fig. 5 shows simulations of the individual-based model conducted in a 200 ha area containing approximately 83,000 trees with, initially, 80 species. There was no immigration. The model parameters were the same for all species, and all species followed exactly the same model rules. We selected β_fi = β_ff to obtain no differences in con- and heterospecific interactions and s_f = 1 (no background mortality), and we adjusted the parameters β_ff = 0.0075 and r_f = 0.1 to yield tree densities (415 ha⁻¹) and an overall 5 yr mortality rate (10%) similar to those of trees with dbh ≥ 10 cm in the BCI plot⁵¹.

The Gaussian kernel used to place recruits around conspecific adults or around random cluster centres had a parameter σ = 10 m. There were 40 random cluster centres in total for each species that had a probability of p_fp = 0.3 of changing location within one census interval. The only difference between the simulation shown in Extended Data Fig. 5b and the one shown in Extended Data Fig. 5c is that in the former, we used a proportion p_d = 0.05 of recruits to be placed around randomly distributed cluster centres (that is, 95% of the recruits were placed close to their parents), but in the latter, we selected p_d = 0.95 (that is, 95% of the recruits were placed around randomly distributed cluster centres). In our simulations, on average, one of these cluster centres received four recruits per time step, which were scattered within a radius of approximately 30 m, and received approximately 13 recruits during its lifetime (at each time step it had a probability of 0.3 of changing location). In contrast, in Extended Data Fig. 5a recruits were placed at random locations within the plot.

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