Accounting for spatial autocorrelation in null models of tree species association

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1 Ecography 34: , 2011 doi: /j x 2011 The Authors. Journal compilation 2011 Nordic Society Oikos Subject Editor: Pedro Peres-Neto. Accepted 14 June 2011 Accounting for spatial autocorrelation in null models of tree species association Michael M. Fuller and Brian J. Enquist M. M. Fuller Dept of Biology, Univ. of New Mexico, Albuquerque, NM, USA. B. J. Enquist, Ecology and Evolutionary Biology, Univ. of Arizona, Tucson, AZ, USA, and The Santa Fe Inst., Santa Fe, NM, USA. A commonly used null model for species association among forest trees is a well-mixed community (WMC). A WMC represents a non-spatial, or spatially implicit, model, in which species form nearest-neighbor pairs at a rate equal to the product of their community proportions. WMC models assume that the outcome of random dispersal and demographic processes is complete spatial randomness (CSR) in the species spatial distributions. Yet, stochastic dispersal processes often lead to spatial autocorrelation (SAC) in tree species densities, giving rise to clustering, segregation, and other nonrandom patterns. Although methods exist to account for SAC in spatially-explicit models, its impact on non-spatial models often remains unaccounted for. To investigate the potential for SAC to bias tests based upon non-spatial models, we developed a spatially-heterogeneous (SH) modelling approach that incorporates measured levels of SAC. Using the mapped locations of individuals in a tropical tree community, we tested the hypothesis that the identity of nearest-neighbors represents a random draw from neighborhood species pools. Correlograms of Moran s I confirmed that, for 50 of 51 dominant species, stem density was significantly autocorrelated over distances ranging from 50 to 200 m. The observed patterns of SAC were consistent with dispersal limitation, with most species occurring in distinct patches. For nearly all of the 106 species in the community, the frequency of pairwise association was statistically indistinguishable from that projected by the null models. However, model comparisons revealed that non-spatial models more strongly underestimated observed species-pair frequencies, particularly for conspecific pairs. Overall, the CSR models projected more significant facilitative interactions than did SH models, yielding a more liberal test of niche differences. Our results underscore the importance of accounting for stochastic spatial processes in tests of association, regardless of whether spatial or non-spatial models are employed. Ecologist frequently rely upon null models to separate the effects of niche-sorting processes from those of random and neutral processes. To paraphrase Gotelli and Graves (1996), a null model for a putative ecological mechanism is constructed by randomizing the elements of an empirical pattern that pertain to a given hypothesis, while holding other elements constant. Although null models often involve comparisons of species occurrences among multiple, geographically distinct sites (Gotelli and Graves 1996), the general approach of building a null model by randomizing an observed pattern is also used to construct null models for a single community (Lotwick and Silverman 1982, Palmer and van der Maarel 1995, Roxburgh and Chesson 1998). As with multi-site null models, single community null models generally depict species as randomly and independently distributed with respect to each other, and are often used to test for nonrandom association. Null models for species coexistence among multiple sites generally do not represent a completely random pattern. Typically, null models account for observed limits on the number of species a site can support, or restrict the frequency of co-occurrences to be no higher than observed frequencies. Similarly, null models for species association within a single community (i.e. at a single site) should also account for nonrandomness that is unrelated to the process under investigation. In tree communities, dispersal processes induce nonrandom variation in species densities, such as the formation of a seed shadow, wherein conspecifics are grouped into patches that show positive spatial autocorrelation (SAC). In these patches, which typically range from 75 to 200 m or more (Condit et al. 2000, Plotkin et al. 2000, Seidler and Plotkin 2006), the density of offspring decreases as a gradient with increasing distance from the parent. The concentration of conspecifics within patches can limit the diversity of local neighborhoods, reduce the number of heterospecific contacts among species, and therefore influence patterns of association (Plotkin et al. 2000, Harms et al. 2001). The autocorrelated residual variation in the density of species that occur in patches, or that show SAC for other reasons, can lead to an inflated rate of type I errors if association is tested against a completely random pattern, or using a statistical test that assumes spatial independence within and among species (Legendre and Fortin 1989, Legendre 1993, Clark et al. 1998, Lennon 2000, Dungan et al. 2002). Several methods have been devised to account for SAC when using spatially-explicit models to test for nonrandom species association within a community (discussed below). However, the bias associated with SAC remains a significant Early View (EV): 1-EV

2 problem for non-spatial models of association, such as those that compare the observed rate at which species form nearest-neighbor pairs to that expected of a well-mixed community. In a well-mixed community, species are assumed to be independently distributed and show a pattern of complete spatial randomness (CSR; Wiegand and Moloney 2004). Therefore, a common method is to base the expected pairing frequencies on the species community proportions, P i n i /N, where n i and N are the number of individuals of the species and community, respectively. If pairing occurs randomly, then the expected rate of pairing for any two species, i and j, in a well-mixed community, is P i 3 P j. The use of non-spatial models permits comparisons of species-pairing frequencies when explicit data on the species spatial distributions is lacking. Yet, whereas many spatiallyexplicit models account for the influence of SAC on association, non-spatial models commonly do not (Lieberman and Lieberman 2007, Kraft et al. 2009, Perry et al. 2009). By failing to consider the influence of stochastic dispersal processes, many non-spatial models risk overestimating the importance of niche differences on patterns of association. To investigate the potential bias of non-spatial null models on tests of species association, we developed a new approach for incorporating SAC into spatially-explicit null models. Our method of constructing null models yields spatially heterogeneous (SH) patterns of association, which incorporate observed patterns of SAC. The SH models are designed to account for the influence of dispersal processes on the rate that species form nearest-neighbor pairs. We used a stem map, compiled for a tropical tree community, to compare tests based upon the SH null models to those based upon non-spatial models, the latter which assume that the community is well-mixed, as described above. We used the comparison to quantify the ability of null models to detect elevated and suppressed rates of species pairing, which may be associated with asymmetrical species interactions and niche differences. Our test statistics included the frequency at which the species formed nearest-neighbor pairs, and the proportion of nearest-neighbor pairs that were conspecifics. In using nearest-neighbor relationships, we focused on the niche-based processes that influence the survival of individuals: competition, facilitation, and suppression. These processes operate at the lowest scales of species association (0 50 m; Silander and Pacala 1985). Thus, we investigated the influence of the traits of individuals on the survival of their nearest neighbors. Approaches for constructing null models of tree communities A variety of spatially-explict approaches have been proposed that, like ours, are intended to control for SAC when investigating the causes of species association within communities. The best known of these approaches include methods for estimating the parameters of general linear models, such as linear regression (Keitt et al. 2002), and methods for constructing null models by permuting or simulating observed patterns. As our focus is on null model construction, we will say no more about methods for parameter estimation. Methods developed for constructing null models from an observed community can be grouped into four categories: 1) toroidal shift and related methods (Lotwick and Silverman 1982, Palmer and van der Maarel 1995), 2) permutation of species among sampling units (Wilson et al. 1987), 3) permutation of the sampling units (Roxburgh and Chesson 1998, Harms et al. 2001), and 4) point process models (Batista and Maguire 1998, Plotkin et al. 2000, John et al. 2007). The purpose of all of the above methods is to construct simulated communities, in which the species are distributed independently of each other. Toroidal methods depict the study plot as a torus, by applying periodic boundary conditions to the plot, in which the edges are wrapped around so that they join. A null model is then constructed by shifting the distribution of each species, as a whole, a random distance and direction on the torus. In the second category, null models are constructed by randomly reallocating species to sampling units. An alternative (category 3) is to permute the sampling units themselves, preserving the original mix of species in each unit, as well as other geometric or statistical properties of each species distribution. By contrast, point process methods generate null models by constructing an artificial point map for each species that has statistical properties similar to the observed distribution. In a point map, individuals are represented as points in the plane. The spatial growth that gives rise to species clusters can be simulated as a point cluster process (e.g. a Neyman Scott process). See Batista and Maguire (1998) and Illian et al. (2008) for details on point process methods. Our construction of SH models differs from the above methods in two important respects. First, we do not permute the spatial arrangement of each species independently of other species. Instead, we jointly permute species labels over short distances. Our approach was specifically designed to preserve the SAC observed for each species, as well as the influence of SAC on species association. It also leaves undisturbed each species spatial distribution across the plot. As a consequence, our approach preserves the correspondence of each species distribution with the environmental conditions of the plot. The disruption of the species environment relationship has been a criticism of toroidal methods (Wilson 1995, Dixon 2002). Second, the spatial scale of permutation we use to construct our null models is much lower than that of most previous methods. For example, toroidal methods shift the entire distribution (e.g. stem map) of each species. Methods that randomly allocate species among sampling units generally permit species to be redistributed across the entire plot. Likewise, methods that permute the sampling units (e.g. quadrats) also utilize the full area of the plot. Such methods are intended to destroy the observed spatial organization of the community. By contrast, in limiting the exchange of labels to stems that occur within a short distance of each other (generally, 40 m), our method constrains permutation to the scale of local neighborhoods. Therefore, our method preserves the larger spatial distribution and co-distribution of the species. Yet, because we perform relabelling within every array cell, the small-scale correspondence of the species (i.e. nearest neighbor identities) is randomized across the entire plot. We are not the first to incorporate observations of SAC into null models of association. In a study of disease transmission, McElhany et al. (1995) parameterized a Markov chain 2-EV

3 model using a range of SAC distances, which bracketed the observed SAC distances. Here, we applied bracketing as well, but for the purpose of scale analysis, not parameter estimation. We are also not the first to employ spatially constrained methods to generate null models of tree communities. For example, Wiegand et al. (2007) used a moving-window approach to construct locally-randomized stem maps. Our comparison of SH models to CSR models is in some ways similar to the comparison by Wiegand et al. (2007) of heterogeneous point process models to homogeneous models. However, we focused on the influence of SAC on small-scale association, whereas the goal of Wiegand et al. (2007) was to separate the independent contributions of environmental conditions and species interactions to the species plot-scale distribution patterns. Methods Study system To investigate the influence of SAC on association, we used a stem map established within a 14 ha rectangular plot ( m) at the San Emilio dry forest of northwestern Costa Rica (Enquist and Enquist 2011). The stem map contains the species identity, stem diameter, and geographic coordinates of all trees with a basal stem diameter at breast height 3 cm ( individual trees, 106 species; for species names and abundance, Supplementary material Appendix 3, Table A1). The topography, degree of deciduousness, and age of the stand changes across the plot. Tree density across the study plot also shows considerable variation, with areas of high and low tree density and a NE-SW trending density gradient on the southern half (Fig. 1a; constructed using the Figure 1. Observed spatial heterogeneity. (a) Kernel-smoothed density of individual stems, depicting the variation in density across the study site. (b) Distribution of the 10 most abundant species on the site, differentiated by color. Circles represent stem crowns, with diameter proportional to stem diameter. R spatstat package (Baddeley and Turner 2005). In addition, the species differ in their distribution on the plot, with some species occurring throughout the plot, while others are confined to certain subregions (Fig. 1b). Estimating the radii of species patches: correlogram analysis Measuring spatial autocorrelation: correlograms We constructed correlograms of Moran s I (Moran 1950) to estimate the mean SAC in the density of each species. A correlogram quantifies the distances, or spatial scale, over which a pattern shows positive and negative autocorrelation. When a species is aggregated into patches due to a contagious process, such as dispersal, its density is highest near the source of individuals (e.g. a parent tree), and declines with distance from the source. At the patch scale, the correlogram reveals the rate at which the patch density decreases from the source, and provides an estimate of the mean patch size. The distance at which Moran s I first reaches zero (i.e. no correlation among samples) is an estimate of the mean patch radius (Legendre and Legendre 1998, pp , Fortin and Dale 2005, pp ). We define the minimum patch radius as the smallest mean radius shown by any of the species examined. To compute stem densities, we divided the stem map into discrete grid cells of equal size. In each cell, we computed each species density as its stem count divided by the area of the cell. The cell densities provided the samples needed to compute the correlograms and to determine the distance classes at which each species showed significant SAC. For each distance class, we determined whether the observed levels of aggregation were statistically significant by comparing the empirical correlograms, representing the observed stem map, to correlograms representing 999 null models, in which the location of each sample of the original map was randomly permuted. We used the R ncf package (ver ) to compute and test the correlograms and establish the confidence envelopes (R Development Core Team 2006). The ncf package provides a resampling function for generating confidence intervals, by permuting samples, and we deemed the ncf method to be suitable to our purpose. For significance testing, we used an a probability of 0.05 (i.e. the 97.5 and 2.5 percentiles of the null models, representing a two-tailed test). We also performed an analysis of scale effects, to determine the spatial scale at which SAC in species density was most apparent (Supplementary Material, Appendix 1, I). Critical-value correction for correlogram analysis It is important to note that the value of Moran s I for each distance class of a correlogram is a cumulative statistic that represents an average over the entire plot. The values recorded for larger classes encompass those of smaller classes, such that the distance classes are not statistically independent. Significance tests for cumulative statistics are susceptible to an elevated probability of type I errors because of simultaneous inference and non-independence among classes (Loosemore and Ford 2006). To compensate for the elevated error rate, we adjusted the a probability level used to test the significance of each distance class, using the method of 3-EV

4 progressive Bonferroni correction (Hewitt et al. 1997). For each of the i (1, 2,, k) classes, we adjusted the percentage of the null model values that the observed value must exceed for a finding of statistical significance, by dividing the a probability by the rank of the class, i, as a/i. Thus, the percentage for the first distance class was a/1, the percentage of the second class was a/2, and so forth. The progressive Bonferroni correction reduced the a level necessary for a positive test to between 0.05 and for the number of distance classes we computed. Null model construction and testing We constructed the CSR models using the same method employed by Lieberman and Lieberman (2007) and Perry et al. (2009) to simulate a well-mixed community. For each instance of a CSR model, we randomly drew pairs of individuals from the observed community, with replacement. The probability of choosing an individual of a particular species was equal to the species community proportion only, with no regard to its spatial distribution. As an alternative to CSR, we constructed spatially-explicit, spatially-heterogeneous (SH) null models, that incorporated SAC in species density. As we did for estimating SAC, we divided the stem map into discrete cells prior to relabelling. To avoid confusion with the grids used to estimate SAC, hereafter we refer to the grids used for relabelling as arrays. Within each array cell, we uniformly randomized the nearest neighbor identities by exchanging the species labels among individual stems. This procedure caused the stem labels to be independently and uniformly distributed among the stems found within each cell, creating a Poisson distribution of stem labels. Because the density of each species changes across the plot, the probability that a stem in a given cell is labeled with a particular identity also changes with location. Formally, this means that the intensity parameter of the Poisson pattern of stem labels is spatially heterogeneous. In other words, we constrained relabelling to small spatial scales, but applied it throughout the community, an approach suggested by Manly (1997) and Wiegand and Moloney (2004) and applied by Wiegand et al. (2007). Using the above procedure, we constructed 999 separate SH models (i.e. null model communities). To determine the optimum array cell size to use for relabelling, we compared the SAC of the observed stem map to the map produced by our SH procedure, with the array cell size initially set to the minimum patch size indicated by the correlograms. If the relabelling procedure significantly altered the mean SAC of the SH models, we decreased the array cell length until the influence of relabelling was reduced to a non-significant difference. The details of this procedure are provided in Supplementary material Appendix 1, II. Test statistics: nearest-neighbor frequencies and conspecific proportions We quantified differences in species association among the observed community and null models using contingency tables. The tables contained the frequency of nearest-neighbor pairs for all S 3 S pair-wise combinations of the S species in the community, including conspecific pairs (Pielou 1961, Dixon 2002). Tallies of nearest-neighbor species-pairs are widely used as a measure of species association. Second order statistics (e.g. Ripley s K) have also been used as a measure of association, but for our purpose K is not suitable, as it yields a cumulative metric (average association) across an entire study plot, and we needed a metric that would detect association within discrete neighborhoods. Moreover, K is sensitive to SAC in species densities, and may therefore give misleading results (Plotkin et al. 2000). We compiled the frequencies of species pairs from the observed and randomized stem maps, by applying, for each individual stem, a two-step process. First, we identified the species identity of an individual stem, which we referred to as the focal stem. Second, we determined the identity of the individual that was the shortest distance to the focal stem, based upon the geographic coordinates and diameter of each stem (Simberloff 1979). Each pair of individuals (focal stem nearest neighbor) represents an ordered pair of species, AB, in which the first species (A) represents the focal stem, and the second (B) its nearest neighbor. The frequency of a nearest-neighbor pair, AB is the number of times a species, B, is a nearest neighbor of a particular species, A (Pielou 1961). In addition to the species-pair frequencies, for each species we also computed the proportion of nearest neighbors that represented conspecifics, which we denote as y. Using a similar approach, Plotkin et al. (2000) computed, as P d, the proportion of trees in a plot, distance d apart, which were the same species. Yet, whereas they computed P d over a range of distance classes, we computed y only for nearest neighbors. The index y serves as an indirect measure of the effect of aggregation on species patterns (an issue discussed in some depth by Dixon (1994) and Plotkin et al. 2000). We computed y separately for the CSR (y csr ) and SH (y SH ) models. The value of y CSR is determined strictly by the relative abundances of the species in the stand, whereas y SH depends on the species relative abundances, and the strength of SAC and other aggregative processes. Therefore, for the short distances involved with nearest-neighbor species pairs, we expect y SH y CSR when SAC influences species aggregation (Plotkin et al. 2000). Differences between the SH models and the observed community indicate the influence of deterministic processes on aggregation, in addition to dispersal. Such differences may also be due to environmental heterogeneity at scales greater than the array cell size. For comparisons, we computed the ranked distribution of y for the observed community and the null models, ranking the species in decreasing order of y. Test statistics: significance tests We compared the observed species-pair frequencies, and the distribution of ranked y, to the mean and 95% confidence intervals of the model distributions, representing the 999 iterations, computed for each of the SH and CSR models. We considered the deviations from the observed distributions to be significant if any of the observed values were outside the 95% confidence intervals of the models (two-tailed test). Yates (1934) recommended samples of 5 4-EV

5 when using contingency tables and the χ 2 test to assess the significance of association. Although we did not use the χ 2 test, the application of a minimum sample size as a threshold for statistical tests is appropriate from the perspective of statistical power (type II error rate). To determine the effect of different threshold levels on the outcome of our tests, we repeated the comparisons, using, for the threshold, a sample size ranging from species-pair occurrences, in steps of 10. To improve our ability to detect weak niche effects, we also restricted our analysis to those species with 50 or more individuals, which included 51 species, representing 27 families. As a group, these 51 species accounted for 94.6% of the stems on the plot. Results Spatial autocorrelation The majority of the 51 species we examined showed distinct patch boundaries as indicated by a steep decline in Moran s I over distances, 150 m. Figure 2 shows the correlograms of two species, chosen to represent the typical pattern we found (see Supplementary material Appendix 3, Fig. A2 for correlograms of all 51 species). All but one of the species, Simarouba glauca (DC), showed significant levels of SAC, although for 12 species the correlogram was significant for only one or two distances classes. Most of the latter 12 species were represented by fewer than 100 individuals, and probably represent species for which recruitment is low. The correlograms also indicated that a few species show almost no SAC in their density, while others showed evidence of long distance trends. A number of species evidenced limited periodicity in their spatial patterns, indicating that for these species, patch size is relatively consistent. The strength and spatial extent of SAC, and the mean patch size, was often, but not always, similar among members of the same taxonomic family (Supplementary material Appendix 3, Fig. A2). Our analysis of scale effects (Supplementary material Appendix 2, II) revealed that SAC was generally strongest over distances in the range of m. SH model construction and testing We used the minimum observed patch size, which we based upon the correlograms, as the initial length of the cells used for relabelling (see section: Null model construction and testing). The distribution of patch sizes of the 51 species examined was right-skewed, with a minimum, maximum, mean, and standard deviation of 40, 230, 94.88, and m, respectively. Thus, the majority (75%) of species that we examined were spatially autocorrelated at a scale of 100 m or less. The patch size distribution indicated that relabelling at distances 50 m would incorporate the observed pattern of local aggregation for most species. However, when we compared the correlograms of the observed community to those of the SH models, we found that relabelling within cells 25 m length led to significant changes in SAC between the observed and modelled communities. Therefore, we set the length of the cells used for relabelling to 25 m. This yielded 35.4 m as the maximum distance between any two stems that swapped labels. Nearest-neighbor frequencies Of the 708 observed species combinations with 5 occurrences, representing nearest-neighbor pairs, only a small percentage were significantly more or less frequent than projected by the models. The effect of increasing the minimum threshold for the occurrence of species pairs (see section: Test statistics: significance tests) was to reduce the number of unique species pairs available for the analysis. When the maximum threshold frequency of 50 was used, only 21 unique pairs of species met the criterion. However, because most pair frequencies generated by the models did not differ significantly from the observed community, changing the threshold frequency had no effect on the absolute number of observed pairs that were significantly different from either the Figure 2. Spatial autocorrelation (Moran s I ) in stem density. Correlograms for two species are shown (see Supplementary material Appendix 3, Table A1 for correlograms of remaining 49 species). Solid horizontal line zero correlation. Open circles values not significant based upon 999 permutations. Closed circles significant values. Values in parentheses correspond to the species plot abundance. 5-EV

6 SH or CSR models. Thus, changing the threshold by an order of magnitude reduced the number of comparisons made between the observed community and the models, but not the statistical significance of the differences that we found. The CSR models projected» 10% more unique speciespair combinations than were observed, whereas the SH models projected» 4% more unique combinations than observed. The models also differed in the rate of pairing for those pairs that were significantly more or less frequent, relative to the observed community. Figure 3 shows the 50 most frequent species-pair combinations (38 heterospecific pairs and 12 conspecific pairs), with the 95% confidence intervals of the models included for comparison (species names and pairing frequencies shown in Supplementary material Appendix 3, Table A2). The CSR models found that 18 (2.50%) of the species pairs occurred at a significantly higher rate, on average. Of these 18, 11 represented heterospecific pairs and 7 represented conspecific pairs. By contrast, the SH models indicated that only 8 species pairs (1.00%) had significantly higher rates of pairing than in the observed community (representing 2 heterospecific pairs and 6 conspecific pairs). Table 1 shows the total number of heterospecific and conspecific frequencies in the observed community, and mean numbers for the null models. There were a total of 673 heterospecific pairs in the observed community with a frequency of at least five occurrences, of which none were symmetric in their rate of pairing (frequency of AB frequency of BA). In the CSR and SH null models, the mean number of heterospecific pairs was slightly higher than observed, with the CSR mean being» 13% higher, and the SH mean» 6% higher, but the differences between the models and observed community were not significant at the 0.05 level (Table 1). In the observed community, 35 species had 5 or more conspecifics as a nearest neighbor. The corresponding mean value of the CSR models was 18.6, which is» 47% fewer than observed, a difference that was significant at the a 0.05 level. The mean of the SH models was 23.5, which is» 33% fewer than observed, but not significantly different. Thus, the number of conspecific pairs projected by the SH models was not statistically different from the observed number, whereas the CSR models significantly underestimated the rate of conspecific pairing for some species. Conspecific proportions, y We found a striking difference between the CSR and SH models in their projections for y, the proportion of nearestneighbors that were conspecifics. Figure 4 shows the distribution of y for the observed community and null models (50 most frequent pair combinations shown). For the group of 35 species with at least five conspecific neighbors, the CSR models indicated that 33 species (94.3% of the group, 31% of the community) had significantly higher values of y. By contrast, the SH models indicated that 18 species (51.4% of this group; 17.0% of the community) had significantly higher y. Thus, while both the SH and CSR models projected significant differences in y, the SH models projected 55% fewer than did the CSR models, clearly showing that incorporating into null models the observed levels of SAC improved their agreement with observed patterns of aggregation. The SH models also projected much greater variation in y among the species (Fig. 4), further revealing the influence of the observed spatial structure on patterns of local aggregation and density. Figure 3. Comparison of SH and CSR models in the frequency of pair-wise species combinations, for pairs with at least 5 occurrences. Shown are the 50 species combinations with the highest frequency of pairing, inclusive of conspecific (plus symbol) and heterospecific (filled circle) pairs. Solid line: mean frequency of 999 models. Dashed lines: 95% percentiles of models. The CSR models underestimate the observed species pair frequencies more often and more strongly than do the SH models. Table 1. The total number of species-pair combinations with a frequency of occurrence 5, recorded for the observed community and null models. Obs observed community. CSR and SH refer to the mean values for 999 CSR and SH models (see text). Types of species pairs: Hetsp heterospecific species pairs; Consp conspecific species pairs. Asterisk indicates significance at the a 0.05 level (twotailed test). Type Obs CSR SH Hetsp Consp * EV

7 Figure 4. Comparison of SH and CSR models in y, the proportion of nearest neighbors that are conspecifics. Shown are the 50 species with the highest number of conspecific proportions. The SH model yields a substantially better fit to the observed proportions. Solid line: mean frequency of 999 models. Dashed lines: 95% percentiles. Discussion We examined the influence of stochastic spatial structure (dispersal-related SAC) on the ability of null models to detect elevated and suppressed rates of species pairing, which may be associated with asymmetrical species interactions and niche differences. Our results revealed 1) how strongly SAC can influence tests of association, and 2) the extent to which spatial structure does not influence species association. This last point is critical to an analysis of the neutrality of species interactions and the relative importance of niche-based processes. The results clearly illustrate that non-spatial models that assume a well-mixed community can yield an elevated rate of type I errors when used in the presence of significant SAC. Because the SH models incorporated observed levels of positive SAC, the number of significant species pairs they projected was less than that of the CSR models, yielding a more conservative test. The comparison of the model patterns suggests that stochastic dispersal processes contributed to the frequency of pairing for many species. However, dispersal limitation is not the only source of positive SAC in tree densities. For example, species interactions may have positive or negative effects on density, while species responses to environmental gradients can induce SAC in the direction of the gradient. The SH models should reveal instances of strong facilitation, yet weak facilitation can increase a species local density to the point of generating SAC. Thus, by preserving local spatial patterns, our relabelling method may have masked the influence of weak facilitation for some species. By contrast, asymmetric competition and Janzen Connell effects, which include frequency-dependent seed-predation, herbivory, and disease (Janzen 1970, Connell 1971), generally reduce species densities. Such negative interactions can diminish or eliminate positive SAC related to dispersal, or change the character of SAC from positive to negative. Therefore, if present, negative interactions could also have interfered with our ability to detect the effects of niche differences for some species pairs. With respect to the potential of environmental gradients to induce SAC, a comparison of the stem density map to the patterns revealed by the correlograms provides some insight into the relative influence of environmental conditions, and dispersal, on patterns of association. The overall pattern of stem density (Fig. 1a) reveals obvious contours and gradients of m in stem density. The contours in stem density suggest that topographical patterns influence growth conditions for all trees generally (Hubbell and Foster 1986, Enquist and Enquist 2011). Yet, most of the species are not equally represented in all parts of the plot (Fig. 1b), suggesting that processes operating at small to intermediate spatial scales ( m), in addition to topography, influence their distribution. By contrast to the stem maps, the correlograms permit us to see statistical patterns in the species small to intermediate scale distributions. The correlograms of a few species revealed trends in SAC that occur at about the same scale as the stem density gradients, suggesting that environmental gradients may have influenced the density of those species. However, for most of the species, the correlograms showed significant SAC over shorter distances ( m), which is strong evidence for dispersal-induced aggregation. Local variation in soil conditions may also have contributed to the observed patterns of SAC. However, in a study on the influence of soil nutrients on tree species patterns, John et al. (2007) found that the scale of SAC in soil conditions was m, with soil variation closely following broad topographic patterns. By contrast, the SAC in tree density in our study was generally strongest over distances of 100 m. Thus, there is evidence that at intermediate scales, environmental gradients contributed to the positive SAC in the density of a few species. But for the majority of the species, positive SAC occurred over the small scales associated with dispersal processes, suggesting that dispersal-generated SAC influenced species densities and therefore contributed to patterns of association. The influence of SAC on the species distribution patterns is further indicated by the large differences in model projections for y, the proportion of nearest neighbors that are conspecifics. The CSR models greatly underestimated the 7-EV

8 observed proportions of the most abundant species, whereas the SH models often agreed with observations. Coupled with the evidence for environmental effects discussed above, the disparity between the null models for y (Fig. 4) suggests that dispersal can account for a substantial amount of the stochastic variation in density that would otherwise be attributed to niche-based processes. The differences indicated by y also show that, while raw species-pair frequencies are often used to quantify patterns of association, such as segregation (Pielou 1961, Lieberman and Lieberman 2007, Perry et al. 2009), y and similar indices (Dixon 1994) are more sensitive to changes in the strength of aggregation within species, and are thus better able to detect nonrandom and non-neutral patterns. In addition to revealing the potential of dispersal to influence statistical tests, our results indicated that, in a few cases, species association is influenced by positive interactions. Specifically, the finding of significantly elevated pairing rates in the SH models suggests that the dynamics of establishment were not neutral for the species involved, but involved facilitation. Our SH models were designed to detect the influence of facilitation (i.e. significantly high species-pair frequencies) as well as suppression (i.e. significantly low frequencies). By contrast, the CSR models indicated a much higher incidence of facilitative interactions. This result reveals the potential for bias when non-spatial models are used as a test of asymmetric interactions and niche differences. Interestingly, the significant differences indicated by the SH models involved only elevated rates of pairing, suggesting that negative interactions may be less important to the community dynamics. However, aggregative processes, such as dispersal, can offset the effects of negative interactions, potentially obscuring their influence. Studies have shown that species interactions typically operate over distances of, 20 m (Hubbell et al. 2001, Uriate et al. 2004), which is shorter than the length of the array cells we used for relabelling. Therefore, our relabelling procedure should have been sufficient to remove the signal of direct interactions, yielding, in the SH models, a suitable null model for the influence of species interactions on association. The method of relabelling individuals over short distances preserves not only the observed patterns of SAC, which are most evident over distances of m, but also the species larger scale distribution patterns. Considering the amount of observed structure that was incorporated into the SH models, one might assume that the better fit of the SH models may chiefly reflect the segregation of the species according to broad scale environmental heterogeneity, rather than dispersal-related SAC in species density. For example, segregation of the species into groups associated with distinct habitat types would strengthen the differences between the CSR and SH models. Yet, we found that nearly all of the 673 observed heterospecific species pairs occurred no more or less often than as projected by either the SH or CSR null models. Of the observed species pair frequencies, only 1% were more frequent than SH projections, while 2.5% were more frequent than CSR projections. That the percentage of significant differences was more than twice as high in the CSR models reveals the extent to which spatial structure influenced the outcome of statistical tests. However, because only a small fraction of the species pairs projected by either model were significantly more frequent than observed, it is evident that segregation related to broad-scale environmental heterogeneity does not strongly influence the rate of species pairing in this community. No general rule has been established for the spatial scale at which dispersal effects become important for patterns of association. The scale, or plot size, at which dispersal influences association depends upon the range of dispersal distances observed, which in turn is influenced by the traits of the particular species that form the community, such as dispersal mode and seed size. In tropical forest studies, Seidler and Plotkin (2006) found enhanced conspecific densities over distances as low as m, while Plotkin et al. (2000) found evidence of clustering over m. In an analysis of temperate and tropical species, Clark et al. (1999) showed that most seeds fall within 40 m of the parent tree. Thus, a growing body of work suggests that dispersal related aggregation is strongest over modest distances, and may influence association on plots perhaps as small as 1 ha. 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