Spatial point pattern analysis of pairwise species association
We test the null hypothesis that species pairs are spatially independent, as opposed to showing patterns of attraction or repulsion. If two species show attraction in their spatial distributions, we will find more points of species j within the neighborhood of speciesi than expected under independence of the two species. Conversely, if the two species show segregation, we will find fewer points of species j within the neighborhood of species ithan expected. To assess pairwise spatial associations, we used seminal techniques of bivariate point pattern analysis based on the distributions of distances of all pairs of points between the two species (Lotwick & Silverman, 1982; Wiegand & Moloney, 2014; Wiegand et al., 2017). Two summary statistics, bivariate pair-correlation function (pcf) gij (r ) and bivariate distribution function Dij (r ) of nearest neighbor distances, were used in this analysis. The bivariate pair-correlation function gij (r ) can be estimated using the quantityλjgij (r ), where λj is intensity (i.e. density) of speciesj in the whole study area, measuring the mean density of trees of species j at distancer away from a tree of the focal species i (Ripley, 1981; Stoyan & Stoyan, 1994). Dij (r ) could be defined as the probability that trees of the focal species i have their nearest species j neighbor(s) within distance r(Diggle, 1983). Dij (r ) can provide additional information of the spatial patterns that is not provided by the bivariate pair-correlation functiongij (r ), especially in the extremely heterogeneous cases for focal species, e.g., many individuals of focal species i have no species j neighbor but few have many species j neighbors (Wang et al., 2010; Wiegand et al., 2007).
The independence of bivariate spatial point patterns is examined through the comparison of the summary statistics of the observed bivariate patterns with those of the null model, i.e., the observed patterns are compared against the simulated null model to test whether the hypothesis holds. In this study, we implemented the null model by keeping the locations of the focal species i unchanged while randomizing the distribution of species j by the method of Toroidal shift, which maintains most of structure of species j (Lotwick & Silverman, 1982). The null model of Toroidal shift removes the effects of environmental heterogeneity and the interspecific interactions, while retains the spatial structures of individual species. If a summary statistic of the observed bivariate spatial pattern significantly differs from the expectation of the null model, it is reasonable to conclude that the departure results from species interactions or environmental heterogeneity.
To assess the magnitude of departures from the null model, for each species pair and for each observed summary statisticS0 (r ) (i.e.,gij (r ) orDij (r )), we computed their standardized effect size z (r ) as:
\(z(r)=\frac{S_{0}(r)-\mu_{\text{null}}(r)}{\sigma_{\text{null}}(r)}\), (1)
where S0(r) is the observed summary function (either gij (r ) orDij (r )), and µnull(r ) and σnull(r ) are respectively the average and the standard deviation of the summary functions for 999 bivariate patterns simulated according to the null models (Chanthorn, Wiegand, Getzin, Brockelman, & Nathalang, 2018; Wang et al., 2018; Wiegand, Grabarnik, & Stoyan, 2016). For a given distancer , the hypothesis of independence for a species pair can then be accepted if -z α(r ) <z (r ) < z α(r ) at a given pointwise significance level of α. For \(\alpha\)= 0.05,z α= 1.96, which is equivalent to testing whether the observed summary statistic is located within the 2.5th and 97.5th percentiles of the corresponding null model distribution. Whenz (r ) > 1.96, the observed summary statistic is larger than the expectation of the null model with error rate α= 0.025, and the species pairs are spatially attracted at distancer . While z (r ) < -1.96 suggests repulsion at distance r . The distance r in this study was chosen to be 5, 30 and 50 m to test the effect of scale on spatial patterns. Because the association between two species might be asymmetric, we analyzed the spatial patterns between two species twice with each species serving as the focal species, i.e. species i versus species j and speciesj versus species i . Specifically, we examined the interspecific spatial associations of 80 × 79 = 6320 species pairs in this study for two different summary statistics of bivariate spatial point pattern analysis: gij (r ) andDij (r ). All the spatial association analyses were conducted in R (R Core Team, 2018) and using the package of “spatstat” (Version 1.62-2, Baddeley, Rubak, & Turner, 2015).