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).