Analysis 3: Spatial distribution of size between
congeners
For the analysis of the plant size as a function of the distance between
individuals, we attributed the quantitative mark DSH to points of plants
location. Mark correlation functions are designed to detect correlations
in the marks of pairs of points i and j that are separated
by distance r (Stoyan, 1984). For example, they allow us to find
out if trees of species 1 tend to be smaller than expected if a tree of
species 2 is located nearby (Wiegand and Moloney, 2014).
For bivariate patterns, there are type v (e.g. species 1) and
type w (e.g. species 2) points. The bivariate r -mark
correlation functionκm 1.(r ) measures the mean
mark of type v points located at distance r from typew points. The dot (“.”) in the subscript
“m 1.” indicates that the mark of the second
point (= m 2) is not used for estimation ofκm 1.(r ). This function is
normalized by μv , the mean mark of type vpoints. If κm 1.(r ) = 1,
the marks m 1 of type v points that have a
type w point at distance r are similar toμv ; ifκm 1.(r ) < 1,
they are smaller on average than the mean μv ,
indicating interspecific competition; ifκm 1.(r ) > 1,
they are larger on average, indicating facilitation or location of
individuals at resource-rich patches. Because the r -mark
correlation function is evaluated at different interpoint distancesr , we can observe how spatial effects decline with increasing
distance between individuals (Wiegand and Moloney, 2014).
In order to generate the null model values representing the absence of
interactions for comparison ofκm 1.(r ), we randomly
shuffled the marks within type v points while keeping unchanged
the marks of type w points; additionally, the original location
of all individuals was kept fixed. This “random marking” null model
assumes that the size of type v points is independent of the size
of nearby type w points (Wiegand and Moloney, 2014).
Additionally to distance between individuals, plant size can be
influenced by the environment where individuals are located. Therefore,
nearby plants may be smaller in less favourable patches and larger in
more favourable patches, independently of the distance between
individuals. In order to tease apart the effects of environmental
heterogeneity and distance between individuals, we used the local random
marking null model. In this variation of the null model, shuffling of
the marks is restricted to points separated only up to a given distanceR , which represents an approximated size of environmental
patches. By doing this, we can check whether departures from the
independent marking null model are caused by environmental
heterogeneity. Conversely, departures from the local random marking
indicate congeners interactions within patches (Wiegand and Moloney,
2014). We stipulated R = 25 m (one-quarter of the 1-ha plot) for
the local random marking null model after testing for different values
of R . The selected value corresponds to an approximate patch size
where most distance effects disappear. Additionally, R = 25 m is
large enough to encompass neighbourhood interactions within patches. In
all analyses, we considered the influence of species 1 on species 2 and
vice-versa.