Analysis 2: Spatial relationship between congeners
To assess the spatial relationship between two congeners in the same
size class, we used the bivariate pair correlation functiong 12(r ) as the summary function (Stoyan and
Stoyan, 1994) and a null model of independence. The pair correlation
function g (r ) is based on the distribution of distancesr between pairs of points belonging either to the same pattern
(e.g. same species) or to two different patterns (e.g. species 1 and
species 2). Therefore, g (r ) can describe the dependence
between points on multiple spatial distances r . It determines the
expected density of points of the same pattern (e.g. species 1;
univariate) or of the other pattern (e.g. species 2; bivariate) within a
ring of radius r and width d centred on an arbitrary focal
point of species 1. In the univariate g (r ), this value is
divided by the overall intensity λ 1 of species 1
in the study site, whereas in the bivariate function, it is divided byλ 2 of species 2 (Wiegand and Moloney, 2014).
We compared the observed values of g 12(r )
to those calculated during simulations of the independence null model.
To detect species associations, either caused by species interactions,
or by shared or opposed habitat associations, the independence null
model breaks the potential associations between species but conserves
the observed aggregation of each one of them. To simplify this task, we
fixed the original location of points of pattern 2 and randomized the
point pattern of species 1. To generate null model patterns of species 1
with the desired properties, we used the technique of pattern
reconstruction (Wiegand et al. , 2013) based on non-parametric
optimization. It reconstructs the point pattern of the species based on
information provided by several summary functions calculated from the
observed pattern. We conducted (homogeneous) pattern reconstruction
based on g 11(r ), the univariateL -function L 11(r ) (a transformedK -function, which is a cumulative version ofg 11(r )), and several univariate nearest
neighbour distribution functions. As a result, the spatial structure of
the reconstructed pattern very closely matches that of the observed
pattern (Wiegand et al. , 2013), but because the location of
species 1 was reconstructed without regard of that of species 2, the
locations of the species are spatially independent. Positive departures
of the observed g 12(r ) indicate species
association, negative departures indicate species dissociation, and
values within the simulation envelope indicate no significant departure
from independence (Wiegand and Moloney, 2014).
To find out if spatial associations were caused by the location of
congeners within the same environmental patches, we used the intensity
function λh (x, y ) of species 1 to
parameterize the pattern reconstruction of species 1 (Wiegand et
al. , 2013; Wang et al. , 2015). In this case, observed values ofg 12(r ) should lie within the simulation
envelopes, whereas departures suggest species interactions not explained
by habitat-association (Wiegand and Moloney, 2014).