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).