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.