Analysis 4: Spatial relationship between conspecifics of different sizes
In order to assess ecological processes driving spatial relationships between conspecifics of different size classes, we used a suite of alternative point process models that represent competing hypotheses, namely independence, dispersal limitation, habitat filtering, and a combination of dispersal limitation and habitat filtering. In all cases, we fixed the pattern of large trees, and randomized the pattern of small and medium trees according to the point process model used. To conserve the univariate pattern of small and medium trees during the simulations, we again used the technique of pattern reconstruction (Wiegand et al. , 2013) with different intensity functions tailored specifically for each point process model.
For the independence hypothesis (lack of small-scale species interactions), we used (as in analysis 2) a spatially constant intensity function λ (i.e. homogeneous pattern reconstruction). To represent dispersal limitation, we used the intensity functionλ d(x, y ) given by the superposition of Gaussian kernels with parameter σ around large trees. This creates patterns where the distribution of small and medium trees follows a normal distribution around the large conspecific trees (Wiegand and Moloney, 2014). The value of the parameter σ was fitted. For the habitat association hypothesis, we used the parametric intensity function\(\lambda_{\text{h\ }}\left(x,\ y\right)\)\(\lambda_{h}(x,y)\) of small trees. Finally, to represent the combined dispersal limitation and habitat filtering hypothesis, we used the geometric mean of the two intensity functions (i.e. [λ d(x, y )λ h(x, y )]0.5\(\sqrt{\lambda_{d}.\lambda_{h}}\)).
To determine the most parsimonious point process model, given the data, we used model selection based on the Akaike information criterion (AIC) and “synthetic” likelihood functions (Wood, 2010). With this method, we reduced the raw point pattern data to three-point pattern summary functions that quantify the spatial structure of the observed bivariate point patterns, namely g 12(r ), the bivariate L -function L 12(r ), and the bivariate nearest neighbour distribution functionD 12(r ). We performed 999 simulations of the point process model to obtain the mean and the covariance matrix of these functions for each radius r (in steps of 3 m), given the vector θ of model parameters. This allows for the construction of the synthetic likelihood to assess model fit. The resulting log-likelihood can then be used to calculate the AIC that balances model fit and model complexity to identify the most parsimonious model (Akaike, 1974; Wiegand and Moloney, 2014). We used here 999 simulations for better estimation of the covariance matrix needed for construction of the likelihood function.
We performed all point pattern analyses with the softwareProgramita (Wiegand and Moloney, 2014), which can be accessed at www.programita.org. Estimators of the summary functions and the edge correction used in Programita are detailed in Wiegand, Grabarnik, and Stoyan (2016). We used a spatial resolution of 1 m, which is much smaller than the study plot, fine enough to answer our questions, and larger than the mapping error of the data (Wiegand and Moloney, 2004, 2014). We selected a ring width d = 3 m in all analyses.