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.