Appendix A. A model for locating regenerated trees.
The pattern of trees was retained over time by first fitting a statistical model to the empirical tree pattern, and then locating new trees using the fitted model. More specifically, we first analyzed the empirical tree pattern using the K-function for point patterns (Diggle 1983, Bailey and Gatrell 1995, Venables and Ripley 1999). The estimated K-function is defined as
where r is the radius of a circle with the centre at a randomly chosen point i, n is the observed number of points, A is the study area, Ih(dij) is an indicator function which is 1 if dij r and 0 otherwise, and wij is the proportion of the circumference of this circle which lies within A, i.e., an edge correction.
For a random pattern with density n/A, the expected number of neighbors within a distance r from an arbitrary point of the pattern is r2n/A. The benchmark of complete randomness is the Poisson process for which Kest(r) = r2. For an aggregated pattern, the points have more neighbors than expected under the null hypothesis, hence Kest(r) > r2n/A, and conversely, for a regular pattern the points have fewer neighbors, Kest(r) < r2n/A.
We constructed 95% confidence envelopes for random patterning of the trees by randomizing the number of trees (n) within A 99 times, estimating Kest(r), and then plotting the highest (upper envelope) and lowest (lower envelope) Kest(r) values (Diggle 1983). This plot (not shown) revealed that trees at Valkrör were spatially aggregated up to a scale of 25 meters.
We next fitted a model to the empirical tree pattern (coordinate system in meters), assuming an underlying Poisson cluster process, according to Diggle (1983). The fitted model was used for locating the new trees ("offspring") which thereby showed the same scale of aggregation as their "parent" trees. We fitted the Poisson cluster model by minimizing
where K(r) is the theoretical K-function with the parameter vector , and r0 and c are "tuning constants" chosen to provide desirable estimation properties. We choose r0 = 45, corresponding to the observed scale of tree aggregation, and c = 0.25, suggested by Diggle (1983) for aggregated patterns. Diggle (1983) showed that for a specific Poisson cluster process, with Poisson number of offspring per parent, and where the probability density function, for the distance between offspring and parent, is a radially symmetric normal distribution,
K(r) = r2 + (1 exp(-r2/42) / .
The parameter 2
is the variance of the normal distribution, and
is the density of the random parent Poisson process. We found that D()
was minimized with
2 = 41.9 and = 0.0019. For each simulation time step, we thus located two new trees using this model for the Poisson cluster process, K(r). An offspring was not allowed to be located closer than 0.1 meter from an existing tree. This procedure retained the scale of tree aggregation during the simulated 100 years.
Bailey, T. C., and A. C. Gatrell. 1995. Interactive spatial data analysis. Longman Group Limited, Essex, UK.
Diggle P. J. 1983. Statistical analysis of spatial point patterns. Academic Press, London, UK.
Venables W. N., and B. D. Ripley. 1999. Modern Applied Statistics with S-PLUS. Third edition. Springer-Verlag, New York, New York, USA.