*Ecological Archives*
E086-007-A1

**Tord Snäll, Johan Ehrlén,
and Håkan Rydin. 2005. Colonization–extinction dynamics of an epiphyte
metapopulation in a dynamic landscape. ***Ecology* 86:106–115.

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, *I*_{h}(*d*_{ij})
is an indicator function which is 1 if *d*_{ij}
*r* and 0 otherwise, and *w*_{ij} 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 *r*^{2}*n/A*.
The benchmark of complete randomness is the Poisson process for which *K*_{est}(*r*)
= *r*^{2}. For an
aggregated pattern, the points have more neighbors than expected under the null
hypothesis, hence *K*_{est}(*r*) > *r*^{2}*n/A*,
and conversely, for a regular pattern the points have fewer neighbors, *K*_{est}(*r*)
< *r*^{2}*n/A*.

We constructed 95% confidence envelopes
for random patterning of the trees by randomizing the number of trees (*n*)
within *A* 99 times, estimating *K*_{est}(*r*), and then
plotting the highest (upper envelope) and lowest (lower envelope) *K*_{est}(*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 *r*_{0} and *c* are "tuning constants" chosen
to provide desirable estimation properties. We choose *r*_{0} =
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*)
= *r*^{2} + (1 –
exp(-*r*^{2}/4^{2})
/ .

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.

LITERATURE
CITED

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.

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