**
In a planar field with many stationary objects, a mosaic of polygons can be drawn or
tesselated whose network of boundaries are the set of points that are equally close to two or
more nearby objects. Each polygon region, called a Dirichlet cell, contains all points that are
closer to an object than to any other objects in the field (Green & Sibson, 1978). The Dirichlet
cell, first proposed in 1850, has been useful in many scientific disciplines and thus is known
under a variety of names including Voronoi, 1909, Thiessen, 1911, Wigner-Seitz, 1933, cell
model, 1953, and S-mosaic, 1977 (Rogers 1964; Mead 1971; Rhynsburger 1973; Upton &
Fingleton 1985; David 1988). More recently, a Dirichlet tessellation algorithm was developed
to define colonization territories of bark beetles (Coleoptera: Scolytidae) under the bark of host
trees (Byers 1992). In addition, simulations of point patterns at increasing spatial uniformity
resulted in a decrease in the variation of Dirichlet cell areas. Based on this relationship, a
method was formulated that estimated a minimum allowed distance (MAD) of spacing between
the attack holes of individual bark beetles. The results suggested that several species of bark
beetle can reduce competition by not boring in areas closer than their species-specific
distance from neighbouring attack sites.
**

Unfortunately, the program for drawing Dirichlet tessellations (Byers 1992) does not
calculate the polygon areas correctly. The program draws the polygons and calculates
average nearest neighbour distances appropriately. However, calculations of the area of a
Dirichlet cell are undervalued by about 10 to 30 percent. The calculation of the Dirichlet area
(always a convex polygon) is done by finding the cell's center x_{c},y_{c}
(the average of x,y coordinate values) and then sorting the vertices of the polygon by angle. The cell area is then the
summed area of all triangles occurring between the center and two successive vertices (x_{i},y_{i}
and x_{i+1},y_{i+1}). The last pair of successive vertices must cause closure (i.e., include the first
vertex). Unfortunately, this was not done in the original program so that the area of the "last"
triangle was not added to the sum. The iterative formula for calculation of the area (Byers
1992):

A = SUM from i=1 to k of ABS(0.5(x_{c}(y_{i} - y_{i+1}) + x_{i}(y_{i+1} - y_{c}) + x_{i+1}(y_{c} - y_{i})))

is correct if k = number of vertices and x_{k+1},y_{k+1} are equal to x_{1},y_{1}.
The relationship between the percentage of maximum point spacing and the CV
(coefficient of variation) of cell areas (Fig. 2 in Byers 1992) was largely unaffected by the error
since the partial sums of triangular areas varied approximately proportional to the real areas.
Thus the MAD (minimum allowed distance) calculations based on this relationship also were
negligibly affected. Simulations according to the previous methods but using the corrected cell
areas gave a curve similar to that reported earlier. The revised cubic equation is:

Y = 0.0000739X^{3} - 0.01301X^{2} + 0.043734X + 51.9243

with r^{2} = 0.999.
The new version of the program uses this equation to calculate the MAD for
a population of objects.

Fortunately, the biological conclusions in Byers (1992) are still valid since a reanalysis
of the spatial attack data gave a MAD for *Dendroctonus brevicomis* of 1.9 cm (0-2.6 cm, 95%
CI, CV = 44.76%) that is close to 2.0 cm reported earlier (Fig. 4, CV =48.04). The revised
MAD for *Tomicus piniperda* of 4.1 cm (3.9-4.2 cm, CI; CV =23.71%) is near to the previous
4.3 cm (Fig. 5, CV = 24.46%). In *Pityogenes chalcographus*, the revised MAD of 1.6 cm (1.5-
1.7, CI; CV = 27.51%) is identical to the former value (Fig. 6a, CV = 30.28%); while for the
random distribution no MAD could be detected (0-0.52 cm, CI; CV = 54.44%) as before (Fig.
6b, CV = 58.76%). The species-specific values for the MAD's are not expected to vary with
density under endemic population levels (Byers, 1984, 1992). However, at low densities and
sample size the MAD can be difficult to reliably evaluate.

The revised calculations show that the average colonization territory of a bark beetle
family group is actually larger than reported. The revised areas of 42.07 ± 6.24 cm^{2} (± 95% CL)
for *D. brevicomis* (Fig. 4) is almost the same as 42.13 ± 2.95 cm^{2} for *T. piniperda* (Fig. 5). This
similarity could occur if the two population samples had reached a limiting attack density as
a result of similar MAD's (as reported above) causing later arriving individuals to leave for lack
of free territory. Ultimately, monogamous mating pairs of both species may utilize comparable
areas of the bark because they are closely related species (Hylesininae, Tribe Tomicini), are
alike in size (5 mm long, 10 mg fresh weight), and feed on the phloem of similar host trees
(Ponderosa and Scots pines, respectively). The colonization territory of *P. chalcographus*
(Scolytinae) is smaller at 8.5 ± 0.5 cm^{2} probably due to the beetle's smaller resource
requirements (its size is only 2 mm long, 1 mg weight), although several females may occupy
the area with a single male. The same density for a random distribution of points (from Fig.
6b) gave an average cell area of 8.35 ± 0.93 cm^{2}.
The undervalued cell areas were not discovered sooner because one usually
tessellates within an inner border to avoid `edge effects'. Thus the total area of the polygons
is variable and difficult to verify. However, the program can be used to place 50 points in an
area of 100 units on a side, without any inner border, causing the polygons to fill the arena.
The average cell area reported with the revised method and these parameters is, as expected,
equal to (100 x 100)/50 or 200 units.

The program software is available below:

Download software: DIRICHLE.ZIP
See
References
Byers, J.A. (1984). Nearest neighbor analysis and simulation of distri-
bution patterns indicates an attack spacing mechanism in the bark
beetle, *Ips typographus* (Coleoptera: Scolytidae). Environmental
Entomology, 13, 1191-1200.
Byers, J.A. (1992). Dirichlet tessellation of bark beetle spatial attack
points. Journal of Animal Ecology, 61, 759-768.
David, C.W. (1988). Voronoi polyhedra as structure probes in large
molecular systems. VI*I.* Channel identification. Computers and
Chemistry, 12, 207-208.
Green, *P.*J. & Sibson, R. (1978). Computing Dirichlet tessellations in the
plane. The Computer Journal, 21, 168-173.
Mead, R. (1971). Models for interplant competition in irregularly distri-
buted populations. Statistical Ecology Volume 2 (Ed. by G.*P.* Patil, E.
C. Pielou & W.E. Waters). pp. 13-30. Penn State University Press,
University Park, USA.
Rhynsburger, *D.* (1973). Analytic delineation of Thiessen polygons.
Geographical Analysis, 5, 133-144.
Rogers, C.A. (1964). Packing and Covering. Cambridge Tracts in Mathematics
and Mathematical Physics. No. 54, Cambridge University Press, UK.
Upton, G. & Fingleton, B. (1985). Spatial data analysis by example. Volume
1. John Wiley and Sons, New York, USA.

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