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The distribution and abundance of organisms can be represented by spatial points in a plane. A Dirichlet tessellation surrounds a point as a planar polygon in which all regions are closer to the point than to any other points. This tessellation was proposed in 1850 by Dirichlet (Upton & Fingleton 1985) and a formal mathematical definition is given by Green & Sibson (1978). The latter authors state that "the Dirichlet tessellation is one of the most fundamental and useful constructs determined by an irregular lattice." The Dirichlet tessellation cell, also known as Voronoi or Thiessen polygons, has been reinvented several times and is useful to research in many scientific fields (Rogers 1964; Mead 1971; Rhynsburger 1973; Upton & Fingleton 1985; David 1988; Galitsky 1990).

Dirichlet tessellations can be thought of as representing the areas of territorial animals, allelochemic-producing plants, or the packing of cells in a tissue. For example, two adjacent points, representing competitive animals of equal strength, bisect the planar area between them as well as with any other nearby animals. In general, competitors that are farther away from an organism will be less likely to interfere spatially unless there are no other organisms in between that can contest the areas. Thus, the areas of Dirichlet tessellations should coincide generally with the areas of the territorial or competitive ranges of the organisms (Tanemura & Hasegawa 1980; Kenkel, Hoskins & Hoskins 1989a, b).

The first computer algorithm for drawing Dirichlet cells was offered by Green & Sibson (1978). These authors developed a program in ANSI FORTRAN for use on mainframe computers that has been subsequently utilized in spatial statistics textbooks (Ripley 1981; Diggle 1983; Upton & Fingleton 1985). Another algorithm has been described in Russian and programmed in FORTRAN IV (Galitsky 1990). Dirichlet (Voronoi) cells have been delineated by algorithms that use Delaunay triangles and circumscribing circles on a HITAC M-180 computer (Tanemura & Hasegawa 1980) or elimination of intersecting circles according to a set of rules (Honda 1978). Unfortunately the above algorithms are described in only general terms, or in the program code, so they are generally difficult to use. Recently, the commercial statistical software, SYSTAT 5.0 (Wilkinson 1990), has offered graphical plotting of Voronoi polygons. Wilkinson (1990) says the algorithms of Green & Sibson (1978) were not used, but no references or algorithms are presented. Since none of the previous methods nor SYSTAT calculate areas and variance of Dirichlet cells, my objectives were both to develop algorithms for drawing tessellations with personal computers and to calculate cell areas. These general procedures could then be used specifically to analyze the spatial distributions of bark beetle `attacks' on their host trees. Statistical regression using the coefficient of variation of cell areas revealed a new method for analyzing spatial point distributions. In addition, these analyses offer a second way of determining species-specific spacing distances, termed the minimum allowed distance (

A computer program, coded in BASIC, implementing Dirichlet tessellation algorithms was developed for personal computer that allows

Once the

The program draws tessellations about each point within the inner border area, although all points including those in the peripheral area are considered. Real coordinates and dimensions are scaled on the monitor screen as well as on laser printers. The algorithms are described in six steps:

(1) The first step is to find the nearest neighbours which might affect the Dirichlet cell outline. Coordinates of all points (

(2) The next step is to calculate the equations of the lines that are perpendicular bisectors between the center point and each of its 35 neighbours (as well as the four boundary lines). The 35 resulting equations have the form

Fig. 1. Dirichlet tessellation (irregular pentagon) composed of perpendicular bisectors (solid lines) between center point and six surrounding points. Dashed lines connect the center point with the 15 intersection coordinates of the bisector lines. Bisector line

(3) The above 35 equations plus the four boundary equations are then compared to each other non-redundantly to obtain a total of SUM from j=1 to n-1 of j, or 741 possible intersection coordinates. The

(4) The program then calculates the equations of the 741 lines between the center point (

(5) The perpendicular bisector lines, found in (2) above, are compared to each of the dashed lines (Fig. 1), found in (4) above, to see if any intersections occur (

(6) The final step takes the coordinates of the true vertices of the Dirichlet cell and sorts them in ascending order by angular direction from the center point. This must be done since it is not yet known what the correct order of drawing is between vertices. The general method for obtaining polar coordinates uses cos à = x/r and the appropriate quadrant (Batschelet 1979, p. 121).

Coefficient of variation of Dirichlet cell areas

The Dirichlet cell area (

A = SUM from i=1 to k of ABS(0.5(x

where

The mean and standard deviation (

The

The patterns of attack entrances of the bark beetles

Simulated placements of 250 points at a density of 0.00125 points per unit area revealed the relationship (Fig. 2) between the minimum distance of separation between points and the coefficient of Dirichlet cell area variation (

Fig. 2. See correction. Relationship between the coefficient of variation (

The minimum separation distance between otherwise randomly placed points was increased in increments of ten percent of the maximum hexagonal spacing distance possible at this density (i.e. 30.39 units, Clark & Evans 1954). It was found that the

Several indexes of dispersion have been proposed to describe the degree of uniformity or aggregation among spatial points representing organisms (Clark & Evans 1954; Pielou 1959; Morisita 1965; Lloyd 1967; Goodall & West 1979). One of the most widely used is the

Earlier I proposed a method called the minimum allowed distance (

It now is apparent that one may also find the

Attack spacing of

The

Fig. 3. Effect of border width on the estimated percentage of maximum spacing value (obtained from the cubic regression in Fig. 2) for the attack patterns of the bark beetles

The estimate of the percentage of maximum spacing was relatively constant (Fig. 3) for different sized areas (and numbers of Dirichlet cells) for the attacks of

The pattern of attacks of

Fig. 4. Dirichlet cell tessellations of 35 attacks of

The average Dirichlet cell area was 33.9± 5.4 cm

The average Dirichlet cell area for

Fig. 5. Dirichlet cell tessellations of 44 attacks of

The

Dirichlet tessellations of

Fig. 6 (a). Dirichlet cell tessellations of 84 attacks of

The

The Dirichlet cell was first proposed in 1850 but has been rediscovered several times and given names such as Voronoi polygons, 1909, Thiessen polygons, 1911, Wigner-Seitz cells, 1933, the cell model, 1953, and the S-mosaic, 1977, (Upton & Fingleton 1985). For a theoretical Poisson forest, the expected number of sides of the Dirichlet cell is 6, the expected area is , and the expected perimeter length is , where

Applications of the Dirichlet cell in plant ecology and forestry have been discussed with regard to interplant competition and prediction of growth for individual trees (Brown 1965; Mead 1971; Cormack 1979; Kenkel, Hoskins & Hoskins 1989a, b; Welden, Slauson & Ward 1990). Dirichlet polygons describe territories of pectoral sandpipers,

Boots & Murdoch (1983) used Monte Carlo procedures (programmed in FORTRAN IV) to investigate the properties of Dirichlet tessellation of random points. Their program, however, is not generally useful to ecologists since "there is no input to the program". It is not known if the algorithms used in the present study are as efficient as those of Green & Sibson (1978), Honda (1978) or Tanemura & Hasegawa (1980). However, the use of a math-coprocessor allows drawing of 500 cells within a few minutes by personal computer (computation time is similar to that for SYSTAT). The computation cost of the algorithm of Green & Sibson (1978) increases roughly as

Dirichlet polygons can represent the competitive interactions of a colony of bark beetles packed onto the bark surface. Most temperate bark beetles (Coleoptera: Scolytidae), including

Bark beetles can minimize potential competition by avoiding areas releasing pheromone components that indicate higher densities of established individuals (Byers et al. 1988; Byers 1989). Another mechanism that may require little time and energy expenditure to gain large advantages in reproductive success is to avoid boring too closely to established attack holes and their galleries. Several bark beetle species, including

Both sexes produce

Another mechanism that has been postulated to regulate density of attack is acoustic stridulation. Compared to males, females stridulate very weakly and it has been reported that females increased their chirping rate when other stridulating females were boring holes in the vicinity (Rudinsky & Michael 1973). However, male chirps can be heard from the onset of colonization, and for several days, from even a meter or more away by the human ear (Byers et al. 1984). Vibrations from the male stridulation could possibly be felt by walking females which would then decide to leave the area. Alternatively, males within holes with females may stridulate to warn walking males not to attempt entry into their tunnels. The function of stridulation is poorly understood.

Nilssen (1978) used nearest neighbour analysis to show that the pattern of attacks of

In contrast to

Bark structure also has been postulated as enforcing spacing in bark beetles (Safranyik & Vithayasai 1971). Ponderosa pine has rather deep furrows running longitudinally that branch in diagonal directions. It was observed that 79 of 97 attacks of

At the base of Scots pine trees the furrows are more numerous and attacks of

The Dirichlet tessellation program and calculation of the

Funding for the work was contributed by the Swedish Forest and Agricultural Research Agency (SJFR). I thank my colleagues Olle Anderbrant and Fredrik Schlyter in the Pheromone Research Group for review of the manuscript.

John A. Byers Department of Animal Ecology, Lund University, SE-223 62 Lund, Sweden Present address: Department of Plant Protection, Swedish University of Agricultural Sciences, SE-230 53 Alnarp, Sweden |
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Anderbrant, O., Schlyter, F. & Birgersson, G. (1985).