J. Anim. Ecol. 65:528-529. at end of this paper)                                                                                                                                                                                                ******************************************************************************* Byers, J.A. 1992. Dirichlet tessellation of bark beetle spatial attack points.       Journal of Animal Ecology 61, 759-768.                                     ******************************************************************************* JOHN  A. BYERS                                                                                                                                                  U.S. Arid-Land Agricultural Research Center, USDA-ARS, 21881 North Cardon Lane, Maricopa, Arizona 85239, USA  (http://www.chemical-ecology.net/software.htm)                                                                                                                       SUMMARY                                                                                                                        (1) Algorithms for Dirichlet tessellation of spatial points are developed and implemented on personal computer. Up to 3000 tessellations of points in an area of any rectangular dimensions can be scaled appropriately and viewed on computerscreen or output to laser printer.                                                                                                                                (2) The program also calculates Dirichlet cell areas and their coefficient of variation (CV) as well as the average nearest neighbour distance between points.                                                                                  (3) Simulations revealed the polynomial relationship between the CV and the   minimum spacing between points. The relationship is used to predict the         percentage of maximum spacing that is exhibited by a population. This value     times the maximum spacing distance possible between objects in an area          (hexagonal arrangement) yields the minimum allowed distance (MAD) that is       characteristic of individuals of some territorial or `inhibitive' species.                                                                                        (4) The program and relationship were used to analyze the spatial attack      patterns of the bark beetles, Dendroctonus brevicomis LeConte, Tomicus piniperda(L.), and Pityogenes chalcographus (L.) and determine their MAD's. All three    species exhibited spacing between attack sites, in agreement with known         behavioural mechanisms that are proposed for avoiding intraspecific competition for food resources.                                                                                                                                             Running Title: Tessellation of bark beetle attacks                                                                                                              Key Words: Voronoi polygons, Scolytidae, Coleoptera, intraspecific competition, algorithms                                                                                                                                                                                      INTRODUCTION                                                                                                                    The distribution and abundance of organisms can be represented by spatial pointsin 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 (MAD) as proposed earlier (Byers 1984).                                                                                                                                                                                             METHODS                                                                                                                          A computer program, coded in BASIC, implementing Dirichlet tessellation     algorithms was developed for personal computer that allows x- and y-coordinates of spatial data in any units to be entered into a file for later retrieval. The data files are compatible with another program for drawing contour maps of pointdensities (Byers 1992). Alternatively, one can generate x,y coordinates at      random, with or without a degree of minimum spacing. For a given rectangular    area (AREA) containing N points, the maximum distance possible to space apart   points is given by 1.0746/SQR(N/AREA) (BASIC terminology),which means the pointsare in a perfect hexagonal arrangement (Clark & Evans 1954). Thus, an input     value of no more than about 70 percent of the maximum distance should be        attempted since the computer otherwise may not find locations for all the pointsdue to constraints from the initial selections. The algorithms for spacing      points have been described earlier (Fig. 1 in Byers 1984).                          Once the x,y coordinates of the points are entered, an inner border area    should be chosen in order to avoid tessellating points on the periphery. Points near the edges of the area are not surrounded by other points so the            tessellation outline would be altered by the boundary of the area. It was found empirically that for randomly distributed points the use of a border width of   at least 1.5 times the distance expected for the fourth nearest neighbour       (        1.0937/SQR(N/AREA)           Thompson 1956) gave good results. The     distance used, however, is arbitrary and can be adjusted.                                                                                                                             Dirichlet tessellation algorithms                                                                                                            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 (N) are scanned to count the  number of points contained within a `box' centered about the point in question. The size of the initial box is smaller than that expected to hold P neighbours  (equal to N-1 or 35 points, whichever is smaller) if they were distributed at   random      0.2*(SQR((AREA/N)*P)/2)       .  However, if less than P points     (hereafter 35 points) fall within the box, then the box is successively enlargedby 10 percent until the required points are obtained. The actual distances from the `center' (xc, yc) point to its neighbours (xn, yn) in the box are then      computed, dn=SQR((xc-xn)^2+(yc-yn)^2)             , and the 35 lowest distances are sorted with an exchange sort algorithm.                                                                                                                     (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  anx + bny + cn = 0, where an = 2xc - 2xn, bn = 2yc - 2xn, and cn = xnxn - xcxc +ynyn - ycyc . In Fig. 1, a simplified case is shown where a center point is     surrounded by six neighbours with six perpendicular bisector lines (solid       lines).                                                                                                                                                         (3) The above 35 equations plus the four boundary equations are then compared   to each other non-redundantly to obtain a total of (no formula), or 741 possibleintersection coordinates. The xj,yj coordinates of the intersection of two such equations, a1x1 + b1y1 + c1 = 0 and a2x2 + b2y2 + c2 = 0, are xj = (-c1b1 +     c2b2)/(a1b2 - a2b1) and yj = (-a1c2 + a2c1)/(a1b2 - a2b1).                                                                                                                                                                                      (4) The program then calculates the equations of the 741 lines between the      center point (xc,yc) and each of the intersection coordinates (xj,yj),          where aj = yj - yc, bj = -(xj - xc), and cj = -ycbj - xcaj. In Fig. 1 these     lines are represented by the 15 dashed lines (lines to the boundary             intersections are not shown).                                                                                                                                   (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 (x,y; method as in part 3 above) but only in the segment from the center  point to and including the intersection point of the two respective bisectors   (xj,yj; from part 3). Thus, if the x-coordinate is greater or smaller than both xc and xj or the y-coordinate is greater or smaller than both yc and yj then no intersection can occur. If the number of intersections is more than two (both   intersecting bisector lines intersect with a dashed line) then the intersection point found in (3) can not be one of the legitimate vertices of the Dirichlet   cell. This can be seen in one case in Fig. 1 where the bisector line b, between point a and the center, intersects the bisector line c at d (a dashed line      connects d to the center) but bisector line e intersects the dashed segment at  f, thus invalidating the intersection coordinates at d as a vertex of the       Dirichlet cell.                                                                                                                                                 (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  = x/r and the appropriate quadrant (Batschelet 1979, p. 121).                                                                                                      Coefficient of variation of Dirichlet cell areas                                                                                                     The Dirichlet cell area (A) is calculated by means of summing the areas of   the triangles constructed from the center point and the vertices (xi,yi):       A = SUM (i=1 to k) .5(xc(yi - yi+1) + xi(yi+1 - yc) + xi+1(yc - yi))          where k = number of vertices and xk+1=x1, yk+1=y1. (this equation has been      revised from that published in the 1992 paper).                                                                                                                    The mean and standard deviation (S.D.),                                      SQR((N*SUM(An^2) - (SUM An)^2)/(N*(N-1)) , of the cell areas are used to        calculate the coefficient of variation, CV = S.D./mean x 100. Computer          simulations were carried out to determine the relationship between the CV and   the degree of uniformity in spacing apart of points. A square area of 447.21    units on a side had 250 points placed within it to obtain a density of 0.00125  per unit area. Points in areas were increasingly spaced apart at distances from 0 (random) to 70 percent of the maximum possible spacing (30.39 units) in       increments of 10 percent. A total of 32 point sets, each of 250 points, at each of the spacing constraints were simulated. The algorithms for spacing have been reported earlier (Byers 1984) and comprise an inhibition model where points are sequentially placed at random unless they are closer to an established point    than the minimum allowed distance (MAD). The inner border width used for the    points was 2.5 times the expected fourth nearest neighbour distance (random     distribution), 77.36 units.                                                         The CV of the cell areas will be distorted if tessellations are attempted   on points near the edges of the area. Thus a border area must be chosen that is a compromise between reducing the "edge effects" and having sufficient points   remaining for statistical analysis. As mentioned above, it was found that a     distance of 1.5 times the expected fourth nearest neighbour distance gave       adequate results. The border can be somewhat larger if more points are available(>100). The effect of changing the border width on the CV and the estimated     percentage of the maximum point spacing was investigated by using increasingly  larger widths during tessellations of natural bark beetle attack patterns       (described below).                                                                                                                                                                 Analysis of bark beetle attack patterns                                                                                                         The patterns of attack entrances of the bark beetles Dendroctonus brevicomis LeConte on ponderosa pine, Pinus ponderosa Doug. ex. Laws., Pityogenes          chalcographus (L.) on Norway spruce, Picea abies (L.) Karst., and Tomicus       piniperda (L.) on Scots pine, Pinus sylvestris L., were recorded from bark      samples in the field. This was done by overlaying a plastic sheet on each of thebark areas and marking the plastic with ink. The marks were then measured for   x and y coordinates. An average nearest neighbour distance analysis was done on the attack distributions (Clark & Evans 1954; Thompson 1956) with a previously  described computer program (Byers 1984). A minimum allowed distance analysis    (MAD) using six simulation runs of 300 points at each of eight spacing steps    also was performed on each of the attack densities (Byers 1984). A simulated    placement of 179 attacks at random was done by the computer program for         comparison with the natural attacks of P. chalcographus above. Dirichlet        tessellations were then done on the point patterns to evaluate the program as   well as the attack distributions.                                                                                                                                                                 RESULTS                                                                                                                                     Coefficient of variation of Dirichlet cell areas                                                                                                  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 (CV). The   minimum separation distance between otherwise randomly placed points was        increased in increments of ten percent of the maximum hexagonal spacing distancepossible at this density (i.e. 30.39 units, Clark & Evans 1954). It was found   that the CV of cell areas was about 55.6 % regardless of the density or number  of points when simulating random point distributions (tests up to 3000 points). At the maximum point spacing possible it is intuitive that the CV of the areas  would be zero since all the cells are perfect hexagons of identical size. This  `point' can not be found by simulation, although the theoretical value was used with the simulated points to find the best fitting cubic equation (Fig. 2).          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 R index of Clark & Evans (1954) which is the     ratio of the observed nearest neighbour distance to the expected nearest        neighbour distance when points are distributed randomly. A value greater than   1 indicates that points are more uniformly spaced than at random. It is         significant that this ratio is independent of density. In Fig. 2, the CV of     Dirichlet cells is also independent of density. Thus, one could tessellate a    spatial point pattern and determine the CV and then use the cubic regression    equation (Fig. 2) to obtain a value for the percentage of the maximum spacing   that organisms exhibit. Solving algebraically for X in polynomial equations is  not possible but must be done by `binary successive approximation' until a valueis found that gives the known Y (CV). This method has been incorporated into theprogram. Fig. 2 has the wrong equation, the new one is:                                                                                                              Y = 0.00007393X^3 - 0.01301X^2 + 0.43734X + 51.9243                                                                                                             Earlier I proposed a method called the minimum allowed distance (MAD) whichpurports to find the preferred or instinctive minimum distance that individuals will space themselves apart from others (Byers 1984). Beyond this species-      specific distance individuals are free to colonize sites at random. The method  relies on construction of a quadratic regression curve from computer simulationsof increasing minimum allowed distances of separation and the resulting average nearest neighbour distances obtained at a density corresponding to natural      spatial data. The observed nearest neighbour distance for the natural data is   then used to solve the quadratic equation to obtain the MAD for the species.    This distance is independent of density since it is based on a behavioural      distance that is relatively constant regardless of density.                         It now is apparent that one may also find the MAD, assuming one exists, fromthe CV of the Dirichlet areas. Cells with a lower CV than expected indicate thatthe points are more uniformly spaced, and from the relationship in Fig. 2 one   may find the percentage of the maximum point spacing at a particular density.   This percentage multiplied by the maximum point spacing distance is equal to theMAD. The results of the two methods will be compared subsequently for several   examples of the bark beetle attacks.                                                                                                                                              Analysis of bark beetle attack patterns                                                                                                          The CV of the Dirichlet cells becomes incorrectly large when the border      within which tessellations are drawn is made too narrow. However, an increase   in the width of the border after a certain amount does not appreciably effect   the magnitude of the CV and the estimate of the percentage of maximum spacing   (Fig. 3). The estimate of the percentage of maximum spacing was relatively      constant (Fig. 3) for different sized areas (and numbers of Dirichlet cells) forthe attacks of Pityogenes chalcographus and Tomicus piniperda as well as the    random point distribution, but not for the attacks of Dendroctonus brevicomis   (data from Figs. 4-6). This indicates that the samples of the former two speciesand the random pattern are consistent at all scales while the data for D.       brevicomis is more variable.                                                        The pattern of attacks of D. brevicomis can be seen in Fig. 4. The average  Dirichlet cell area was 42.076.24 cm^2 (95% confidence limits, C.L.). The     coefficient of variation in the cell areas was 44.8% (39.0-52.6%, 95% confidenceinterval, C.I.) which yields a spacing value of 27.6% (0-37.8%, 95% C.I.), of   the maximum possible spacing as estimated from cubic regression (Fig. 2). The   estimated MAD using the Dirichlet CV method is then                             0.276*(1.0746/SQR(97/(90*45)) = 1.92 cm (0-2.63 cm, 95% C.I.). The observed     average nearest neighbour distance was 3.630.39 cm (95% C.L.) and the expected corresponding distance if points were random is 3.230.34 which gives a R =     1.12, indicating a significant degree of spacing (P=0.02, Clark & Evans 1954).  The alternative analysis using the nearest neighbour distances and simulation   of spaced points (Byers 1984) gave a MAD = 1.67 cm (0.38-2.56 cm, 95% C.I.). Thetwo methods give slightly different estimates of the MAD possibly because the   spatial distribution of attacks was not consistent at different border widths   (Fig. 3).                                                                           The average Dirichlet cell area for Tomicus piniperda (Fig. 5) was 42.132.9cm^2 with a CV = 23.7% (22.2-25.5%, C.I.) and a maximum spacing percentage of   59.9 (57.5-62.1%, C.I.). The MAD was thus estimated to be 4.08 cm (3.91-4.22,   C.I.). The average nearest neighbour distance was 4.620.19 cm (C.L.), the      expected distance was 3.160.31, giving a R = 1.46, indicating significant      spacing (P<0.001). The nearest neighbour simulation analysis estimated the MAD  to be 3.71 cm (3.4-4 cm, C.I.).                                                     Dirichlet tessellations of Pityogenes chalcographus attacks (Fig. 6a)       produced cells with an average area of 8.500.50 cm^2 and a CV = 27.51% (26.0-  29.3%, C.I.) yielding a maximum spacing percentage of 54.7% (52.2-56.8%, C.I.). The MAD was thus calculated to be 1.61 cm (1.54-1.7, C.I.). The average nearest neighbour distance was 1.990.1 cm (C.L.), the expected distance was 1.370.11, giving a R = 1.45, indicating significant spacing (P<0.001). The nearest        neighbour simulation analysis estimated the MAD to be 1.58 cm (1.42-1.75 cm,    C.I.). The estimates of the two methods are very close. In comparison, the      random distribution at the same point density (Fig. 6b) gave a CV = 54.4% which,as expected, was not different from random distributions (Fig. 2 at 0). The     average nearest neighbour distance was 1.47 cm (greater than the expected       distance) and gave a R = 1.07, but this was not statistically significant from  R = 1 for a random pattern (P=0.08).                                                                                                                                                             DISCUSSION                                                                                                                        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   1/d     , and the   expected perimeter length is   4/SQR(d)  , where d is the density of `trees'    (Meijering 1953). Matrn (1979) calculates the expected length of border areas  between Dirichlet cell mosaics of two species, if the distributions of the      species are random.                                                                 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 & Hoskins1989a, b; Welden, Slauson & Ward 1990). Dirichlet polygons describe territories of pectoral sandpipers, Calidris melanotos, male mouthbreeder fish, Tilapia     mossambica, and nest areas of Royal terns, Sterna m. maxima (Grant 1968; Barlow 1974; Buckley & Buckley 1977). The cellular patterns of coenobial green algae,  Pediastrum boryanum, as well as cultured epithelial cells of chicks (retinal andlung), rat (intestine) and mudpuppy (gallbladder) show Dirichlet packing (Honda 1978).                                                                              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 timeis similar to that for SYSTAT). The computation cost of the algorithm of Green  & Sibson (1978) increases roughly as n^1.5 (Diggle 1983), while the computationatime for the algorithm presented here increases as n^1.36 (geometric regression,n=7).                                                                               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 Dendroctonus brevicomis, Tomicus piniperda, and Pityogenes chalcographus attack the outer bark and bore into the thin layer of phloem/cambium covering the woody xylem tissue of trees. The beetles         construct a two-dimensional system of tunnels or galleries within the layer     where they feed and reproduce. The thickness of the layers is similar to that   of a beetle so competition is expected to be severe for this limited food       resource. In fact, reports of intraspecific competition in several species in   the genera Ips, Dendroctonus, Scolytus and Tomicus have shown that brood output per female decreases at higher attack densities (Miller & Keen 1960; McMullen   & Atkins 1961; Eidmann & Nuorteva 1968; Ogibin 1973; Beaver 1974; Mayyasi et al.1976; Wagner et al. 1981; Light, Birch & Paine 1983; Anderbrant, Schlyter &     Birgersson 1985).                                                                   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 Tomicus piniperda, are known to space    their attacks (Nilssen 1978; Byers 1984) and this is evident also for           Dendroctonus brevicomis and Pityogenes chalcographus (Figs 3, 4 and 6a).            Dendroctonus brevicomis is the most important pest bark beetle of forests   in California (Miller & Keen 1960). The female initiates the attack and bores   a sinusoidal tunnel under the bark in the phloem layer. She produces a pheromonecomponent, exo-brevicomin, which attracts primarily males (Silverstein et al.   1968; Byers 1989). The male arrives and joins a female in her gallery and he    releases a second attractive pheromone component, frontalin, which is           synergistic with exo-brevicomin (Kinzer et al. 1969; Wood et al. 1976). This    causes beetles to aggregate en masse and overcome the tree's resistance,        resulting in its death and successful reproduction by the beetle. Over-crowding would result if not for several mechanisms, only partly understood, to avoid    severe competition (Byers 1989).                                                    Both sexes produce trans-verbenol which at close-range inhibits the female  sex from entering holes releasing attractive pheromone components (Byers 1983). Verbenone, is produced by males and this compound as well as trans-verbenol     inhibits both sexes from flying to sources releasing attractive pheromone       (Bedard et al. 1980; Byers et al. 1984). Still another compound, ipsdienol, is  produced in small amounts by males and inhibits both sexes (Byers et al. 1984). Thus, these compounds may function together to limit the overall attack density as well as close-range spacing.                                                     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 forseveral 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 Tomicus piniperda was more uniform than a random pattern. Presumably the spacing of attacks is to avoid competition between larvae as has been shown in simulation models (DeJong & Saarenmaa 1985). The MAD of from 3.7 to 4.3 cm   indicates that T. piniperda spaces further apart than either D. brevicomis (1.7-2 cm) above or the European spruce bark beetle, Ips typographus (2.5 cm, Byers  1984).                                                                              T. piniperda, although a pest of Scots pine in Europe, does not produce an  aggregation pheromone, as do most bark beetles that aggregate on trees, but     instead is attracted to host volatiles. A combination of the monoterpenes -    pinene, 3-carene, and terpinolene emanating from wound oleoresin of storm-      damaged trees serves in a mechanism for recognition of the host as well as its  susceptibility to attack (Byers et al. 1985). Both sexes contain a small amount of verbenone in their hindguts (Lanne et al. 1987) which is probably released   with the faecal pellets as are pheromone components in other bark beetle species(Byers 1989). Verbenone, at a range of a few mm to cm, could inhibit beetles    from boring nearby. Beetle infested logs of Scots pine increasingly released    verbenone with time while a control log released a constant and very low amount (Byers, Lanne & Lfqvist 1989). Release of verbenone at rates comparable to     several infested logs significantly reduced the attraction of flying beetles to host monoterpenes. This indicates that verbenone may function in spacing of     attacks as well as a cue that the host is now unsuitable for colonization       (Byers, Lanne & Lfqvist 1989). Males of T. piniperda also stridulate audibly   so this could be part of a mechanism for spacing. Stridulation appears to play  a role in mate recognition, and undoubtedly in male-male fighting (Byers 1991).     In contrast to D. brevicomis and T. piniperda where the female attacks, the male of Pityogenes chalcographus chooses the attack site on Norway spruce and   thus is responsible for avoiding competition. The males produce two pheromone   components, (E,Z)-(2,4)-methyl decadienoate and chalcogran, which are attractiveto both sexes (Francke et al. 1977, Byers et al. 1988). Several host            monoterpenes, including 3-carene, - and -pinene, stimulate entering of        artificial holes when the pheromone components are present, indicating the      monoterpenes play a role in host recognition (Byers et al. 1988). There are no  known olfactory inhibitors which might regulate spacing. However, higher releaserates of the attractive pheromone components cause the males, but not apparentlythe females, to be less attracted (Byers et al. 1988). This mechanism may       inhibit males from boring too closely to established attacks. Neither sex seems to be able to stridulate. Therefore, a suitable explanation for spacing in this species is lacking.                                                                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 D. brevicomis were in crevices, visible as the longitudinal       'dotted-lines' in Fig 4. The deep crevices in ponderosa pine allow a more rapid and efficient entry into the phloem, this combined with competition probably hasprovided the selection pressure for evolution of the sinusoidal gallery making. The boring of a gallery parallel to the wood grain, as in many other bark       beetles, would be disadvantageous for D. brevicomis since it would soon         encounter attacks of neighbours. However, a sinusoidal gallery would avoid      nearby neighbours immediately above and below in the crevice.                       At the base of Scots pine trees the furrows are more numerous and attacks   of T. piniperda are associated with crevices (all 67 attacks in an area of 64   x 56 cm), but higher up the trunk the attacks occur more often under bark flakes(as in Fig. 5). It appears that the density of suitable sites for attack on the bark is higher than the observed attack density. Also, if the number of bark    flakes is limiting then one would expect clumping of attacks - which does not   appear to occur (Fig. 5). However, in the case of P. chalcographus (Fig. 6a) thebark of Norway spruce was like that of a fine-grained `sandy' surface so bark   irregularities seem unlikely to have caused the spacing between attacks.            The Dirichlet tessellation program and calculation of the MAD of spacing    will allow many other plant and animal species to be analyzed. The program is   available from the author, please send a formatted disk and mailer for IBM-     compatible personal computers.                                                                                                                                                                 ACKNOWLEDGMENTS                                                                                                                    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.                                                                                                                           REFERENCES                                                                                                                     Anderbrant, O., Schlyter, F. & Birgersson, G. (1985). 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Spatial Statistics, John Wiley & Sons, New York, U.S.A.    Rogers, C.A. (1964). Packing and Covering. Cambridge Tracts in Mathematics and      Mathematical Physics. No. 54, Cambridge University Press, UK.                                                                                               Rudinsky, J.A. & Michael, R.R. (1973). Sound production in Scolytidae:              stridulation by female Dendroctonus beetles. Journal of Insect Physiology,      19, 689-705.                                                                                                                                                Safranyik, L. & Vithayasai, C. (1971). Some characteristics of the spatial          arrangement of attacks by the mountain pine beetle, Dendroctonus ponderosae     (Coleoptera: Scolytidae), on lodgepole pine. 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Ecology, 37, 391-394.                                                                                                  Upton, G. & Fingleton, B. (1985). Spatial data analysis by example. Volume 1.       John Wiley and Sons, New York, USA.                                                                                                                         Wagner, T.L., Feldman, R.M., Gagne, J.A., Cover, J.D., Coulson, R.N. &              Schoolfield, R.M. (1981). Factors affecting gallery construction,               oviposition, and reemergence of Dendroctonus frontalis in the laboratory.       Annals of the Entomological Society of America, 74, 255-273.                                                                                                Welden, C.W., Slauson, W.L. & Ward, R.T. (1990). Spatial pattern and                interference in pinon-juniper woodlands of northwest colorado. Great Basin      Naturalist 50, 313-319.                                                                                                                                     Wilkinson, L. (1990). SYGRAPH: The System for Graphics. SYSTAT, Inc., Evanston,     IL., USA.                                                                                                                                                   Wood, D.L., Browne, L.E., Ewing, B., Lindahl, K., Bedard, W.D., Tilden, P.E.,       Mori, K., Pitman, G.B. & Hughes, P.R. (1976). Western pine beetle:              specificity among enantiomers of male and female components of an attractant    pheromone. Science, 192, 896-898.                                                                                                                           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 b between the center and point a is included in   the calculation but in this case was not necessary for the drawing of the       Dirichlet cell (see text for more details).                                                                                                                     Fig. 2. Relationship between the coefficient of variation (CV) of Dirichlet cellareas and the minimum spacing between points in any area expressed as the       percentage of the maximum possible point spacing if the points were hexagonally arranged. The vertical lines represent 95 percent confidence limits (n = 32     simulations of 250 points each per spacing increment; about 100 tessellations   per simulation were analyzed).                                                                                                                                  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 Pityogenes chalcographus (P. c.), Tomicus piniperda (T. p.),   and Dendroctonus brevicomis (D. b.) and for a simulated pattern of randomly     placed points (R). Vertical lines represent 95 percent confidence limits. The   numbers along the dashed line are the number of attacks that were both          tessellated and within the area inside the border.                                                                                                              Fig. 4. Dirichlet cell tessellations of 35 attacks of Dendroctonus brevicomis   inside a border width of 1.5 times the expected fourth nearest neighbor distancein an area of 90 x 45 cm (n=97). The average cell area is 33.9 cm^2 5.4 (95%   confidence limits) and the coefficient of variation (CV) is 48%. The percentage of maximum spacing was 28.1 (0-39.3, 95% confidence interval, Fig. 2).                                                                                          Fig. 5. Dirichlet cell tessellations of 44 attacks of Tomicus piniperda inside  a border width of 1.5 times the expected fourth nearest neighbor distance in an area of 86.5 x 50 cm (n=108). The average cell area is 33.4 cm^2 2.4 (95%      confidence limits) and the coefficient of variation (CV) is 24.5%. The          percentage of maximum spacing was 62.8 (60.3-64.9, 95% confidence interval, Fig.2).                                                                                                                                                             Fig. 6 (a). Dirichlet cell tessellations of 84 attacks of Pityogenes            chalcographus inside a border width of 1.5 times the expected fourth nearest    neighbor distance in an area of 30 x 45 cm (n=179). The average cell area is    6.87 cm^2 .44 (95% confidence limits) and the coefficient of variation (CV) is 30.3%. The percentage of maximum spacing was 55 (52.2-57.5, 95% confidence      interval, Fig. 2); (b) Dirichlet cell tessellations of 91 points placed at      random inside a border width of 1.5 times the expected fourth nearest neighbor  distance in an area of 30 x 45 cm (n=179). The average cell area is 6.74 cm^2   .81 (95% confidence limits) and the coefficient of variation (CV) is 58.8%. Thepercentage of maximum spacing was 0 (0-18.2, 95% confidence interval, Fig. 2).                                                                                  ------------------------------------------------------------------------------   John A. Byers is a Ph.D. graduate of the University of California at Berkeley   in entomology. Currently (Oct. 1992) he is a Hgskolelektor (Assoc. Professor)  in the Department of Plant Protection, Chemical Ecology, Swedish University     of Agricultural Sciences, Box 44, S-230 53 Alnarp, Sweden. His interests        include chemical ecology of bark beetles and computer simulation of behavior    and ecology.                                                                   ------------------------------------------------------------------------------                                                                                  Correction to above paper:                                                      ******************************************************************************* Byers, J.A. 1996. Correct calculation of Dirichlet polygon areas. J. Anim. Ecol.    65, 528-529.                                                                *******************************************************************************                                                                                 In a planar field with many stationary objects, a mosaic of polygons can be     drawn or tessellated 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 xc,yc (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 (xi,yi and xi+1,yi+1). The last pair of successive vertices must cause closure (i.e., include the first vertex). Unfortunately, this was notdone 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 =       .5(xc(yi - yi+1) + xi(yi+1 - yc) + xi+1(yc - yi))             is correct if k = number of vertices and xk+1,yk+1 are equal to x1,y1.                                                                                                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 areasgave 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 calculatethe 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%) asbefore (Fig. 6b, CV = 58.76%). The species-specific values for the MAD's are notexpected 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 abark beetle family group is actually larger than reported. The revised areas of 42.076.24 cm2 (95% CL) for D. brevicomis (Fig. 4) is almost the same as 42.132.95 cm2 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 8.50.5 cm2 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       8.350.93 cm2.                                                                                                                                                        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 beused 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 1002/50  or 200 units.                                                                                                                                                         The corrected version of the Dirichlet program is available by sending a  formatted 3.5" or 5" IBM disk to the author. The software also can be downloadedfrom the Internet (http://www.chemical-ecology.net/software.htm).                                                                                               References                                                                                                                                                      Byers, J.A. (1984). Nearest neighbor analysis and simulation of distribution         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. VII. 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 distributed        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.                                                                                                                        