 Byers, J.A. 2000. Wind-aided dispersal of simulated bark beetles flying             through forests. Ecological Modelling 125:231-243.                                                                                                                                     John A. Byers                                                            Department of Plant Protection                                            Swedish University of Agricultural Sciences                                                 S-230 53 Alnarp                                                                     Sweden                                            Abstract                                                                                   Larger bark beetles such as Ips typographus (Coleoptera: Scolytidae) at about 2 m/s for up to several hours. Computer simulations in two dimensions  showed that bark beetles are capable of dispersing from a brood tree over wide  areas while drifting with the wind. For example, if beetles take an angle of    maximum turn (AMT) at random up to 10 either left or right each second, about  90% of the beetles become distributed over a 31.9 km area after 1 hour of      flight. Larger maximum turning angles by beetles decrease the area of dispersal in proportion to the reciprocal of the square of the AMT. An increase in the    dispersal time causes a linear increase in dispersal area and downwind drift    distance, while increases in wind speed have no affect on the ultimate dispersalarea but do increase the drift distance. Dispersal of bark beetles in a 10 x 10 km forest of 5 million trees of 0.15 m trunk radius, corresponding to the       natural density and trunk size of a 70-year-old Norway spruce forest (Picea     abies), was simulated by spacing trees at appropriate density in a 50 m radial  area centered on a beetle. A new area with trees was constructed similarly      whenever the beetle left the former area. These simulations showed trees reducedthe size of the dispersal area by 11% and downwind drift by 18% after 1 hour of flight due to the effect of turning some beetles back toward the release point, similar to the effects of increasing the AMT. The average dispersal distance anddownwind distance decreased as linear functions of trunk density. Given step    size, number of steps, and AMT, the correlated random walk equation of Kareiva  and Shigesada (1983) predicts mean squared dispersal distance. This can be      transformed to the more meaningful average dispersal distance by taking the     square root and multiplying by a proportion obtained from a three dimensional   surface equation fitted from simulation results.                                                                                                                Keywords: Dispersal; Migration; Host selection; Populations; Correlated Random            Walk                                                                                                                                                                                                                                   1. Introduction                                                                                                                                                 Bark beetles (Coleoptera: Scolytidae) are important predators of coniferous     forests. For example, Norway spruce (Picea abies), predominating in many regionsof Europe and Asia, is attacked by Ips typographus, the tree's most serious     enemy (Austar et al. 1984). The adults of `aggressive' bark beetle species in  the genera Ips and Dendroctonus must kill the host tree so that it does not     continue to produce toxic resin that can also entrap the beetles and their      larvae (Byers, 1995). Thus, newly emerged adults emerge from the brood tree or  overwintering sites and fly in search of the usually rare hosts that are more   susceptible due to disease and abiotic factors such as drought and storm damage.A few beetles of the population are termed `pioneers' since they presumably are first to locate and `attack' a susceptible tree and begin the colonization. If  the tree is of low resistance, then insufficient resin is produced to repel the pioneer beetle so it has time to feed and produce pheromone. This causes a largepart of the flying population in the vicinity to aggregate in response to the   pheromone and exploit the food and mate resource (Byers, 1996a). The host-      selection process by pioneers and the population dispersal flight are still not well understood.                                                                                                                                                Insects disperse when their habitat is becoming unsuitable. This can be from a  lack of food resources, mating possibilities, territories and suitable domicilesor from the need to escape the local buildup of parasites and predators (c.f.   Ricklefs, 1990). Apparently for the same reasons, bark beetles emerge from the  dead brood tree, or litter near the brood tree, and begin a dispersal flight    that probably can range from a few meters to several kilometers. Evidence from  the laboratory has shown that bark beetles can fly remarkably far. For example, Jactel and Gaillard (1991) flew Ips sexdentatus on rotary flight mills connectedto computer and found that 50% of the beetles could fly more than 20 km based onabout 50 interrupted flights (a total of at least 2.5 hours of flight). In      another study where I. typographus were placed on flight mills, the longest     continuous fight was 6 h and 20 min (Forsse and Solbreck, 1985). This indicates that a few I. typographus flying at 2 m/s (Byers et al., 1989) could travel up  to 45.6 km without the aid of wind. Strong individuals of the Douglas-fir       beetle, Dendroctonus pseudotsugae, flew up to 8 h uninterrupted on flight mills (Atkins, 1961), and the southern pine beetle, D. frontalis, has flown up to 6 h on a flight mill (G. Birgersson, personal communication).                                                                                                       Knowledge of how far and where bark beetle populations disperse is mainly from  (1) mark-release-recapture studies using pheromone traps and from (2) the       geographical occurrence of new infestations relative to previous ones. Both     lines of investigation are inconclusive since (1) only a few pheromone traps    were used, usually some tens to hundreds of meters from the release site, so    that a large proportion of released beetles escaped, or (2) the origins of      attacking beetles were uncertain. Several studies have placed various sized     rings of pheromone traps around a source of marked beetles. For example, the    spruce bark beetle of Europe, I. typographus, was recaptured at various outer   distances from 120 to 1000 m (Botterweg, 1982; Zumr, 1992; Zolubas and Byers,   1995; Duelli et al., 1997). In California, I. paraconfusus was recaptured in    outer traps at 2 km (Gara, 1963). The ambrosia beetle, Trypodendron lineatum,   was recaptured at 500 m (Salom and McLean, 1989). As expected, a small          proportion of the released beetles were recaptured by the widely-spaced outer   traps, and the large gaps between traps probably allowed many to slip through   as they drifted with the wind (e.g., gaps of 785, 1257, and 393 m in Zumr, 1992;Gara, 1963; and Salom and McLean, 1989; respectively). An adverse effect of     marking, although discounted, might also influence the dispersal.                                                                                               Anecdotal evidence of long-range dispersal (Nilssen, 1978; Miller and Keen,     1960) is inconclusive since it is difficult to rule out all possible sources of beetles. The best evidence of this type is found in Miller and Keen (1960) who  summarize results of studies by the US Forest Service in California on the      western pine beetle, Dendroctonus brevicomis. This beetle infested `islands' of ponderosa pine, initially free of beetles, that were separated from the main    forest by open sagebrush areas. They concluded that significant numbers of bark beetles must have flown a minimum of 3.2 km in one study, and 9.6 or even 20 km in another study, to reach the infested trees and kill them.                                                                                                    Little is known about the flight paths of bark beetles since they are small and dark, thus difficult to observe for any significant distance. Ips typographus   and some other Ips and Dendroctonus species have been caught primarily under    10 m in height or under the forest canopy (Gara and Vit, 1962; Forsse and      Solbreck, 1985; Duelli et al., 1986; Byers et al., 1989). I once ran after      several individual I. typographus that had taken flight from a brood log pile   in a clearcut in a 3-4 m/s wind in which they flew or drifted generally downwind(none flew crosswind) at 2-3 m height in an approximately straight path for some60 meters. Bark beetles, including I. typographus, usually fly away from releasesources in all directions unless winds are strong where they appear to drift    with the wind (Meyer and Norris, 1973; Botterweg, 1982; Helland et al., 1984;   Byers et al., 1989; Salom and McLean, 1989; Thoeny et al., 1992; Zolubas and    Byers, 1995; Duelli et al., 1997).                                                                                                                              The first objective was to simulate dispersal of bark beetles using various windand flight parameters in order to visualize how natural dispersal distributions might appear that otherwise are nearly impossible to observe. A second          objective was to simulate the occurrence of trees at the density of a Norway    spruce forest to see what effects they might have on flight dispersal. The      ability to construct theoretical distribution patterns based on realistic       parameters may allow a better understanding of the dispersal ecology of bark    beetles and the probability of them killing trees next to outbreak centers.     Finally, simulations can be used to test results from previous studies          proposing equations that predict dispersal distances of populations based on thedistribution of turning angles, number of steps, and average step length (e.g.  Kareiva and Shigesada, 1983). It might also be possible to modify or correct    such equations if they are found to diverge from the simulated reality.                                                                                         2. Methods                                                                                                                                                      Wind-aided bark beetle dispersal in forests or clearcuts. The algorithms for    simulating insect flight movement in two dimensions have been developed in      earlier models (Patlak, 1953; Rohlf and Davenport, 1969; Kitching, 1971; Byers, 1991, 1993, 1996a, b). Briefly, modelled insects take steps in a forward        direction with possible random deviations up to an angle of maximum turn (AMT), either right or left at random. The flight path of an insect, from x0,y0 to x,y is determined by calculating the insect vector using polar coordinates from the former position based on the step size (s), or distance travelled in one second,and former direction ( in radians) plus the random angle of turn (-AMT <  <   AMT). The wind vector, wind speed (w) and direction (), is then added to the   polar coordinates of the insect vector to obtain the resulting path [i.e.,      x = x0 + COS(+)s + COS()w  and  y = y0 + SIN(+)s + SIN()w ]. The input    parameters of the model are dispersal time, average insect speed and step size, coordinates of the brood tree, wind direction and speed, number of insects and  the area length and width. Initial directions of insects are chosen randomly    (0 to 360).                                                                                                                                                    In all simulations, flight speed was 2 m/s which is about what larger bark      beetles such as Ips typographus can maintain in still air (Byers 1996a). Most   simulated dispersal periods were limited to 1 hour, although these beetles on   flight mills have flow up to 6 hours (Forsse and Solbreck, 1985; Forsse, 1991). In one simulation, the dispersal times of the population were varied about a    mean of 1 hour according to a normal distribution with standard deviation of 15 minutes (Walker, 1985). The AMT was either 10 or 20 unless varied from quite  straight (2) to highly random and circuitous (90). At the end of each         simulation period the positions of all insects were recorded for plotting and   analysis.                                                                                                                                                       The dispersal patterns of distribution were visualized by constructing isolines that encircle approximately 90 percent of the points (N). This was done by usingthe coordinates of all points to calculate a center of mass (averages of x's    and y's). From this center usually N/20 pie-shaped sectors, evenly dividing     360, were calculated, and the angles to all points were found to determine     which points were within each sector. The distances to enclosed points within   each sector then were calculated and sorted based on increasing distance from   the center. For a 90 percent isoline, an average distance was calculated from   the distances of the two points less than, and greater than, the 90th percentileof distances. This distance was then used as a radius from the center along the middle of the sector to find an endpoint. These endpoints were used in a three- point rolling average to form a polygon whose area was found by summing the     areas of the polygon's triangles about the center.                                                                                                              The dispersal patterns were further analyzed by centering the points just insidea rectangle and then constructing a grid of cells (30 x 30) in which points werecounted. The grid cell counts were smoothed by a surrounding 9-cell rolling     average and plotted as bars in three-dimensions without perspective. The        possible effects of random wind directions and speeds were investigated with    this analysis method. For example, the dispersal of 500 beetles was simulated ina 10 x 10 km area with a grid of 50 x 50 cells (200 x 200 m each) in which each cell had a random (0-360), but consistent, wind direction. A variation of this model had cells with consistent wind vectors up to 90 left or right, at random,of an eastward direction. Finally, wind direction and speed were varied (0-360 and 0-2 m/s) for each beetle at each step, but with an average wind speed of    1 m/s.                                                                                                                                                          Trees could affect the dispersal patterns of bark beetles. This is difficult to simulate because there are 50,000 Norway spruce trees (0.15 m radius, 70 year   plantation) in a square km (Magnussen, 1986) or 5 million in a 10 x 10 km area  needed for simulating dispersal for an hour. The array needed to hold and searchthese tree coordinates requires more memory and speed than possible with programsoftware and personal computers. However, the task was accomplished by          simulating one beetle at a time, many times, and placing a radial area of forestcentered on the beetle at the start. The area's radius was 50 m and within this area 393 trees are expected (Magnussen, 1986). The trees were spaced apart at   least 50 percent of the maximum hexagonal spacing possible, a distance equal to a minimum allowed distance (MAD) of 2.4 m between trees (Byers, 1984, 1992).    Trees were also spaced this distance or more from the beetle at the center, but otherwise the trees were placed at random within these constraints. The beetle  was then allowed to move as above. However, if it would have struck a tree trunk(algorithm in Byers, 1991) according to the flight and wind vectors, then an    algorithm picks an angle  (0.57) either left or right at random from the      former flight angle and tests this flight angle. If the beetle still would      strike the tree, then the angle  is expanded incrementally (0.57), but        alternating left and right from the former flight angle, until the beetle missesthe trunk. This means that beetles will pass by the tree either left or right ina realistic way usually according to which side of the tree they tended toward  initially.                                                                                                                                                      The beetle continues until eventually passing out of the circular area,         whereupon a new set of 393 spaced trees is centered about the beetle (using the same memory array). This requires little memory and speeds the searches of tree coordinates by 12,723 times compared with searching 5 million pairs (requiring  80 MB memory). All simulations and graphical analyses were done using a         combination of QuickBASIC 4.5 and PostScript 2.0 programming languages.                                                                                         Equations predict mean dispersal distance. During the studies, I wondered       whether it is possible to use an equation to predict the average distance of    dispersal of a population of animals from a release point given: (1) the step   size (or average step size), (2) the number of steps, and (3) the AMT. The      average distance of dispersal and variance can be found by simulation to check  the validity of any such equation. The equation of Kareiva and Shigesada (1983) uses move lengths, turning angles, and total moves to calculate dispersal       distances. However, their formula gives the expected mean squared dispersal     distance, E(R):                                                                                                                                                                         c          1 - c                                      E(R) = nE(L) + 2E(L)    (n -    )             (1)                                          1 - c        1 - c                                                                                                                      where L is the step size (since L is constant then E(L) = E(L) = L ), n is   the number of steps, and c is the average of the cosines of all possible turningangles (in radians) from a specified random distribution:                                                                                                            AMT                                                                         c =  cos  g() d                                      (2)                                                                                                     -AMT                                                                                                                                                         The mean squared dispersal distance, unfortunately, is difficult to compare to  the intuitively more meaningful mean dispersal distance. Assuming a uniform     distribution of random angles between -AMT and AMT, I calculate a mean c from   the AMT (converted to radians, AMT*/180) by summing the cosines of  in        incremental steps of 2*AMT/i (where i = 20,000) from -AMT to AMT and dividing   the sum by the number of iterations:                                                                                                                                 AMT                           S                                             S =  cos           and     c =                      (3)                       -AMT                           i                                                                                                                            The problem is now to use the mean square distance of Kareiva and Shigesada     (1983) from equation (1) using c from (3) to find the mean dispersal distance.  A first approximation is to take the square root, but this overestimates the    actual mean distance found by simulation by up to 12.4%.                                                                                                        Simulations using various step sizes, and varying both the number of steps (X)  and the AMT (Y, from 0 to 180) stepwise, were used to calculate the mean       distance of dispersal with the pythagorean formula (averages of 4 simulations   of 1000 points at each X,Y). It is then possible to compare the resulting       distances to square roots of expected mean square distances based on equations  (1) and (3) using appropriate parameters. The comparisons were used to find     correction factors (which vary with number of steps and AMT) based on the       ratio of the simulated values and the calculated square root values.                                                                                            3. Results                                                                                                                                                      Wind-aided bark beetle dispersal in forests or clearcuts.  An increase in the   angle of maximum random turn (AMT) caused the area of distribution of simulated bark beetles enclosed by a 90% isoline to decrease proportional to the                                                  -2.03                                   reciprocal of the AMT squared (Y = 3609X      , R=1.00), but had no affect on   the center of the distribution relative to the brood tree source                Press [F10] for Figures. (Fig. 1). The average distance beetles flew away from  the center of mass (or from the brood tree in still air) after 1 hour is        expressed as a reciprocal relation of the AMT,                                              -1.02                                                               (Y = 21.377X     , R=1.00). The constant wind speed and direction affected all  beetles similarly so the patterns were all symmetrical and drifted to the same  position as shown by the equivalent positions of the centers of mass (Fig. 1).  Simulations showed that the patterns were the same relative size and symmetry   regardless of wind speed, which only caused the `centers of mass' to drift more rapidly. The convex polygon area containing about 90% of the 500 simulated      beetles ranged from 31.9 km (or about 6.4 km diameter) for reasonably "normal- flying" beetles (10 AMT) to only 0.4 km (0.7 km diameter) for the highly      circuitous-flying ones (90 AMT).                                                                                                                               Given that beetles all had random turning angles less than 20 right or left,   an increase in time of dispersal caused a linear increase in the area of        distribution of simulated bark beetles (Fig. 2) Y = -0.46 + 9.547X, R=1.00.     As expected, the average distance traveled in 1 hour by beetles away from the   center of mass, or brood tree in still air, is expressed in relation to the                                   0.516                                             square root of time (Y = 1.05X     , R=1.00; Gamov and Cleveland, 1969). The   90% isoline polygon at 0.25 hours dispersal covered an area of 2.2 km which    increased to 18.3 km after 2 hours of dispersal. The centers of mass also      drifted farther downwind with more time (about 7.2 km in 2 hours, Fig. 2). If   the AMT was held constant at 20 and the time at 1 hour, then not surprisingly  an increase in wind speed has no affect on the area of distribution, being about9.00.6 km (95% C.L.) at any wind speed (Fig. 3). However, the centers of the distributions drifted farther with higher wind speeds (Fig. 3).                                                                                                 Initially beetles dispersing from a point source would fly out in all           directions. If the flight direction was nearly straight (AMT = 2) then a       concentric ring of points would flow outward as seen after 1 hour in Fig. 4A.   The density of points in the ring along a radial transect approximates a normal curve. Eventually, beetles by random turns can fly back toward the source. This happens more often if the AMT is larger at 10 in which the density distributionbecomes a three-dimensional bell-shaped curve (Fig. 4B). A constant wind        direction and wind speed does not affect the dimensions of the density          distributions.                                                                                                                                                  An attempt to influence the density distributions by variable wind directions   was done by placing a grid of 2500 cells (200 m square) in which each had a     random wind direction that was constant during the period. The density          distribution was virtually unaffected when wind speed was 1 m/s (compare Fig. 5 to Fig. 4B). Press [F10] for Figures. If the wind speed is made more than the   flight speed of 2 m/s then beetles can be forced along the narrow boundaries of the cells when two wind vectors oppose each other. This situation seems         unnatural and so is not considered further. Finally, a highly random scenario   was simulated in which wind direction varied at random for each beetle at each  step, and also the wind speed was varied at random up to 2 m/s (average 1 m/s). Again, the density distribution or diameter of the area was not significantly   affected.                                                                                                                                                       Individuals of a population of bark beetles are expected to vary in fat content and flight range according to a normal distribution. Therefore, simulated       insects were allowed to vary in flight duration about a mean of 1 hour with a   standard deviation of 15 minutes (Fig. 6). Compared to an exact flight duration of 1 hour, the variable range insects became distributed over a slightly larger area (90% isoline of 36.8 km vs. 30.7 km; mean distance from center of mass of2.14 vs 2.01 km). Otherwise there were little differences in spatial            distribution or downwind movement (Fig. 6).                                                                                                                     The presence of 5 million Norway spruce trees (0.15 m radius, 70 years) in a 10 x 10 km area (Magnussen, 1986) had a subtle affect on the paths and reduced     somewhat the dispersion area of the simulated beetles. After 1 hour of dispersalin a 1 m/s wind, 100 bark beetles in the simulated forest had a 90% isoline areaof 25.3 km and moved downwind an average of 2.85 km (Fig. 7A). In comparison,  the same beetles in an open field covered more area 30.8 km and moved further  downwind at 3.53 km (Fig. 7B). Using 10 simulations of 500 beetles each for eachtype, the `forest' beetles covered an average 90% isoline area of 30.940.98 km(95% C.L.) and drifted on average 2.930.05 km downwind compared to dispersal  in an open area covering 34.811.13 km and moving 3.570.06 km downwind (means significantly different P<0.001, t-test). The average dispersal distance from   the center of mass was 1.980.03 km in the forest and was significantly less    than 2.080.04 km in the open (P<0.001). These results indicate that the trees  reduced the dispersal area by about 11%, the downwind drift by 18%, and the     average dispersion distance from the center of mass by 5%.                                                                                                      Simulations of varying densities of Norway spruce trunks, from 0 to 1000 trees  within the 50 m radius about a beetle, shows that the average dispersal distancedownwind (m) decreases as a linear function of tree density                     (Y = 588.2 - 0.1326X, R=0.97, Fig. 8). Also, the average dispersal distance (m)from the center of mass (Y = 762.5 - 0.0549X, R=0.98) and the 90% isoline area (km) (Y = 3.0576 - 0.000232X, R=0.96) decreases linearly with tree density    (Fig. 8). The paths of beetles were more twisting at the higher trunk densities (Fig. 8) due to the need to more often avoid trees, the same effect as if the   angle of maximum turn (AMT) had been made larger (as in Fig. 1).                                                                                                Equations predict mean dispersal distance. The equations of Kareiva and         Shigesada (1983) for mean square dispersal distances gave square root values    very similar, but not always, to simulation results (actual mean dispersal      distances).  The comparisons show that for a large number of steps (n > 5000),  the actual dispersal distance is actually about 0.89 of the square root of the  expected mean square distance from the formula. Also, when the AMT is above 30 the correction factor stabilizes at 0.89. Interestingly, there is a complex     interplay between the AMT and number of steps which makes it necessary to       describe the correction factor (about 0.89 above) as a three-dimensional surfaceat angles < 30 and steps < 5000 (Fig. 9).                                                                                                                      The step size surprisingly has no affect on the correction factor. The surface  equation of the correction factor (Z) can be described reasonably well by a     multivariate least squares cubic polynomial:                                           3     2          2             2          2     3                         Z = aX  + bX  + cX + dX Y + eXY + fXY  + gY + hY  + iY  + 1.021        (4)                                                                                     where a = -1.123E-12, b = 9.27E-9, c = -2.663E-5, d = 1.464E-10, e = -2.38E-6,  f = 6.449E-8, g = -5.695E-3, h = 1.677E-4, and i = -3.227E-6 (MATHCAD, MathSoft Inc.). However, the fit is best over a limited range, therefore five constraintsapplied in order improve the fit: (1) if Z < 0.89 then Z = 0.89, (2) if Z > 1   then Z = 1, (3) if AMT > 30 then Z = 0.89, (4) if number of steps > 5000 then  Z = 0.89, and (5) if AMT > 20 AND steps > 2000 then Z = 0.89.                                                                                                   Thus, the mean dispersal distance is found from equations (1), (3) and (4). For example, if L = 2 m, n = 1000 steps, and AMT = 30 or 0.5236 radians, then      c = 0.9549 from equation (3) and the mean square dispersal distance is 169,752 mfrom (1) and the mean dispersal distance (MDD) is:                                        _____          ______                                                  MDD = Z E(R)  =  0.89169752  =  367 m                               (5)                                                                                    where Z = 0.89 from equation (4) and constraints. Five simulations with the     same parameters and 1000 insects each gave a mean dispersal distance of         364.0 5.3 m (95% C.L.).                                                                                                                                        4. Discussion                                                                                                                                                   The dispersal patterns shown in Figs. 1-6 are similar to expected point         distributions based on earlier studies of correlated random walks and diffusion models (Okubo 1980). The results of dispersion of bark beetles differ only      because of the specific parameters for flight duration (number of steps), step  size (or frequency of possible turn) and the angle of turn taken at random,     either left or right, within an angle of maximum turn (AMT). Earlier simulationsof animal movements have either used random turns of increments of 45 or 90 on a lattice (Rohlf and Davenport, 1969; Gries et al., 1989; Johnson et al., 1992),a uniform random distribution within a range of AMT (Byers, 1991, 1993, 1996a,  b, 1999; Kindvall, 1999), or random turns using a normal distribution with a    specified standard deviation (Cain, 1985; Weins et al., 1993). At least for the latter two methods, the resulting distributions after dispersal can be made     nearly identical by adjusting the AMTs and standard deviations of turning anglesappropriately (Byers, unpublished).                                                                                                                             The turning angle and step size parameters are difficult to measure for flying  bark beetles and may be complicated by the scale chosen for measurement due to  habitat heterogeneity and periodic behavioral changes (Kaiser, 1983; Cain, 1985;Turchin, 1991; Johnson et al., 1992; Crist et al., 1992; Weins et al., 1993;    With and Crist, 1996). However, if the movement is regular as might occur in a  uniform habitat, then the scale chosen is not critical over large ranges since  smaller divisions of the path give smaller angles of turn while larger division yield larger angular deviations. This can be easily seen in simulations where   approximately the same paths can be constructed from larger steps and larger    possible turning angles as from smaller steps and appropriately smaller turning angles.                                                                                                                                                         The simulation of dispersal in a `natural' forest of tree trunks shows that the trunks deflect beetles enough to reduce the dispersal area of the population    about 11% compared with no trunks (Fig. 7A and B). This is similar in effect to increasing the AMT of beetles (Fig. 1). Interestingly, the dispersal distance   downwind, the dispersal distance from the center of mass, and the 90% isoline   area (not shown) all decrease as linear functions of trunk density (Fig. 8).    Part of the observed reduction in dispersal rates of populations and the        encounter rates between predators and prey in heterogenous environments         (Kaiser, 1983; Johnson et al. 1992; Crist et al., 1992; With and Crist, 1996)   can be due to avoidance of obstacles as shown here, or due to attractive or     arresting properties of the obstacles (e.g., food items).                                                                                                       The results of flight mill studies with larger bark beetles (Atkins, 1961;      Forsse and Solbreck, 1985; Forsse 1991; Jactel and Gaillard, 1991) indicate thatthese beetles which fly at about 2 m/s could travel up to 45 km. The models herereveal the potential extent that bark beetles, and similar insects, can dispersein a relatively short time of one hour. In regard to bark beetle epidemics,     truly laminar wind of either consistent or variable direction (even highly      random in patches) has no affect on the shape or extent of the dispersal area   other than causing the point pattern to drift in unidirectional wind (Fig. 3).  In nature, of course, wind-aided dispersal is probably more complicated. First  of all, wind is usually not laminar but because of topography may flow in ways  to separate and transport beetles into different regions. Beetles may settle    after different flight durations which will tend to increase the dispersal area in wind (Fig. 6). There is some evidence from field traps that beetles avoid    both clearcuts and deep forest, preferring the edges of forests - thus further  disrupting the theoretical dispersal patterns (Botterweg, 1982; Byers,          unpublished). Recently, the spruce bark beetles Ips typographus and Pityogenes  chalcographus have been shown to avoid volatiles of nonhost birch trees (both   from bark and leaves) which suggests the possibility that beetles may not enter areas of primarily birch (Byers et al. 1998).                                                                                                                   The dispersal flight of a bark beetle may vary from only a few meters (as       observed during epidemics) to possibly several kilometers. Several factors      interact to cause the dispersal flight distance to vary between individuals.    The most obvious is that a beetle encounters a susceptible tree early in the    dispersal flight. However, whether this tree is attacked may depend on the levelof fat reserves that can be mobilized for flight (Atkins, 1966, 1969; Byers     1999). A beetle should have higher reproductive fitness if it flies rather far  from the brood tree since it can both avoid inbreeding with siblings and, more  importantly in my view, escape predators and parasites that are locally more    dense near the brood tree. Thus, the dispersal distance has been optimized over evolutionary time to balance the probably logarithmically increasing benefits offlying farther against the probably exponentially increasing likelihood of      exhaustion and failing to find a host. The fat level required for lengthy       dispersal will depend on the conditions in the brood tree during larval         development, for example, disease, insect, and climatic factors will affect the nutritional quality of the host. Severe competition among the larvae will       reduce the size of adults as well as their fat content (Atkins, 1975; Anderbrantet al., 1985). Parasites would reduce the size and fat content of some adults   while predators would lessen competition for those remaining locally, thereby   increasing the variability of dispersal range in the population. The population density of bark beetles should be stabilized by a frequency-dependant           competition for the susceptible trees that would produce increasingly stronger, longer-flying individuals with decreasing attack and larval density while givingweaker, shorter-flying ones with increasing competition.                                                                                                        Pioneer bark beetles find susceptible trees either by landing at random or by   response to volatiles from damaged or weakened trees. Most species that have an aggregation pheromone appear not attracted, or only weakly, by host volatiles   (Byers, 1995). For these species, it is not known if there are two types of     beetles, one that behaves as a pioneer and tests trees for susceptibility, and  another type that only searches for aggregation pheromone and trees undergoing  colonization. Most likely, all beetles have a strategy that depends on the levelof their fat reserves (Byers, 1999). At higher fat reserves during the period   immediately after emerging, the beetle disperses and ignores host trees and     pheromone, but as the fat reserves are depleted both trees and pheromone become increasingly attractive (Atkins, 1966; Gries et al., 1989; Borden et al., 1986; Byers, 1999). Finally, if no pheromone is present the beetle may test any tree  at random in the desperate hope of landing on a susceptible tree (Byers 1995,   1996a, 1999).                                                                                                                                                   If the pioneer beetle is fortunate to land on a tree of low resistance that can not produce sufficient resin to repel the beetle, then it has time to feed and  excrete pheromone components with the fecal pellets. This then functions as a   beacon to the population in the surrounding area that a weakened host can be    exploited as a food and mate resource (Byers, 1996a). Aggregation pheromone is  an evolutionarily adaptive signal since only trees too weak to vigorously repel beetles with resin will allow beetles to produce pheromone and joining beetles  will likely suffer little mortality. Some species, usually termed less          aggressive ones, such as the European pine shoot beetle, Tomicus piniperda, are attracted to volatiles produced after injury to the host tree by biotic or      abiotic factors that indicate susceptibility (Byers 1995; 1996a).                                                                                               Whether larger bark beetles can disperse in the manner described in the models  is not known since these beetles have not been observed in flight over any      appreciable distance while seeking hosts. However, in addition to the accounts  about D. brevicomis mentioned earlier (Miller and Keen, 1960), I. typographus   appears to have migrated as in the models. An infestation of these beetles in   several hectares on the side of a valley in the Harzt Mountains, Germany, was   surrounded by uninfested spruce forest for several km until the following seasonwhen the beetles probably left en masse and flew about 1.5 to 2 km downwind to  attack an area of some 20 ha killing hundreds of trees in several scattered     groups on the other side of the valley (H. Niemeyer, personal communication).                                                                                   Control strategies that attempt to contain epidemic populations of bark beetle  by use of trap trees or pheromone-baited traps in border zones must consider    the potentially wide dispersal areas, especially in mild winds, that can result.A border area of only 500 to 1000 m width, as proposed for containment of the I.typographus outbreak in the Bayerischer Wald National Park of Germany (Schrter,1998), may likely be inadequate. However, the dispersing population from an     epidemic area, where severe competition has reduced the fat content for extendedflight, may respond to host and pheromone after a short flight so that narrower border zones of treatments would be effective in stopping the spread.                                                                                           Earlier studies have investigated whether it is possible to use an equation to  predict the average distance of dispersal of a population of animals from a     release point given (1) the step size (or average step size), (2) the number of steps, and (3) the AMT.  The well-known diffusion equation for two dimensions   (Pielou 1977, Okubo 1980, Rudd and Gandour 1985) predicts the density of        organisms at any distance from the release point after a certain time or number of steps, but only for random walks (AMT = 180). Insects and many other        organisms do not exhibit truly random or Brownian movement but rather show      correlated random walks in which the previous direction influences the          direction of the next step. Patlak (1953) reports a modification of the         Fokker-Planck equation that can predict densities of points at any distance     and time for correlated random walks where the average angle of turn is known.  However, his equation (42) is exceeding complex in my view and thus has not beenused in practice. Turchin (1991) took the Patlak equation for one dimension and "simplified" it in his work on patch density transitions. His equation is still complex and difficult to use, and it is not as yet applicable in two dimensions.                                                                                The equation of Kareiva and Shigesada (1983) comes close to calculating the meandistance of dispersal when move lengths, turning angles, and total moves are    known. However, their formula gives the expected mean squared dispersal distancewhich obviously is much larger than the intuitively meaningful, mean dispersal  distance. By taking the square root of the mean squared dispersal distance, thisvalue still overestimates the actual mean dispersal distance of a population    (simulated population) by up to 12.4%. Only for animals with a straight path    (AMT = 0) do the formula square root and simulation values become identical,   for all other turning angle distributions and numbers of steps, the square root of the formula values give incorrect results. However, correct mean dispersal   distances can be found by multiplying correction factors from the three         dimensional surface equation from simulated results (equations 4 and 5) by the  square root of the formula values. This means that instead of using longer      running simulations of many points, the mean dispersal distance can be predictedusing equations with only the mean step size, number of steps, and AMT.                                                                                         The dispersal program software is available from the author for IBM-compatible  personal computers by downloading (BB-DISP.ZIP) from the Internet URL           (http://www.vsv.slu.se/johnb/software.htm).                                                                                                                     Acknowledgements                                                                This work was initiated after discussions supported by the "Bayerische          Landesanstalt fr Wald und Forstwirtschaft" about the large outbreak of Ips     typographus in the Bayerischer Wald National Park. I am grateful to Jrgen      Jnsson who encouraged my further refinements of the model. 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