JOHN A. BYERS Department of Crop Science Swedish University of Agricultural Sciences SE 230-53 Alnarp, Sweden |
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Larger bark beetles such as

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Insects disperse when their habitat is becoming unsuitable. This can be from a lack of food resources, mating possibilities, territories and suitable domiciles or 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

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,

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,

Little is known about the flight paths of bark beetles since they are small and dark, thus difficult to observe for any significant distance.

The first objective was to simulate dispersal of bark beetles using various wind and 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 the distribution 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.

In all simulations, flight speed was 2 m/s which is about what larger bark beetles such as

The dispersal patterns of distribution were visualized by constructing isolines that encircle approximately 90 percent of the points (N). This was done by using the 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 percentile of 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 inside a rectangle and then constructing a grid of cells (30 x 30) in which points were counted. 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 in a 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 search these tree coordinates requires more memory and speed than possible with program software and personal computers. However, the task was accomplished by simulating one beetle at a time, many times, and placing a radial area of forest centered 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).

15 insects disperse outward at 10 m steps in a 100 m radius area which has 100 trees, each of 4 m radius, spaced at least 50% of the maximum spacing possible (1.0746 / SQR(#trees/area) * 0.5) or at least 9.52 m apart in this case. The flight paths in blue show the insects avoiding the green trees (image at left was made from an actual simulation in BASIC, note that this GIF and yellow text box are not part of original paper). |
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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.

where

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

The problem is now to use the mean square distance of Kareiva and Shigesada (1983) from equation (1) using

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.

Fig. 1. Dispersal patterns of 500 simulated bark beetles flying from a source (pluses at left) in 1 m/s wind (from left) for 1 hour with different flight angles of maximum turn (AMT). Beetles took steps of 2 m (2 m/s speed), and at each step they could turn either left or right up to a maximum of 10, 20, 45 or 90° at random. The largest polygon includes 90% of the beetles having an AMT of 10° in an area of 31.9 km

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 (Y = 21.377X

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).

Fig. 2. Dispersal patterns of 500 simulated bark beetles flying from a source (pluses at left) in 1 m/s wind (from left) for different time periods. Beetles took steps of 2 m (2 m/s speed), and at each step they could turn either left or right up to a maximum of 20° at random. The largest polygon includes 90% of the beetles having flown for 2 hours to cover an area of 18.3 km

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 square root of time (Y = 1.05X

Fig. 3. Dispersal patterns of 500 simulated bark beetles flying from a source (pluses at left) for 1 hour in different wind speeds, from left to right. Beetles took steps of 2 m (2 m/s speed), and at each step they could turn either left or right up to a maximum of 20° at random. The polygons include 90% of the beetles having flown in various wind speeds of 0, 0.5, 1, and 2 m/s to disperse over similar areas of 9.3, 9.2, 9.0, and 8.4 km

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 distribution becomes a three-dimensional bell-shaped curve (Fig. 4B). A constant wind direction and wind speed does not affect the dimensions of the density distributions.

Fig. 4. A. Cell counts (9-cell rolling average) of 500 simulated bark beetles after dispersal from a point source for 1 hour. Beetles flew at 2 m/s and at each step of 2 m they could turn either left or right at random up to a maximum of 2° .

Fig. 4. B. The same conditions except that beetles took random turns with deviations up to 10° from straight ahead.

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). 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.

Fig. 5. Cell counts (9-cell rolling average) of 500 simulated bark beetles after dispersal from a point source for 1 hour. Beetles flew from a source centered in a 10 x 10 km area with a 50 x 50 grid of 200 x 200 m cells each with a random, but consistent, wind direction of 1 m/s velocity. Beetles flew at 2 m/s and at each step of 2 m they could turn either left or right up to a maximum of 10° at random.

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

Fig. 6. Dispersal patterns of 500 simulated bark beetles flying from a source (pluses at left) in 1 m/s wind (from left) for 1 hour (top polygon) or for a mean of 1 hour with normal variation (SD = 15 min., bottom polygon). Inset frequency histogram pertains to bottom polygon. Beetles took steps of 2 m with random turns at each step either left or right up to a maximum of 10° . The top and bottom polygons include 90% of the beetles and cover an area of 30.7 and 36.8 km

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 dispersal in a 1 m/s wind, 100 bark beetles in the simulated forest had a 90% isoline area of 25.3 km

Fig. 7. A. Dispersal paths of 100 simulated bark beetles flying from a source (at cross in lines) for 1 hour in 1 m/s wind (left to right) in a 10 x 10 km forest of 5 million trees. Beetles took steps of 2 m (2 m/s speed), and at each step they could turn either left or right up to a maximum of 10° at random. The 90% isoline polygon of dispersion was 25.2 km

Fig. 7. B. Dispersal paths of 100 simulated bark beetles flying from a source (at cross in lines) for 1 hour in 1 m/s wind (left to right) in a 10 x 10 km open area. Beetles took steps as in A. The 90% isoline polygon of dispersion was 30.8 km

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 distance downwind (m) decreases as a linear function of tree density (Y = 588.2 - 0.1326X, R

Fig. 8. Average dispersal distances either downwind from a source or from the calculated center of mass of 200 simulated bark beetles flying in Norway spruce forests of different trunk densities. Bars represent 95% confidence limits for a mean of 5 simulations. Beetles flew at 2 m/s for 10 minutes, taking steps of 2 m with turns at each step either right or left up to a maximum of 10° at random. Beetles turned when encountering spruce tree trunks of 0.15 m radius, as shown in circular regions of 50 m radius. Initial paths of 10 beetles, released uniformly, are shown in regions with 200 and 1000 trees.

Equations predict mean dispersal distance.

Fig. 9. Correction factor as a three dimensional surface function of the number of steps and the angle of maximum turn (AMT) at each step. The mean dispersal distance is obtained by multiplying the correction factor by the square root of the expected mean square dispersal distance. Points represent averages of four simulations of 1000 points each. See text for the best-fitting multivariate cubic polynomial equation.

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:

where

Thus, the mean dispersal distance is found from equations (1), (3) and (4). For example, if

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.).

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 simulations of 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 angles appropriately (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 that these beetles which fly at about 2 m/s could travel up to 45 km. The models here reveal the potential extent that bark beetles, and similar insects, can disperse in 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

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 level of 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 of flying 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; Anderbrant et 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 giving weaker, 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 level of 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,

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

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

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 been used 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 mean distance of dispersal when move lengths, turning angles, and total moves are known. However, their formula gives the expected mean squared dispersal distance which obviously is much larger than the intuitively meaningful, mean dispersal distance. By taking the square root of the mean squared dispersal distance, this value 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 predicted using 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.wcrl.ars.usda.gov/cec/software.htm .

This work was initiated after discussions supported by the "Bayerische Landesanstalt für Wald und Forstwirtschaft" about the large outbreak of

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