 Byers, J.A. 2001. Correlated random walk equations of animal dispersal                resolved by simulation. Ecology 82:1680-1690.                                                                                                                                                                                                            CORRELATED RANDOM WALK EQUATIONS OF ANIMAL                                           DISPERSAL RESOLVED BY SIMULATION                                                                                                                                        John A. Byers                                                                                                                                            Department of Crop Science                                               Swedish University of Agricultural Sciences                                                  230 53 Alnarp                                                                      Sweden                                                                                                                            Abstract. Animal movement and dispersal can be described as a correlated        random walk dependent on three parameters: number of steps, step size, and      distribution of random turning angles. Equations of Kareiva and Shigesada       (1983) use the parameters to predict the mean square displacement distance      (MSDD), but this is less meaningful than the mean dispersal distance (MDD)      about which the population would be distributed. I found that the MDD can be    estimated by multiplying the square root of the MSDD by a three-dimensional     surface correction factor obtained from simulations. The correction factors     ranged from 0.89 to 1 depending on the number of steps and the variation in     random turns, expressed as the standard deviation (SD) of the turning angles    about 0 (straight ahead). Corrected equations were used to predict MDDs for    bark beetles, butterflies, ants, and beetles (parameters from literature) and   the nematode Steinernema carpocapsae (Weiser). Another equation (Bovet and      Benhamou 1988) finds the MDD directly, and this agreed with the MDD obtained    by simulation at some combinations of SD of turning angles and numbers of       steps. However, their equation has an error that increases as a power function  when the SD of turning angles becomes smaller (e.g. < 6 at 1000 steps, or <    13 at 250 steps). Lower numbers of steps also increases the error. Equivalent  values of AMT (angle of maximum turn) in uniform random models and of SD in     normal random models were found that allowed these two models to yield similar  MDD values. The step size and turning angle variation of animal paths during    dispersal and host- and mate-searching were investigated and found to be        correlated, thus, use of different measured step sizes gives consistent         estimates of the MDD.                                                                                                                                           Running Head: MEAN DISPERSAL DISTANCE                                                                                                                           Key Words: correlated random walk, dispersion, host searching, mate searching,  mean dispersal distance, insects, Nematoda, nematodes, Scolytidae, Steinernema  carpocapsae, Steinernematidae, bark beetle, Coleoptera                                                                                                          Key Phrases: correlated random walk, host searching, mate searching, mean       dispersal distance, Steinernema carpocapsae (Nematoda: Steinernematidae), bark  beetle (Scolytidae: Coleoptera)                                                                                                                                                               INTRODUCTION                                                                                                                              Prior to 1950, the dispersal of animals was attributed to a steady      leakage of individuals out of an area, a forced exodus due to overcrowding,     and accidental transport (Wellington 1979). The risks of dispersal were         emphasized more often than the benefits, and population ecology usually         ignored dispersal since it is difficult to observe. Insects and other animals   often disperse from their brood habitat due to a lack of food resources,        suitable mates, and territories, or from the need to escape the local buildup   of parasites and predators (Johnson 1960, Kennedy 1975, Southwood 1977,         Wellington 1979, Root and Kareiva 1984, Ricklefs 1990). Evolution has favored   life strategies that can take advantage of changing mosaics of suitable         habitats (Cain 1985). Obligatory dispersal, an innate programmed dispersal      regardless of current habitat conditions, also is adaptive if this prevents     injury that otherwise would occur by staying in areas likely to rapidly         deteriorate in resources and increase in competition.                                     The role of dispersal of animals in population ecology has been       investigated from two directions: (1) theoretical analyses using diffusion      models and (2) empirical studies that employ computer simulations (Turchin      1991). Increasingly, these two approaches have been compared or integrated      (Kareiva and Shigesada 1983, Bovet and Benhamou 1988, Turchin 1991, Byers       1999, 2000, Hill and Hder 1997). The advantage of equations as models is that  they can be easily applied and they provide nearly instantaneous answers. The   disadvantage is that equations are not often intuitive and usually difficult    to verify. Simulations, on the other hand, always take much longer to produce   results and require program software. However, the supposed validity of         equations may be corroborated or falsified by simulations that can be           visualized on computer screen. Often simulations are the only way to begin to   understand natural processes that appear intractable to modelling by            equations.                                                                                The well-known diffusion equation for two dimensions (Pielou 1977,    Okubo 1980, Rudd and Gandour 1985) accurately predicts the density of           organisms at any distance from the release point after a certain time or        number of steps, but only for completely random walks. This type of random      walk would have a uniform random distribution of turns with a maximum angle     of turn (AMT) of 180 right or left. 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 (Fig. 1, Kitching 1971, Kareiva and Shigesada 1983, Bovet and Benhamou     1988, Byers 1991, 1996, 1999, 2000, Hill and Hder 1997).                                                                                                             press [F10] to see figures.                                                                                                                                Here I want to consider whether it is possible to use a correlated random      walk equation to accurately predict the average distance of dispersal of a      population of animals from a release point given any values for (1) the step    size (or average step size), (2) the number of steps, and (3) the distribution  of possible random turning angles at each step. The average distance of         dispersal and variance can be found by simulation to check the validity of any  such equation.                                                                            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 extremely complex 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) uses move lengths, turning angles, and total       number of moves to calculate an expected mean square displacement distance      (MSDD). However, the MSDD is not a meaningful distance since it is much larger  than the mean dispersal distance (MDD), the latter around which the population  would be distributed (McCulloch and Cain 1989, Crist et al. 1992). Taking the   square root of the MSDD only approximates the MDD, and depending on the         parameters used can vary in its inaccuracy by up to 12 percent. The             calculation of an MDD seems difficult as stated by McCulloch and Cain (1989)    who derived an approximate formula which was again very complex to compute for  a limited number of moves. However, a formula by Bovet and Benhamou (1988)      claims to solve the MDD based on the mean vector length of turning angles,      step size, and number of steps.                                                           The first objective was to compare two models of simulated dispersal  with regard to turning angles, (1) a uniform random distribution and (2) a      normal random distribution, and see how the models affect the mean dispersal    distance (MDD) and its variation. The second objective was to find correction   factors that would correctly convert the mean square displacement distance of   Kareiva and Shigesada (1983) to an MDD. Finally, the simulated results were     used to test the formula of Bovet and Benhamou (1988) and show how it becomes   increasingly inaccurate at low angular turning distributions and lower numbers  of steps.                                                                                                                                                                                        METHODS                                                                                                                                Simulation of animal dispersal by correlated random walks                  The algorithms for simulating animal movement in two dimensions follow the   ideas of Skellam (1973) as described for computer (Kitching 1971, Kareiva and   Shigesada 1983, Bovet and Benhamou 1988, Byers 1991, 1996, 1999, 2000).         Modelled animals take steps with random angular deviations right or left from   the former step's direction. The path of an animal is determined by first       calculating the movement vector using polar coordinates from the former         position based on the step size (or distance travelled in one second) and       former direction plus the random angle of turn depending on the chosen angular  distribution. Two types of turning angle models were employed (Fig. 1): (1)     a uniform random distribution of angles within a left or right angle of         maximum turn (AMT), or (2) a normal random distribution of angles with          standard deviation (SD). The random angles from a normal distribution are       chosen proportional to their Gaussian probability (Walker 1985).                          A computer program was used for simulations with input parameters of  (a) dispersal time (or number of steps in some cases), (b) average speed and    step size, (c) distribution range of turning angles (AMT for uniform or SD     for normal distributions), (d) number of animals, and (e) the area length and   width. All animals were released from the origin with initial directions        chosen randomly (0 to 360). All simulations and graphical analyses were done   using QuickBASIC 4.5 (Microsoft Corp.) and PostScript 2.0 (Adobe Systems Inc.)  programming languages unless otherwise noted.                                                                                                                       Comparison of uniform and normal distributions of random turning angles               An equation of Kareiva and Shigesada (1983) calculates the expected   mean square displacement, E(R), when move lengths, turning angles, and total                                                                                  moves are known.  The E(R), here called the MSDD, can be calculated below                                                                                     assuming the random turning angles are about equally distributed right/left     from the previous direction:                                                                                 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        turning angles (in radians) from a specified random distribution [g()]:                                                                                                                                                                           c =    cos  g() d                                          (2)                                                                                                -                                                                       Assuming a uniform distribution of random angles  between -AMT and AMT, the    average cosine value c can be found either by computer iteration or more        accurately with the integral (Mathcad 7, MathSoft Inc.):                                                                                                                  A  cos()       sin(A)                                                    c =       d =                                      (3)                       2A            A                                                          -A                                                                                                                                                       where A = AMT. The value c for a normal distribution with a specified standard  deviation (e.g.  = 10) is found either by computer iteration or by            integration [see equation (6) below]:                                                                                                                                       cos()   -/2     -/2                                             c =       e        =  e                                 (4)                      (2)                                                                    -                                                                                                                                                       Equations 3 and 4 were used to find equivalent values for AMT and SD that       should give similar MDDs in simulations.                                                                                                                                       Calculating mean dispersal distance (MDD)                            The problem is now to use the mean square displacement distance (MSDD) of   Kareiva and Shigesada (1983) from equation (1) using c from (3) or (4) to find  the MDD. A first approximation is to take the square root of the MSDD, but      this may overestimate by up to 12.4% the actual MDD. In order to find           correction factors that could by multiplied by the square root of the MSDD to   obtain an MDD, simulations were performed where the number of steps and the     AMT or SD (0 to 60) were varied. After each simulation run, an MDD was         calculated from the average of 1000 Euclidean distances from the start point    to the final positions. These MDDs were divided by the square root of the MSDD  calculated from the equations above with relevant parameters to yield           correction factors. The correction factors depend on two variables, number of   steps and distribution of turning angles, and thus are fit by surface           equations in three dimensions (Mathcad 7).                                                An equation for the MDD was presented by Bovet and Benhamou (1988):                                                                                            MDD = L (0.79 n (1+r)/(1-r))                            (5)                                                                                              where                                                                                                                                                                 r = exp(-(SD)/2)                                        (6)                                                                                           and SD is as above but expressed in radians, with L and n as above. This        equation was found to report values different from the corrected Kareiva and    Shigesada equation or from simulation results. In order to describe which       angular distributions and numbers of steps gave errors in excess of 1% and 5%,  stepwise calculations from 5 to 1000 steps at SD in 0.06 increments from 0     to 30 at each step were done for the two equation types. Values were recorded  when transitions in error limits occurred.                                                                                                                           Using turning angles and steps of organisms to estimate Mean Dispersal     Distance                                                                           The angular distributions of path segments and their lengths reported in     the literature were used to estimate dispersal distances for various            organisms. In addition, several hundred insect-killing nematodes, Steinernema   carpocapsae (Weiser) (Byers and Poinar 1982), were released from a central      point on 1% ion agar covered with a film of carbon particles (india ink).       After 18 hours of dispersal at 20 C, the tracks were photographed, the slide   scanned (ES-10, Olympus Optical Co., Ltd., Tokyo, Japan), the image printed,    and then coordinates along tracks recorded with a digitizing tablet (D-9000,    ACECAD Inc., Taipei, Taiwan) connected to an IBM-compatible personal computer.  A computer program in BASIC calculated the angles and distances between         coordinates.                                                                                                                                                            Effects of turning angles and step number on variation in dispersal     distances                                                                                 The average dispersal distance from the origin is derived from the    individual distances which have a variation that depends on both the turning    angles and the total number of steps. The variation in distances can be         expressed as the coefficient of variation (CV), which is the standard           deviation (sd) as a percent of the mean (sd is used instead of the SD of turn   angles). Simulations were used to find the relationship between the AMT or SD   and the CV of a population if the animals took either 180 or 3600 steps.        Similarly, the relation between the number of steps and the CV was found for    populations with AMT or SD turning angles of 4 and 20 using logistic and       Gompertz non-linear regressions (STATISTICA, StatSoft Inc., Tulsa, Oklahoma;    formulas shown in figures below).                                                                                                                                        Effects of arbitrary path segmentation on calculating the MDD              In practice, a path of an organism is often segmented into steps based on   constant distance or time and the turning angles measured between successive    moves (as was done above for the nematode tracks). It might be that the choice  of the step size or time unit would affect the estimated MDD. This was tested   by generating the coordinates of a path of 2001 steps of 2 m each and random    turning angles with an SD of 30. The path then was segmented into fewer steps  by connecting every second, third, and so on, coordinates and measuring the     resulting step sizes and angular turns (cosine of radian) by computer. These    two values and the resulting number of steps were used to calculate the MDD     using the equations derived in the results section.                                                                                                                                                                                                                              RESULTS                                                                                                                              Simulation of animal dispersal by correlated random walks                     Simulations using different movement parameters were performed in order     to test the equations for calculating mean dispersal distance (MDD) and mean    square displacement distance (MSDD). The simulation models were validated in    part by watching pixel movements on the computer screen that appeared to mimic  natural dispersal of animals. Two models were used, with either a normal or     uniform distribution of turning angles. These models when given equivalent      parameters could produce similar density distributions of 1000 "bark beetles"   released from a point source and "flying" for 1 hour (Fig. 2). In these         examples, beetles flew at 2 m/s (Byers 1996) and at each step of 2 m they       could turn at random either left or right with an AMT (angle of maximum turn)   of 10 (uniform) or with an SD (standard deviation) of 5.776 (normal). The     SD of the normal and the AMT of the uniform distributions were chosen           appropriately to yield equivalent mean dispersal distances (MDD) as explained   subsequently.                                                                   The MDD for the uniform distribution was predicted at 2103.7 m (using the       correction factor method, explained below) while the simulated MDD was 2054.4   m (sd = 1008.4 m, CV=49.08%). The MDD for the "equivalent" normal distribution  was predicted to be 2130.8 m (correction factor method) or 2115.9 m (Bovet and  Benhamou 1988) while the simulated value was 2137.9 m (sd=1057.4 m, CV=49.46    %). The movement and distribution of "animals" in the two types of models       appear nearly identical, although it is possible for a larger turn with the     normal distribution.                                                                                                                                                Comparison of uniform and normal distributions of random turning angles               The average cosine of turning angles from a uniform distribution of   angles decreases from 1 to 0 with an increase in AMT from 0 to 180 (equation   3, Fig. 3). In the case for a normal distribution, the average cosine also      decreases as the SD (turning angles) is increased to 180, but the curve is     more sigmoid in shape (equation 4, Fig. 3). The question to solve is whether    an organism that turns with a uniform random distribution of angles between     left/right turns of, for example, an AMT of 10 might disperse to an average    distance identical to another type that has a normal random distribution of     turns with perhaps an SD of 5.8. The value of SD () that gives an equivalent  c for a specified AMT is solved by equating equations (3) and (4):                                                                                                    = (-2 ln(sin(A)/A))                                    (7)                                                                                              which results in the curve shown in Fig. 4. However, an algebraic solution for  AMT given SD is not possible except by using iterative calculations such as     with binary successive approximation. This method is employed in the computer   program available on the internet (see end of paper). The relationship for SD   versus AMT (equation 7, Fig. 4) was used to convert the AMT = 10 to an         equivalent SD = 5.776 to obtain the similar density distributions of           simulated bark beetles after 1 hour of flight (Fig. 2).                                                                                                                         Calculating mean dispersal distance (MDD)                                 Simulations using various step sizes, and varying both the number of  steps and the AMT or SD (from 0 to 60) stepwise, were used to calculate the    MDD with the Pythagorean formula. The resulting MDD were compared to square     roots of expected mean square distances based on equations (1), (3) and (4).    The comparisons showed that for a large number of steps and larger turning      angles, the actual dispersal distance is about 0.89 of the square root of the   expected MSDD. This proportion (MDD/MSDD) is called the correction factor      (Fig. 5) since its multiplication by the square root of the expected MSDD       gives the simulated MDD (equivalent to the natural MDD). The step size has no   affect on the correction factor, but does scale the MDD. However, there is a    complex interplay between the AMT or SD (Y) and number of steps (X) on the      correction factor which makes it necessary to describe it as a three-           dimensional surface, especially at angles < 30 and steps < 5000 (Fig. 5). 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 + eXY + fXY + gY + hY + iY + j              (8)                                                                                        where a = -2.191E-13, b = 4.472E-9, c = -3.31E-5, d = -1.803E-11, e = 4.611E-   7, f = -2.095E-9, g = -.01, h = 2.604E-4, i = -2.206E-6, and j = 1.043 for AMT  and a uniform random distribution (Mathcad 7). However, the fit is best over    a limited range, therefore five constraints applied in order are needed: (a)    if AMT < 1 then interpolate Z between Y = 1 and 0 at X, i.e. Z(X,Y) = Z(X,1) + (Z(X,0) - Z(X,1))(1-Y), where Z(X,0) is 1, (b) if AMT > 30 then Z = 0.89,    (c) if AMT > 20 AND steps > 2000 then Z = 0.89, (d) if Z > 1 then Z = 1, and    (e) if Z < 0.89 then Z = 0.89. Equation (8) and constraints also calculate      correction factors for normal distributions of turning angles, where a = -      1.772E-13, b = 3.823E-9, c = -2.786E-5, d = -2.657E-11, e = 5.515E-7, f = -     3.253E-9, g = -0.012, h = 3.759E-4, i = -3.531E-6, and j = 1.028. Careful       inspection of the correction factors from the two models, uniform and normal,   does reveal small differences (Fig. 5).                                              In practice, if the average step size is 2 m (L), 1000 steps are taken     (n), and the AMT is 30 or 0.5236 radians, then c = 0.9549 from equation (3)    and the expected MSDD is 169,752 m from (1) so the mean dispersal distance      (MDD) is:                                                                                                                                                           MDD = Z(E(R)) = 0.89(169752) = 367 m                (9)                                                                                                 where Z = 0.89 from equation (8) and constraints. Five simulations with the     same parameters and 1000 insects each gave a MDD of 364.0 5.3 m (95% C.L.).    This means that instead of using time-consuming and complex simulations to      find the MDD, it can be calculated as accurately by the equations above with    the mean step size, number of steps, and AMT or SD.                                       Equations (5) and (6) given above (Bovet and Benhamou 1988) usually   give nearly the same MDD as the application of the correction factor to the     equations of Kareiva and Shigesada (1983). For example, using the parameters    above and converting AMT=30 to an equivalent SD with equation (7) gives        17.40 or 0.304 radians, their equation predicts an MDD of 370.2 m (compared    to 370.1 m for the correction factor method). Unfortunately, their equation     becomes increasingly incorrect for turning angle distributions at smaller SD,   for example, at 1000 steps the error becomes apparent at an SD less than about  6 degrees (Fig. 6). A large range of SD of turning angles as well as lower      numbers of steps give errors in excess of 1% and 5% (Fig. 7), indicating the    Bovet and Benhamou equation should only be used for x,y-values above the 1%     shaded area (Fig. 7).                                                                                                                                                   Using turning angles and steps of organisms to estimate Mean Dispersal  Distance                                                                                  The twisting paths of organisms can be segmented into small steps of  x,y coordinates and the angles between coordinates measured. The path is then   described as segment lengths with a mean and variance as well as the            distribution and mean of turning angles from the previous direction. In the     simulations here, I have used the SD or standard deviation in the expression    of the turning angles from the previous direction. In addition to this way,     other authors have used different values such as: (1) the mean or variance of   absolute angular deviation (AAD), (2) the mean vector length (r), and (3) the   K value of turning angle concentration (Batschelet 1981, Cain 1985, Casas       1988, Wallin and Ekbom 1988, Turchin et al. 1991, Hill and Hder 1997,          Kindvall et al. 2000). These different values can be converted to SD. The       relationship between the SD of turning angles and an absolute angular           deviation (AAD) was found by increasing the SD in increments and taking some    hundreds of random numbers from the appropriate distribution (Walker 1985) and  determining the mean and sd of the AAD. This relationship is linear: mean of    AAD = -0.31 + 0.8136(SD) or sd of AAD = -0.2 + 0.611(SD). However, this         assumes that the data are normally distributed.                                     The mean vector length (r), a simple expression of path sinuosity, can be   converted to SD by solving equation 6:                                                                                                                                SD = (-2 ln(r))  and  r not equal 0                (10)                                                                                                  The K value can be converted to SD using table C in Batschelet (1981). SD is    expressed in degrees when using the correction factor method or in radians      (e.g. SD/180) for the exact method of Bovet and Benhamou (1988).                        The MDDs of various populations of organisms after 1000 steps were    calculated with equations 5 and 6 using data from the literature, or from       nematode tracks on agar, and converted or averaged when necessary to obtain     a step size and SD of turning angles (Table 1).                                                                                                                 Table 1. Prediction of mean dispersal distance (MDD) after 1000 steps for the   nematode Steinernema carpocapsae and various insects (data from literature)     based on mean step length and SD of turning angles.                                                                                                              Organism (study)   Mean step length (N)    SD of angles    MDD after 1000 steps                                                               (time)a                         nematode:                                                                       Steinernema                                                                     carpocapsae                                                                     on 1% agar                                                                      (original data)     0.3515 mm (1403)          19.66              5.76 cm                                                                         (4.88 h)                                                                                      ant:                                                                            Serrastruma lujae                                                               foraging (A.                                                                    Dejean in Bovet                                                                 and Benhamou 1988)    1.5 cm (664)            46.98              1.03 m                                                                          (67.8 min)                                                                                    ant:                                                                            Tapinoma nigerrimum                                                             foraging (Lpez                                                                 et al. 1997)          2.68 cm (29)            33.7                2.56 m                                                                          (33.3 min)   darkling beetle:                                                                Eleodes extricata                                                               in grass                                                                        (Crist et al. 1992)    4 cm                   53.75 b              2.42 m                                                                          (83.3 min)  ground beetle:                                                                  Pterostichus                                                                    melanarius                                                                      (Wallin and                                                                     Ekbom, 1988)          1.28 m (86)             44.68 c             92.62 m                                                                        (10.42 days)  butterfly:                                                                      Pieris rapae                                                                    ovipositing (Root                                                               and Kareiva 1984)     2.56 m (327)c           48.06 c            172.44 m                                                                         (5.56 h)     butterfly:                                                                      Euphydryas editha                                                               (Turchin, et                                                                    al. 1991)             7.16 m (140)            56.02 b            415.5 m                                                                                       weevil:                                                                         Hylobius abietis                                                                on sand (Kindvall                                                               et al. 1999)          8.72 cm (78)            23.99 d             11.71 m                                                                          (2.78 h)    hymenopteran                                                                    parasitoid:                                                                     Pnigalio soemius                                                                (Casas 1988)          2 mm (55)               41.43 d             15.59 cm      a based on average velocity.                                                    b converted from mean vector length (or mean cosine) with equation 11.          c generated from histograms of turning angles or step sizes using random          variates                                                                      d converted from absolute angular deviation (AAD), SD = (AAD+0.31)/0.8136.                                                                                      Both the equation of Bovet and Benhamou (1988) and the transformed MSDD         equation (9) gave essentially the same results (Table 1). However, as           mentioned earlier, the equation of Bovet and Benhamou (1988) is incorrect at    low turning angles. For example, if organisms take 1000 steps of 2 m each with  a 2 SD of turning angles, then they predict an MDD of 3220 m (2000 m is the    theoretical maximum). Equation 9, the corrected square root MSDD predicts an    MDD of 1784 m while simulation gave 1796 m (N=1000).                                The expected number of steps (n) required to reach a certain MDD based on   turning angles (r) and step sizes (L) is derived here from equation 5:                                                                                                  n = 1.265823 MDD/(L(-1/(r-1) - r/(r-1))             (11)                                                                                              Also, the expected step size needed to reach a certain MDD based on the         turning angle distribution and number of steps can be derived:                                                                                                          L = 1.125088 MDD/(n(1+r)/(1-r))                      (12)                                                                                              The expected mean vector length of turning angles necessary to reach an MDD     based on the step size and number of steps is determined by:                                                                                                                -7.9E39 n L + 1E41 MDD                                                    r =                           (13)                          7.9E39 n L + 1E41 MDD                                                                                                                             and use equation 10 to convert r to SD. Equations 11-13, derived from Bovet     and Benhamou's equation, must be used with caution since they will be           inaccurate at low numbers of steps or low angles of turn (as seen in Fig. 7).   In this regard, the equation Y = 2.29 + 5836.3/X, where Y is the SD and X is    the number of steps, can be used to find SD values below which an error >1%     will result (R = 0.95).                                                                                                                                               Effects of turning angles and step number on variation in dispersal      distances                                                                                 The longer animals disperse away from a source and the larger their   turning angles, the more variation is expected in their spatial distribution    with time - but what are the precise relationships? The coefficient of          variation (CV) of the population's spatial distribution is the standard         deviation of dispersal distance as a percentage of the mean distance. After     a certain number of steps the CV is near 0% if the animals proceed nearly       straight (low AMT or SD) and increases as the variation in possible angular     turns increases (Fig. 8). In all cases, the relationships are logistic and      increase toward the asymptote more rapidly at higher numbers of steps (Fig.     8). The maximum CV at the asymptote is probably the same for all combinations   of steps and turning angles regardless of whether from a normal or uniform      distribution, and is about 51% (Fig. 8).                                                  If the number of steps is varied in increments instead, and turning   angles are taken at random from normal and uniform distributions, then the CV   begins near 0% and eventually reaches the same asymptote of 51% regardless of   the turning angles (Fig. 9). The "slope" for smaller angle AMT and SD turning   distributions is less since the population, dispersing initially in an          expanding ring, takes longer to become spatially randomized due to the          relatively straight paths (Fig. 9). Surprisingly, the relationships are not     sigmoid (or logistic) but can be fit well by non-linear Gompertz regression     (formula in Fig. 9).                                                                                                                                                    Effects of arbitrary path segmentation on calculating the MDD                  The segmentation of a 2001 step path into larger and larger segments     consisting of multiples of the steps from 1 to 200 resulted in increasing       average step size and average cosine of turning angles (Table 2).                                                                                               Table 2. Prediction of the mean dispersal distance (MDD) based on different     segmentations of a 2001 step walk (2 m steps of 30 SD of turns) resulting in   different numbers of steps, mean step sizes, and mean cosine of turning         angles. The segmentations, after the first step, were done by connecting every  coordinate, or every second or more as indicated (divisor) to obtain the        number of steps.                                                                   Steps (divisor)   Mean step size  sd     Mean cosine  sd          MDD                                                                                         2000 (1)            2.000  0             0.8710  0.1737           303         1000 (2)            3.856  0.203         0.8200  0.2385           344          500 (4)            7.328  0.654         0.6938  0.3854           342          333 (6)           10.488  1.278         0.5710  0.4902           325          250 (8)           13.456  2.165         0.4589  0.5519           310          200 (10)          16.067  3.061         0.3656  0.5981           296          100 (20)          26.242  8.159         0.1828  0.7016           280           40 (50)          45.38  20.17          0.1457  0.6664           294           20 (100)         71.24  32.20          0.0973  0.6852           311           10 (200)         98.47  63.15          0.0641  0.7117           294                                                                                                   However, the increasing step size, decreasing numbers of steps, and      increasing average cosine of angles covaried so that the MDD estimate was       largely unaffected (Table 2). Further increases in the segmentation size        finally did increase the variation in MDD estimates.                                                                                                                                                                                                                            DISCUSSION                                                                                                                            Animal dispersal is often difficult to observe for any extended period    of time. For example, flying bark beetles can not be observed for more than     a few meters, although marked beetles can be recaptured in pheromone-baited     traps at various distances up to several km from a release site (Byers 1999).   For this reason, simulation is used to understand the effects of component      processes and leads to hypotheses that can be tested both by simulation and     in the field. Simulation of dispersal is considered here as virtual reality.    This means that if organisms behave according to several assumptions, then the  resulting simulated displacements are how real populations with the same        movement parameters would perform. Thus, the model assumptions are crucial to   the proper testing of hypotheses. In this regard, the graphical representation  of organisms moving about on the computer screen is important to the intuitive  analysis of the appropriateness of assumptions.                                           For simulation of animal dispersal and the correlated random walk     equations, the assumptions are that the habitat must be rather homogeneous and  that the organisms have similar and consistent behavior with time. This         implies that the distributions of step lengths and turning angles do not        change during the course of dispersal. Furthermore, consistent behavior means   that the animals are not constrained by a home-range or the need to return to   a nest. Examples of homogeneous habitats would be sand (beaches, deserts),      grass (prairies) or agricultural crops for walking animals. Habitats also       might appear homogeneous to surface-swimming or flying organisms (many insects  fly relatively unobstructed within a few meters of the ground). Even obstacles  in a uniform to random pattern at a larger scale could be considered            homogeneous. However, the obstacles would tend to reduce the mean dispersal     distance (MDD) in proportion to their density and size. For example, simulated  bark beetles flying under the forest canopy and avoiding tree trunks (at the    size and density of 70-year old Norway spruce, Picea abies) flew 5% less far    from the source over an hour than when unhindered (Byers 2000). This was        because avoidance of tree trunks required an occasional increase in the         beetle's turning angle about once every 67 m of flight (Byers 1996).                      The angle of maximum turn (AMT) has been used in simulating animal    movement because it is the most easy to implement (Byers 1991, 1993, 1996).     Other models have used                                                          the variance or SD of the normal distribution of turning angles since these     presumably better reflect movement of organisms (Cain 1985, Bovet and Benhamou  1988, Crist et al. 1992). However, the two models appear to give similar        results when using equivalent values of AMT and SD (Figs. 2 and 4). Byers       (1991, 1993, 1996) found little difference between simulations using either     a 5 or 30 AMT in causing similar proportions of a bark beetle population to   find mates while walking, or susceptible host trees while flying, due to        boundary effects. Nonetheless, increasing the AMT or SD significantly reduces   the search area of an individual. The relationship between the AMT and SD was   established (Fig. 4 and equation 7) to find values that produced dispersal      patterns yielding equivalent MDDs in both models (Fig. 2). However, this does   not mean the models are identical since the normal distribution allows turns    of any angle, while the AMT limits the range.                                             This paper develops equations (8 and 9) that utilize the MSDD of      Kareiva and Shigesada (1983) to find a mean dispersal distance (MDD) of a       population moving outward from an origin for any set of movement parameters     that include the turning angle distribution (SD), average step size, and        number of steps. The MDD can also represent the average displacement of         organisms distributed at random or uniformly over an area. The MDDs from the    equations are the same as those obtained by simulation but are more readily     and rapidly calculated. The calculated MDD can serve as a comparison to field   observations of MDD, or in prediction of expected population dispersal given    knowledge of movement parameters.                                                         Equations 1 and 2 (Kareiva and Shigesada 1983) which find the mean    square displacement distance (MSDD) have the advantage that any distribution    of turning angles, including Gaussian, can be used by finding the average       cosine of the turning angles. If the average cosine from a chosen angular       distribution can not be found by integration, then computer iteration methods   are used. The angular distributions are assumed to be symmetrical about 0 or   straight ahead. The MSDD is proportional to the number of steps or time         (Kareiva and Shigesada 1983). However, the MSDD is not intuitively meaningful   since it is considerably larger than the MDD; for example, an MSDD of 122,500   m implies an MDD less than 350 m. The MDD can be found approximately by taking  the square root of MSDD, but the true value of MDD is up to 12.4% less,         depending on the turning angles and number of steps. The MDD can be found       precisely by multiplying correction factors (Fig. 5) from an appropriate 3D     surface equation by the square root of MSDD (equations 1, 4, 8, and 9).                   A formula that calculates the MDD directly was presented by Bovet and Benhamou (1988) as represented in equations 5 and 6. These equations are less   well known since their paper was about calculating the "sinuosity" of paths     and they did not draw attention to the MDD prediction. Also, there appear to    be some limitations for their equations, first the turning angle distribution,  represented by r (the mean vector length) assumes a normal distribution. Even   if another distribution could be used, r (equal to the mean cosine of turning   angles) would have to be calculated. Second, the equations overestimate the     MDD significantly at high r (or low turning angles). For example, a 2 SD, 2    m average step, and 1000 steps gives an MDD of 3220 m, which is much more than  the maximum possible if animals go straight for 1000, 2 m-steps to disperse     2000 m out from the release point. The discrepancy between actual MDD (from     simulation) and the MDD from Bovet and Benhamou (1988) is seen clearly in Fig.  6. Their MDD is related as a negative power of SD (R = 1) which becomes        increasingly incorrect for decreasing turn angle variation. In fact, the error  depends on both the SD of turning angles and the number of steps (Fig. 7). For  example, at 100 steps a greater than 5% error begins for an SD below 25. This  problem is avoided by the surface equation correction of the square root of     MSDD.                                                                                     The coefficient of variation (CV) in the distance of dispersing       individuals after various numbers of steps from the origin was related          logistically to increasing values of AMT or SD turning angle distributions      (Fig. 8) or as a Gompertz function to numbers of steps (Fig. 9). In both these  figures, the CV appears asymptotic at about 51-52%. Bovet and Benhamou (1988)   give the sd of the MDD as 0.52(MDD) which is the same as a CV of 52%. However,  Figs. 8 and 9 show that the CV is not constant but ranges from 0 to 0.52 (or    52%), being lower at lower turning angles and lower numbers of steps,           precisely when the Bovet and Benhamou equation is inaccurate.                             In practice, a path of an organism is often segmented into steps base on constant distance or time, and the turning angles measured between           successive moves. It would seem that the choice of the step size or time unit   would affect the estimated MDD.  In addition, a calculation of the MDD based    on analysis of initial movement paths might be in error due to heterogenous     environments or changes in behavior during dispersal (Johnson et al. 1992,      Crist et al. 1992, Wiens et al. 1993). However, if there are no changes in the  turning rate and step sizes, other than random variations as in the simulation  models, then the value calculated for the MDD is surprisingly stable over a     large range of segmentations of the path (Table 2). This implies that the       precise segmentation of natural paths in homogeneous environments is not        critical until very large steps are measured that obscure the path.                       The software for simulation, MDD prediction, and analysis (MEAN-      DD.ZIP) is available on the Internet at                                         http://www.vsv.slu.se/johnb/software.htm.                                                                                                                                                     ACKNOWLEDGMENTS                                                                                                                   The work was supported by a grant from the Swedish Agricultural and Forest      Research Council (SJFR). Several reviewers, editors (F. Adler), and B.          Holmqvist were helpful in revision of the manuscript. F. Adler provided the     integrals used in equations 3, 4 and 7 to replace iterative methods used in     earlier drafts.                                                                                                                                                                              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