Copyright Ó 2007 by the Genetics Society of America DOI: 10.1534/genetics.107.077206

The Genetic Structure of Clonal Organisms Generated by Exponentially Bounded and Fat-Tailed Dispersal

Luzie U. Wingen,*,1 James K. M. Brown* and Michael W. Shaw† *Department of Disease and Stress Biology, John Innes Centre, Norwich NR4 7UH, United Kingdom and †School of Biological Sciences, University of Reading, Reading RG7 6AS, United Kingdom Manuscript received June 12, 2007 Accepted for publication July 10, 2007

ABSTRACT Long-distance dispersal (LDD) plays an important role in many population processes like colonization, range expansion, and epidemics. LDD of small particles like fungal is often a result of turbulent wind dispersal and is best described by functions with power-law behavior in the tails (‘‘fat tailed’’). The influence of fat-tailed LDD on population genetic structure is reported in this article. In computer simulations, the population structure generated by power-law dispersal with exponents in the range of 2 to 1, in distinct contrast to that generated by exponential dispersal, has a fractal structure. As the power- law exponent becomes smaller, the distribution of individual genotypes becomes more self-similar at different scales. Common statistics like GST are not well suited to summarizing differences between the population genetic structures. Instead, fractal and self-similarity statistics demonstrated differences in structure arising from fat-tailed and exponential dispersal. When dispersal is fat tailed, a log–log plot of the Simpson index against distance between subpopulations has an approximately constant gradient over a large range of spatial scales. The fractal dimension D2 is linearly inversely related to the power-law exponent, with a slope of 2. In a large simulation arena, fat-tailed LDD allows colonization of the entire space by all genotypes whereas exponentially bounded dispersal eventually confines all descendants of a single clonal lineage to a relatively small area.

HE importance of long-distance dispersal (LDD) 2003; Davies et al. 2004; Bialozyt et al. 2006). Com- T for the distribution and evolution of organisms puter models of the process explained the patchy has long been recognized and was acknowledged as genetic patterns, observed in modern oak , early as 1859 by Darwin (Darwin 1859). Until recently, by LDD founder events (Davies et al. 2004; Bialozyt however, population genetic and evolutionary studies et al. 2006). have concentrated mainly on short-distance dispersal Dispersal by wind is a major mechanism of LDD. that is easier to measure but nonetheless has important Fungal spores causing severe agricultural diseases are consequences for local . Owing to dispersed in rare events over hundreds or even thou- better methodology for assessing LDD and increased sands of kilometers (Brown and Hovmøller 2002). awareness of its importance, interest in it has risen Transport through the air is profoundly affected by again in the last 15 years (Nathan et al. 2003). turbulence over a wide range of spatial scales (Aylor LDD plays an important role in colonization of 2003; Nathan et al. 2005). islands (Cain et al. 2000 and references therein; There is an ongoing debate about what kind of Gittenberger et al. 2006), in range expansion (Cain dispersal kernel, the function that describes the prob- et al. 1998; Clark 1998), and in the rates of population ability that a will be deposited at a given expansion and spread of epidemics (Shaw 1995; Kot distance, is best suited to describe LDD. Several studies et al. 1996). However, LDD is rare and difficult to analyze have modeled LDD of or by a mixture of in detail in the field. Modeling of LDD has thus become two exponential dispersal distributions, one with a short an instrument to investigate its importance in evolu- median dispersal distance and one with a very long one tionary and ecological processes. One example is the (e.g.,Nichols and Hewitt 1994; Bialozyt et al. 2006). range expansion of oak trees to the north during the Inferring true dispersal curves from small, wind- postglacial recolonization of Europe. LDD plays an dispersed biological objects like spores or pollen is dif- important role in explaining the speed of the expansion ficult. Measured dispersal distributions are frequently (Le Corre et al. 1997; Austerlitz and Garnier-Ge´re´ leptokurtic or fat tailed, meaning that they have greater density in their shoulders and tails than a Gaussian distribution with the same variance (references in Kot 1Corresponding author: Department of Disease and Stress Biology, John Innes Centre, Colney, Norwich NR4 7UH, United Kingdom. et al. 1996). Many pollen dispersal data are best fitted E-mail: [email protected] with an inverse power-law function (Bullock and Clarke

Genetics 177: 435–448 (September 2007) 436 L. U. Wingen, J. K. M. Brown and M. W. Shaw

2000; Austerlitz et al. 2004; Devaux et al. 2005; Klein genetic structure of populations with a roughly stable pop- et al. 2006; Shaw et al. 2006). For fungal spores, the ulation size. This scenario corresponds to a natural population question of the best-fitting dispersal function is hindered that is restricted in growth, e.g., by limited space or limited nutrients. The simulation arena used was very large and so the by the necessity of large experimental plots free from too occupying population should be very large as well. As com- much background infection. A recent study addressing puter memory was inevitably limited, only sample lineages the above problems showed that dispersal of the wheat were simulated. A large population was assumed to be present stripe or yellow rust (Puccinia striiformis) fitted a in the background of these individuals, competing for re- power-law model well if enough sufficiently distant sources and thus limiting the expansion of the simulated ackett undt individuals. A birth process with Poisson-distributed progeny traps were used (S and M 2005). Moreover, number of one individual per parent and a fixed death age of although several physical processes underlie wind dis- one generation were used to simulate these sample lineages in persal, theoretical arguments strongly propose that LDD a fluctuating, nonexpanding population. of small objects can be modeled by a single function that General model settings: Simulations were initiated with 30,000 will have inverse power-law behavior in the tails (Shaw individuals, initially all of different genotypes, each repre- ot tockmarr ylor sented by a 32-bit number, in effect 32 biallelic loci. The initial 1995; K et al. 1996; S 2002; A 2003; individuals were placed randomly in a simulation arena of Shaw et al. 2006). Uplift is the most important factor for 108 3 108 square units in size, with 1 unit corresponding to the heavier but many factors are equally impor- closest distance allowed between two individuals. Individuals tant for smaller objects such as spores (Nathan et al. gave birth to offspring at the beginning of each time step, 2005). A simplification of the dispersal process which were immediately dispersed according to the chosen dispersal function. The genotype of a new individual was either by a single negative power-law dispersal function is a the same as that of the parent or mutated by conversion of one useful basis for theoretical modeling. random bit of the 32-bit genotype. Mutation took place at random Recent simulation studies have addressed either the with a frequency set by the mutation rate m ¼ 104. Individuals influence of LDD, modeled as a dual exponential func- died after 1 time step and thus had the chance to produce tion, on population genetic structure (Bialozyt et al. progeny only once in their lifetime. Offspring were not placed outside of the simulation arena or closer to other offspring than 2006) or the influence of power-law dispersal function the minimal interaction distance. If the simulation generated on spatial distribution of (Cannas et al. 2006). such an event, a new location was calculated until a legitimate one This article investigates the influence of LDD, mod- was found. Individuals possibly adjacent to a given point were eled as a negative power law, on population genetic found quickly using the indexing algorithm in Shaw (1996). The structure of populations in quasi-equilibrium. Inverse simulations were assumed to be a part of a huge population of uniform density. All ‘‘background’’ lineages were assumed to power-law functions with exponents in the range of 1 , disperse in the same way as the simulated lineages and thus b # 2 were used as dispersal functions to simulate fat- result in a similar population genetic structure. The main tailed dispersal. The resulting population structures simulations were aimed to run for 50,000 generations. were compared to those generated by a negative expo- Simulations that were used mainly to calculate the fractal nential dispersal function or a global dispersal (uniform dimensions were run for 10,000 generations only. Dispersal functions: The dispersal of the spores was modeled random) function. Widely used statistics from by an inverse power-law probability density function with the and were applied to the resulting spore concentration t(r j u) at distance r from the source along populations. Some of them were more suitable than a given bearing u given by others to describe the population structures and to dis- 1 tinguish the outcomes of different modes of dispersal. tðr j uÞ } ð1Þ 1 1 r b The simulations reported here used an arena several orders of magnitude larger than the median dispersal with b . 1(Shaw et al. 2006). Of special interest were values distance of the dispersal function. We used a novel sim- of b # 2. Theoretical and experimental results suggest this is the relevant range in wind dispersal of small particles like ulation strategy that allowed us to investigate a range of fungal spores and pollen (Mccartney 1987; Mccartney and spatial scales covering nine orders of magnitude. This Bainbridge 1987; Ferrandino 1993; Sackett and Mundt usage of a large arena may be of paramount importance 2005; Klein et al. 2006; Shaw et al. 2006). If not stated for a thorough investigation of the consequences of otherwise, simulations used values of 1.2 # b # 4. The main LDD as it is hypothesized that the population structure simulations were performed with values of b ¼ 2.5, 2.0, and 1.5. The case of b ¼ 2 corresponds approximately to the well- of organisms with small wind-dispersed propagules is in- known Cauchy distribution. Quasi-random power-law variates fluenced over scales ranging from centimeters to several were generated as described in Shaw (1996). Two rational hundred or even thousands of kilometers (Pedgrey 1986; function approximations were derived using the algorithm in O’hara and Brown 1998; Brown and Hovmøller 2002). Mathematica 4.0, calculating the median of a power-law distribution with any exponent by numerical integration. For b between 1.2 and 2,

MATERIALS AND METHODS 0:2816583 0:4939602b 1 0:3358465b2 log ðmedianðbÞÞ ¼ : Computer simulations: An individual-based, spatially ex- e 1 1:7971499b 1 0:8078619b2 plicit model of haploid individuals without sex in a continuous ð2Þ was used, developing the models of Shaw (1995, 1996). The simulation aimed at investigating the spatial For b $ 2, Population Structure Due to Different Types of Dispersal 437

26 7 1:7316499 1:1767354b 1 1:1298891b2 side length of 2 (6.7 3 10 ) units, randomly placed within log ðmedianðbÞÞ ¼ : the simulation arena. The analysis area was divided into e 1 1 2:0186175b 2:2337080b2 smaller and smaller squares by repeated equal divisions into ð3Þ four. The smallest square had a side length of 2 units. Each step For comparison, simulations were also done following the of this subdivision formed one scale of analysis. The units of commonly used dispersal model of an exponential decline of analysis were the squares of subdivision (‘‘boxes’’), which can spore concentration with a characteristic scale of k: also be regarded as subpopulations. If boxes contained one or more individuals they were counted as ‘‘occupied’’ and were tðr j uÞ } er=k : ð4Þ analyzed, and if they had none they were counted as empty. A fractal distribution of individuals is reflected by a re- Simulations with global or uniform random dispersal were lationship of the type performed for further comparison, in which offspring were placed randomly in the arena, with both x and y coordinates 1 D0 N ðsÞ } ð5Þ drawn from a uniform distribution. Simulations were repeated s 20 times for the negative exponential dispersal and 12 times for other dispersal functions. with N(s) the number of boxes needed to cover all individuals Parameter settings: Distances in our model were chosen to fit at the scale s, the side length of the boxes (Hastings 1993). experimental data, especially on wind dispersal of powdery The exponent D0 is the box-counting dimension. By introduc- mildew (Blumeria graminis) and yellow rust (P. striiformis)of ing a constant k (5) can be linearized to cereals. The minimal interaction distance between individu- 0 als, 1 unit, corresponding to the minimum distance between 1 two distinct rust or mildew lesions on one leaf, was assumed for logðN ðsÞÞ ¼ D0log 1 logðk0Þ: ð6Þ the sake of simplicity to be 10 cm. Other authors have assumed s it of similar magnitude (2 cm was used as the size of a single yellow rust lesion by Lett and Østerga˚rd 2000). The median A practical approach to determine D0 was to cover the arena dispersal distance was set to 30 units for the exponentially of the simulated populations with grids of different box sizes bounded function. The power-law functions were adjusted to and do box counts for the different scales (Halley et al. 2004), approximate the same median value, corresponding to 3 m in counting the number of boxes, Nj(s), needed to cover indi- real-world dimensions. The median dispersal for powdery viduals of a given genotype j and at each scale (Figure 1). This mildew conidiospores is estimated to be 3.1 m for field disease was done for all common genotypes of frequency of at least 10, gradients (half the distance of the exponential model; Fitt partly to reduce the computational demands and partly be- et al. 1987) or 1 m to several meters (modeled from deposition cause rare genotypes will most likely not be detected in a velocities and impaction efficiency, with 50% deposition rate; population study. Moreover, rare genotypes contribute little to Mccartney 1987). Modeled estimates for wheat yellow rust calculation of D0 as their box-occupation pattern is inevitably are 2.8 m–6.5 m (calculated from exponential fits to the three similar at most scales. D0 was then estimated by the negative largest data sets for upwind dispersal in Sackett and Mundt slope of all log(Nj(s)) on log(s) by applying a regression anal- 2005). Estimates by O’hara and Brown (1998) are similar. ysis to the linear region of the graph ( judged by eye). The median dispersal distance setting was thus in concor- The correlation dimension D2 is related to D0 and either dance with field data. The extension of our simulation arena equal to or smaller than it (Grassberger 1983). It detects a of 108 units can then be regarded as a distance of 104 km, e.g., power-law relationship between the concentration of individ- spanning further than the whole of Europe. uals in the boxes and the scale. The concentration is measured Software and hardware: Simulation software was written in by the Simpson index, IS, calculated from the proportionP of N ðsÞ 2 Kylix (Borland Delphi for Linux, version 14.5, Open Edition) individuals that fall in each box, pi,asISðsÞ¼ i¼1 ðpi Þ using the Free Pascal Compiler (version 1.9.8). Simulations (Simpson 1949), with N(s) the number of boxes at that scale. were run on a GNU/Linux platform with four 2.8-GHz The relationship is thus processors. Statistical analysis of populations and graphical output were performed using the free R software suite (version 1 D2 I ðsÞ} : ð7Þ 2.1.0; R Development Core Team 2004). S s Characterization of populations: Statistical characterization of populations was performed at different time steps. Results By introducing a constant k2 (7) can be linearized by analogy are shown for generation 6000 and in some cases for genera- to (6) as tion 40,000. If not stated otherwise, results from later genera- 1 tions did not notably differ from those from generation 6000. logðI ðsÞÞ ¼ D log 1 logðk Þ: ð8Þ Results of repeated runs were summarized as box plots. S 2 s 2 The spatial genetic structure of the simulated lineages was analyzed using graphs of genetic dissimilarity against genetic D2 was determined from the linear region of the graph of alley distance. For this, 20,000 individuals were drawn randomly logðISj ðsÞÞ against log(s) for all common genotypes j (H from the population and grouped into pairs. The geographic et al. 2004). distance and the genetic dissimilarity (1, dissimilar; 0, similar) The conditional incidence IC is a measure of self-similarity, haw haw of each pair were determined. The distance range was divided effectively the derivative of D0 (S 1995; S et al. 2006). into 100 intervals and the percentage of dissimilar pairs in IC is calculated as the ratio of two incidences at neighboring each interval was plotted against the logarithm of the mid- scales, point of the distance interval. Spatial statistics: Several statistics were calculated at different N ðsÞ=Ns I ðsÞ IC ¼ ¼ ; ð9Þ spatial scales. These were fractal dimensions D0 and D2, con- N ðs 1Þ=Ns1 I ðs 1Þ ditional incidence IC, conditional Simpson incidence IC2, and a measure for population subdivision GST. Analysis was done with I(s) the proportion of occupied boxes N(s) among total for a square area a little smaller than the simulation arena with boxes Ns at scale s. IC is constant for a truly self-similar 438 L. U. Wingen, J. K. M. Brown and M. W. Shaw

RESULTS Quasi-equilibrium: The simulation assumed a large background population of which only 30,000 lineages were followed. The simulated individuals were restricted in growth, as the total population size was assumed to be regulated in a density-dependent manner. Natural populations generally fluctuate in size, modeled here simplistically by a fixed mean birth rate and death after 1 time step. The numbers of individuals fluctuated during the course of the simulations and in a considerable number of cases all lineages died out. This was expected because all branching processes with one expected de- scendant per individual are certain to die out (Feller 1968). For further analysis simulations were chosen that ran for 50,000 time steps with individual numbers be- tween 15,000 and 85,000 to make results from different dispersal functions and time steps comparable. The mean population size varied slightly between sets of these ‘‘long runs’’ with different dispersal functions: 39,400 6 13,200 for power-law (PL) r1.5, 42,800 6 Figure 1.—Example of the box-counting procedure. (A) 12,900 for PL r2.0, 40,700 6 12,800 for PL r2.5, 42,400 6 The big squares represent the simulation arena with grids 13,500 for exponential, and 40,900 6 15,000 for global of different box sizes superimposed. The arena is populated dispersal (mean 6 standard deviation); the differences by individuals of two genotypes (solid and shaded dots). A box count for each genotype and each box-side length, or scale, is of the means were statistically significant. These num- performed. The box count is the number of boxes containing bers were products of large numbers of calls of the one or more individuals of a given genotype. In the above ex- Poisson generator. Since the Poisson generator and the ample, all boxes that contribute to the box counts for the dispersal module were tested independently and had no ‘‘solid’’ individuals are shaded. Scales and counts for those in- shared variables or parameters, variations in population dividuals are listed under the arenas and used for the plot in B. (B) The results from the box counting are plotted as log size must have been due to slight remaining long-range (box number) against log(scale). If the resulting graph, or correlations in the underlying random-number gener- parts of it, shows a linear region with a measurable gradient, ator, coupled with differences in the number of calls the negative of this gradient is the box-counting dimension D0 made to it. The generator, Knuth’s subtractive method of that genotype over the extent of the linear region. D0 is an to generate uniform random deviates (Press et al. 1996), indication of a fractal distribution of that genotype. is widely recommended and has passed all the usual tests. However, no generator is truly random and there distribution. For constant IC and each scale being half the will always be some remaining signatures of the un- length of the next larger one (which results in a subdivision into quarters), the relation between D and I is thus derlying deterministic nature of the algorithm. Since 0 C the imbalance between large and small random Poisson 5 D0 ¼ 2 1 log2ðIC Þð10Þ deviates was of the order of only 1 in 10 , we believe the general patterns in the data were robust. This belief was (Shaw 1995). supported by the observation that in general results The conditional Simpson incidence IC2 was calculated in a similar way but using the Simpson index divided by the total from simulations that went extinct were very similar to number of boxes as a measure of incidence at a scale: the long-run results. Exceptions were runs where the P population size dropped dramatically under 10,000 and N ðsÞ 2 i pi =Ns I ðsÞ=N recovered, causing a dramatic drop in genotype number I ¼ P ¼ S s : ð11Þ C2 N ðs1Þ to near uniformity. p2 =N ISðs 1Þ=Ns1 i i s1 Initially, all 30,000 individuals were of different geno-

IC and IC2 were calculated separately for each common types. Most of the genotypes died out within the first genotype at each scale. 4000 generations in a process similar to exponential A common way of summarizing population genetic struc- decay as is expected for a Poisson birth process with ture over a wide range of spatial scales is to use F-statistics constant death rate. Without mutation, all genotypes (Wright 1951), especially the coefficient of inbreeding, FST. GST, the coefficient of gene differentiation, is used as an analog but one died out (data not shown). With mutation, the ei of FST (N 1973) and often used for haploid organisms and number of genotypes stabilized after 4000 generations was applied here by regarding the occupied boxes as sub- at a mean value of 62 6 20 (Figure 2), indicating an populations at different scales. The GST values estimated from the simulated lineages are proportional to those that would be equilibrium between mutation and drift. In addition to found by unbiased sampling from the whole populations. similar genotype numbers, the distribution of genotype Population Structure Due to Different Types of Dispersal 439

row, different time steps). It was hard to define a cluster unambiguously with PL r1.5 dispersal, so it was not pos- sible to analyze this situation quantitatively. However, the number of ‘‘clusters’’ observed was determined by two processes. The first process was the rate at which individuals within the cluster radius became extinct and the second one the rate at which new lineages reached a distance larger than the cluster radius and were able to multiply to form a new cluster. The second rate de- pended strongly on the probability of LDD and was much greater for smaller b. The difference between dis- persal functions was greater the larger the cluster radius being considered. Cluster numbers were highest with PL r1.5 dispersal of all reasonable values for the cluster Figure 2.—Decrease of genotype number with time. The radius. logarithm of genotype number of inverse power-law ½PL Spatial genetic structure: Population genetic theory r1.5 (black), PL r2.0 (dark shading), and PL r2.5 (medium states that limited dispersal results in individuals located shading) and exponentially (light shading) and globally close to each other being genetically more similar than (very light shading) dispersed lineages is plotted against time. individuals further apart (Male´cot 1969). However, Box plots correspond to the statistics of 20 (exponential) or 12 replicates (all others). many of the presented simulations were using fat-tailed LDD. The relationship of genetic dissimilarity and geo- graphic distance at quasi-equilibrium was thus of interest frequencies was also similar between time steps and and investigated as the percentage of genetic dissimilar dispersal functions. pairs found at different distances (Figure 4). Individuals Spatial patterns: The different dispersal functions were initially genetically dissimilar and were distributed generated different spatial population patterns (Figure 3). randomly over the whole arena with wide spacing be- Differences became more prominent over time. Initially tween them. Over time, genetically similar individuals all individuals were randomly distributed. In the case appeared as particular lineages grew by chance. In line- of global dispersal, that distribution persisted (not ages dispersing with an exponential dispersal function shown). For the other dispersal functions, the com- these individuals were found near one another. In these bination of dispersal and the Poisson birth process led lineages genetically dissimilar individuals appeared near to an aggregation of individuals in clusters. Individuals one another as well, as mutants arose within lineages of the same genotype tended to cluster together because of spatially adjacent individuals. The graphs of genetic the dispersal patterns were inversely related to distance. dissimilarity reflect the same clustering tendencies of This clustering process was strongest for the exponen- genotypes as discussed above for the whole population tial simulations, in which tight clusters arose over time, (Figure 3). The strong clustering tendency of the of which most died out (Figure 3, bottom row) and only exponential dispersal resulted in the dissimilarity graph one cluster survived in the analyzed simulations after having a short extension on the log(geographic dis- 40,000 generations. The exponential dispersal pattern tance) axis (Figure 4). The extension of the graphs was did not enable new clusters to form as the maximum longer for the power-law dispersal functions, reaching distance over which dispersal was at all probable was in higher geographic distances. This trend increased with the same range as the spatial size of an individual cluster. decreasing b. The extremely fat-tailed distribution PL The structures that arose from PL r2.5 and PL r2.0 r1.5 dispersed sufficient individuals to distances compa- dispersal also led to a few distinct clusters, but the pro- rable to the arena size to cause pairs of similar and cess of reducing the number of clusters was slower for dissimilar individuals to be found at far distances at all PL r2.5 and even slower for the PL r2.0 dispersal. The time steps. With all dispersal functions but the global resulting clusters were less dense; the PL r2.5 dispersed dispersal an increase of genetic dissimilarity with geo- lineages were clustered more densely than the PL r2.0 graphic distance was found (Figure 4). dispersed ones (Figure 3, middle rows). LDD was suf- Fractal dimensions D0 and D2: Many natural objects ficiently frequent with PL r1.5 dispersal to counteract have relevant features on a variety of different scales. If the concentrating effect of the Poisson birth process their properties are similar across a wide range of scales, (Figure 3, top row). After an initial loss of single indi- they are self-similar over that range and have a fractal viduals and small clusters within the first 100 gener- structure. Natural objects are often not ideal fractals, ations, the lineages remained scattered in clusters of a but are sufficiently fractal that the tools of fractal geo- wide range of sizes over the entire arena. The number metry, including characterization by noninteger dimen- and sizes of clusters appeared stable in that state over sions, can be used to describe them. Fractal measures, the whole duration of the simulations (Figure 3, top such as the box-counting dimension, have been used for 440 L. U. Wingen, J. K. M. Brown and M. W. Shaw

Figure 3.—Examples of spatial structures of simulated lineages. Top three rows, inverse power-law dispersal (PL r1.5,PLr2.0, and PL r2.5, respectively); bottom row, exponential dispersal. From left to right: generations 6000, 40,000, and a close-up of 40,000 (magnifications of factor 10 for PL r1.5,100forPLr2.0, 1000 for PL r2.5, and 10,000 for exponential). Numbers on the arena scale (left and bottom) are multiples of 107. Individuals are shown as dots; different colors indicate different genotypes in that simulation. describing the spatial distribution of species (e.g.,Kunin separately to individuals of the same genotype in the 1998) and power laws, which are closely connected with simulated samples at different scales and summed up in alley fractals, are a major descriptive tool in ecology (H the graph of log(D0) against log(s) (Figure 5). For the et al. 2004). exponentially dispersed lineages the maximum scale, The most basic fractal dimension is the box-counting the point where the regression line meets the log(s) dimension (Equation 6). This box count was applied axis, was a little above 104 and much smaller than those Population Structure Due to Different Types of Dispersal 441

Figure 4.—Percentages of genetically dissimilar pairs in intervals of geographic distance. Percentages calculated for typical simulations of PL r1.5,PLr2.0,PLr2.5, and exponentially dispersed lineages after 6000 (top row) and 40,000 (bottom row) gen- 1 2 erations are shown. Geographic distance scale is given as log10 (1 ¼ 10 ,2¼ 10 ,...). for the inverse power-law dispersed lineages. This simply tude (Figure 5). The distribution of genotypes therefore reflects the limited spatial extent of any genotype with shows a very strongly fractal pattern, over most of the exponential dispersal. In PL r2.5 dispersed lineages, the observed area. In PL r2.0 dispersed lineages, the fractal 5 2.0 6 2 maximum spatial scale was 10 ,inPLr it was 10 , pattern is a little weaker, D2 ¼ 0.777 6 0.006 (R ¼ 0.92), while in PL r1.5 dispersed lineages, the maximum scale and the linear region stretched over four orders of mag- was larger than the simulation arena (Figure 5). This nitude, which, though less than for PL r1.5, is still quite was confirmed with a limited number of simulations in a large by comparison with reports in natural ecological larger arena where still the line did not meet the log(s) systems (Halley et al. 2004). Results for the PL r2.5 axis (data not shown). dispersed lineages indicate an even weaker fractal 2 A regression analysis for the box-counting graphs of structure with D2 ¼ 1.008 6 0.07 (R ¼ 0.94). The linear the exponentially dispersed lineages produced a co- region of the log(Simpson index) against the log(s) plot efficient of determination of R 2 ¼ 0.87. The linear for the exponentially dispersed lineages stretched over region stretched over a spatial range of about two orders only two orders of magnitude and so did not indicate a of magnitude only, which is too short to merit de- fractal genotype distribution. As expected, globally dis- scription as a truly fractal structure (Halley et al. 2004). persed lineages did not show patterns of any kind as D0 By contrast, broader linear regions, spanning three or- and D2 were both zero. ders of magnitude and three to four orders of magni- The fractal dimensions were determined for five tude, were found in PL r2.5 and PL r2.0 dispersed further power-law dispersal functions (1.2 # b # 4) at lineages, indicating fractal spatial patterns over several generation 6000. Linear relationships between the frac- orders of magnitude with fractal dimensions of D0 ¼ tal dimensions and the inverse of the power-law expo- 2 2 0.994 6 0.008 (R ¼ 0.95) and D0 ¼ 0.760 6 0.006 (R ¼ nent were found (Figure 6). The regression equations of 0.93), respectively. As ranges were limited these can be fractal dimension on 1/b were D0 ¼2:07ð1=bÞ 1 1:82 2 2 classified only as a quasi-fractal spatial pattern. The re- (R ¼ 0.99) and D2 ¼1:94ð1=bÞ 1 1:81 (R ¼ 0.99). 1.5 gression analysis of the box-counting curve of PL r Conditional incidence (IC) and conditional Simpson in- 2 dispersed lineages found a poorer fit with R ¼ 0.85 but cidence (IC2): The self-similarity measures were applied to the linear region stretched over a much wider scale, six individuals of the same genotypes of the simulated line- 1.5 orders of magnitude. Hence PL r populations had a ages. Stable patterns of IC and IC2, respectively, against more extended fractal structure than PL r2.0 popula- log(s) were found, which were characteristic of the tions but the pattern seemed to be less homogeneous. dispersal function (Figure 7). As found by Shaw (1995) The fractal dimension was D0 ¼ 0.439 6 0.004 (Figure 5). the IC values describing exponentially dispersed line- However, the box-counting dimension does not con- ages were much larger than those in PL r2.0 dispersed sider the number of individuals in each box, which can lineages. vary enormously. To capture this aspect of population The IC graphs of the exponentially dispersed lineages structure, the correlation dimension D2 was applied were characterized by a sharp rise from 0.25, which is the impson (Equation 8). This uses the Simpson index (S minimum possible IC value for a grid based on a division 1949) that quantifies the evenness of occurrence be- of each square into four smaller squares, to a maximum tween subpopulations (boxes). value around 0.75 over one to two orders of magnitude In PL r1.5 dispersed lineages, a fractal dimension of followed by a little less steep decline (Figure 7). The fat- 2 D2 ¼ 0.534 6 0.004 was calculated (R ¼ 0.90). The tailed power-law dispersed lineages showed flatter IC linear region stretched over nearly six orders of magni- graphs, with a rise extended over more scales, around 442 L. U. Wingen, J. K. M. Brown and M. W. Shaw

Figure 5.—Estimates of fractal dimensions. Box-counting dimen- sion D0 (left column) and correla- tion dimension D2 (right column) of clonal descendants of inverse power-law (PL r1.5,PLr2.0, and PL r2.5) and exponentially and globally dispersed lineages at gen- eration 6000. D0: log–log plot of the number of ‘‘occupied’’ boxes against scale. D2: log–log plot of the Simpson index of occupied boxes against scale. Box plots cor- respond to the statistics of 20 (ex- ponential) or 12 replicates (all others). Population Structure Due to Different Types of Dispersal 443

contained only one main genotype. In the PL r2.0 and 1.5 PL r dispersed populations, GST values stayed at very high values over many scales, approaching zero only at very large scales. The population structure seemed quite homogeneous over a wide range of scales. For expo- nentially dispersed populations the GST values fell much earlier and reached a plateau around scale 104, the same maximum scale discovered before and again simply showing that the sample was concentrated in a few areas with distinct genotypes. The exact value of GST and the length of the plateau depends on the number of geno- types present in the population and the location of 2.5 population clusters. GST graphs from the PL r dis- igure persed lineages showed an intermediate behavior be- F 6.—Relationship between fractal dimensions and 1.5 1/b (b: power-law exponent). Data points were plotted as solid tween PL r and exponential GST graph trends. In the circles (D0) or open squares (D2) and regression lines were case of global dispersal, GST values were either 1 for fitted to the two fractal dimension types (solid line, D0; dashed small and medium scales where only single individuals line, D2). were present per box or 0 at large scales, revealing per- fect mixing at those scales. A transition phase with 2.5 two and three orders of magnitude for PL r and PL intermediate GST values is very short. r2.0 dispersal functions, respectively, and more or less covering all observed scales for PL r1.5 dispersal. This confirms that the spatial genetic structure generated by DISCUSSION fat-tailed inverse power-law dispersal had a much larger extent than that produced by exponential dispersal. Several experiments and theoretical arguments sug- 4.0 However, IC graphs for PL r dispersed lineages looked gest that LDD dispersal of small propagules by wind is identical to those for exponential dispersed lineages best described by inverse power-law dispersal functions. and thus had the same spatial extent (graphs not shown). Here, the importance of such LDD dispersal events on IC graphs for globally dispersed lineages were, as ex- population genetic structure in quasi-equilibrium is in- pected, uninformative. vestigated by computer simulation of lineages of clonal For the very fat-tailed inverse power-law dispersal haploid organisms, which were dispersed following 1.5 function PL r , the rise of the IC against the log(s) inverse power-law dispersal functions with an exponent curve was very shallow over about two to three orders of 1 , b # 4. Our most important finding is a fractal dis- magnitude. A flat IC curve signifies a self-similar struc- tribution of genotypes in the resulting populations for 1.5 ture, so the underlying genetic structure of PL r small b, which is best detected by the D2 fractal dimen- lineages was close to self-similar over a limited but sub- sion, applying the Simpson index to subpopulations stantial range. at different scales (Figure 5). In contrast, if dispersal The results from the IC2 analysis were in most respects is modeled by an exponentially bounded function, no similar to those from the IC analysis. IC2 against log(s) fractal population genetic structure is produced. The D2 graphs had the same modes as the IC graphs. The IC2 statistic therefore is well suited to differentiate popula- graphs of the global and exponential dispersal were tions dispersed by inverse power-law functions from nearly identical to the IC graphs. However, the two ex- those dispersed by exponentially bounded functions. treme fat-tailed dispersal functions under investigation These results corroborate results from studies of focal resulted in the IC2 graphs having a plateau, reflecting a expansion with fat-tailed dispersal, which showed that self-similar structure in the lineages. For PL r2.0 the LDD produces a patchy, fractal-like, spatial population plateau stretched about two orders of magnitude. For structure very different from the structure of a homo- PL r1.5 the plateau stretched to all scales above 102. geneous front found with exponentially bounded dis- ˚ Although IC2 analysis results were very similar to the IC persal (Shaw 1995; Lett and Østergard 2000; Cannas results, the IC2 was more sensitive to the self-similar et al. 2006). With the present quasi-equilibrium setting we structure present in the lineages. were able to quantify differences in fractal structure. Gene differentiation GST values can be interpreted It was furthermore possible to find a simple relation- as characterizing the population structure by the pro- ship between the fractal dimension of the population portion of total gene diversity between subpopulations. structure and the migration rate (Figure 6). The linear High GST values near 1 were found for all dispersal relationship between the D2 fractal dimension and the functions at small scales (Figure 8). This means that inverse power-law exponent was D2 ¼ 1.8 2.0 3 1/b. almost all the variation in the populations occurred If the necessary data were available, it should be possible between subpopulations at small scales. Subpopulations to estimate the rate of migration, including the power-law 444 L. U. Wingen, J. K. M. Brown and M. W. Shaw

Figure 7.—Self-similarity statistics. Conditional incidence (IC, left column) and conditional Simpson incidence (IC2, right column) of inverse power- law (PL r1.5,PLr2.0, and PL r2.5) and exponentially and globally dispersed lineages at generation 6000 plotted against log10 of scale. Box plots corre- spond to the statistics of 20 (exponen- tial) or 12 replicates (all others).

exponent, from the fractal structure observed in field The power-law dispersal distribution used for this studies. The factor of 2 appearing in this equation sug- study is in essence very similar to Le´vy-flight process gests that the relationship has a simple mathematical (Shlesinger 1996). The resulting genotype distribu- explanation, but we cannot deduce it. tion should have a Le´vy-stable distribution (scale free) Population Structure Due to Different Types of Dispersal 445

igure F 8.—GST at different scales. GST of in- verse power-law (PL r1.5,PLr2.0, and PL r2.5) and exponentially dispersed lineages at genera- tion 6000 plotted against log10 of scale is shown. Box plots correspond to the statistics of 20 (expo- nential) or 12 replicates (all others).

in good agreement with the fractal population struc- (Figure 7), implying D2 ¼ 0.52, which is near the value of tures we report here. The difference between power-law 0.54 from D2 statistics. However, it seems preferable to and exponentially dispersed populations is also dem- use the IC2 graphs for detecting self-similarity only and onstrated by self-similarity statistics IC and IC2 (Figure 7), determine the correlation dimension in case self-simi- which are effectively derivatives of D0 and D2, respec- larity was found. To test the distinctness of the self- tively. The IC statistic was suggested by Shaw (1995) and similarity statistics, some preliminary simulations with a very similar statistic but for the factor of subdivision, different median dispersal distances were performed. the species-level self-similarity parameter a, was recently The mode of the IC or IC2 curves moved to larger scales suggested by Ostling et al. (2003) as helpful for for higher median dispersal distances. However, the characterizing in ecology. In partic- shape of the graph was not changed as long as the mode 1.5 ular, if IC2 is applied to PL r dispersed populations, it still lay within the arena. An exponentially bounded is constant over several orders of magnitude of scale and distribution induced a different pattern from that of reveals the highly self-similar structure of those popula- an inverse power-law distribution with b # 2. Even if the tions (Figure 7). IC2 values for exponentially dispersed median dispersal distance of the exponential function populations are very different in showing a distinctive was very long range the IC graph still showed a distinc- peak and no self-similar structure is detected. The self- tive peak, underlining the potential of this statistic to similarity statistics can pick up other patterns besides discriminate between fat-tailed LDD and exponential self-similarity in the spatial structures. Peaks are a sign of dispersal. visual patchiness, they are caused by areas with more Applying our statistics to field data in the future will occupied boxes, and the scale immediately greater than show how robust they are to noise and sparsity of data at the peak is the scale of greatest patchiness (Cousens larger scales. Data at many different scales are necessary et al. 2004). IC2 may be the most suitable tool if applied for using the new statistics and to detect the different to noisy field data to detect fractal structures or self- structures we predict. We therefore suggest that sam- similarity. The expected graphs may indicate differ- pling schemes for population genetic analyses in which ences stronger than the fractal dimension analysis. In the nature of dispersal between subpopulations may be theory, IC2 values are interchangeable with the fractal significant should routinely include as wide a range of dimension by the relation D2 ¼ 2 1 log2(IC2). However, spatial scales as possible. Sampling on a wider range of it seems difficult to estimate IC2 from the graphs as spatial scale does not necessarily mean that more sam- plateaus are short for all values of b $ 2. From the PL r1.5 ples are needed but that the sampling scheme has to be simulations, the value for IC2 can be estimated as 0.36 optimized for this task. 446 L. U. Wingen, J. K. M. Brown and M. W. Shaw

With all dispersal functions except global dispersal, Hovmøller 2002; Hovmøller et al. 2002) or the an increased genetic dissimilarity with geographic dis- regular reinfection of wheat-growing areas in northern tance was found (Figure 4). Some population studies China with wheat yellow rust from more southerly include the estimation of mean dispersal distances from regions (Brown and Hovmøller 2002). The ability of the genetic structure of the population. This strategy is the PL r1.5 dispersal to generate a wide-ranging distri- problematic for power-law dispersal with b # 2 because bution of an individual genotype over the whole arena mean and variance are not defined for these dispersal was clear. The observations of LDD of wheat yellow rust functions. As this is a substantial problem it cannot be are so far in accordance with those of our model. Our dealt with in the frame of this work. study is consistent with the presence of fractal structure In contrast to the fractal or self-similar measures, the in natural rust or mildew populations but more data on common population genetic statistic GST is not very use- several scales than are usually collected for a population ful for differentiating between different dispersal types genetic study for confirmation would be needed. (Figure 8). Patterns of change were rather similar for all The fractal analysis of the PL r1.5 dispersed popula- dispersal functions. To apply the statistic to field data to tions does not determine a maximum scale of dispersal determine the underlying dispersal function would be as there may be none with this kind of dispersal func- difficult, as clear differences are small and apparent tion. Realistically, the range of dispersal will be finite, only at very long scales, at distances from the source because of processes such as deposition or death, but where data will normally be very scarce. nevertheless very large (Shaw et al. 2006). This was not The emergence of fractal or scale-invariant patterns considered in the present study as the population ge- in simulations of power-law dispersed biological objects netic structure will be dominated by the effective dis- was shown previously by Cannas et al. (2006). They persal function at scales well below the arena size. investigated the expansion of an invading species from a For larger wind-dispersed particles like seeds, a power- central focus, such as a newly established weedy species. law dispersal function with b . 2 will apply (Cannas et al. Using exponents of 2 , b # 3, the resulting spatial 2006). However, inverse power-law distribution func- structure showed a fractal character, but only at some tions may also be useful to model nonwind LDD, as, times and at the border of the main patch. In contrast, e.g., the dispersal of small birds, tits, which followed our study showed a fractal pattern of genotype distribu- an inverse-square (Cauchy) law (Paradis et al. 2002). tion developing from an initially uniform random dis- Larger negative power-law exponents than b . 4 may tribution of genotypes for a population with power-law not be very useful for simulations of biological processes dispersal and stable size. Once developed, the fractal because the spatial pattern produced cannot be distin- pattern remained present during the whole remaining guished from that arising from a negative exponential time of simulation because mutation in our model acts and thus the less computationally expensive model of as a continual source of new invasions superimposed on negative exponential dispersal can be chosen. the existing population. Moreover, fractal patterns ap- From the present study it appears to be inappropriate peared most strongly with small power law coefficients, ever to use exponentially bounded dispersal functions of magnitude ,2. with a scale parameter much smaller than the size of the Fat-tailed dispersal of haploid individuals results in a available arena to model biological dispersal processes. fractal distribution of haploid genotypes over several With exponential dispersal alone a lineage of a dispers- scales. However, true nonsexual propagation is not very ing clonal organism (or, we propose, an allele in a sexual frequent in nature. Even organisms that use clonal re- organism) will inevitably stay in an area a few orders of production for the rapid production of many descend- magnitude larger than the median dispersal distance ants generally have occasional sexual reproduction, (Figure 3). In a nongrowing population this leads to such as many ascomycete and basidiomycete fungi. a concentration of each genotype in a single location. The sexual phase leads to an uncoupling of genes and Such populations would be much more prone to ex- loss of linkage disequilibrium. We predict, therefore, tinction in the case of changes in the environment. If that our main conclusions will apply equally to alleles of the only means of dispersal is an exponentially bounded a single gene in a sexual population. In particular, we process, any species is in fact evolving as strongly iso- predict that power-law dispersal will result in a fractal lated populations, even if it occupies a large homoge- distribution of alleles of a gene in a sexual population, neous, suitable habitat. It therefore seems likely that the just as it does of genotypes in a clonal population. majority of organisms have a LDD means of dispersal, This study investigates LDD in a simulation arena that which will have allowed their ancestors to avoid restric- is large enough to reflect the dispersal of wind-blown tion to a limited area. This argument may seem to depend small particles like fungal spores at scales ranging on the short median dispersal distances we assumed. from leaves to continents. Such LDD is observed in na- However, long-distance (e.g.,105 km) exponential dis- ture, e.g., the spread of recently mutated genotypes of persal is inconsistent with all observations of dispersal of P. striiformis (wheat yellow rust) from the United King- spores, pollen, or seeds. If a mixture of two exponential dom to Denmark, France, and Germany (Brown and functions is used as in Nichols and Hewitt (1994), the Population Structure Due to Different Types of Dispersal 447 model still exhibits a distinct scale, albeit the larger one, Cain, M. L., H. Damman and A. Muir, 1998 dispersal and the with quite distinct isolated patches evolving indepen- Holocene migration of woodland herbs. Ecol. Monogr. 68: 325–347. dently. Assuming a high proportion of long-distance Cain, M. L., B. G. Milligan and A. E. Strand, 2000 Long-distance exponential dispersal in the mixture conflicts with the in populations. Am. J. Bot. 87: 1217–1227. annas arco ontemurro lack of such observations in field data, as stated before. C , S. A., D. E. M and M. A. M , 2006 Long range dispersal and spatial pattern formation in biological inva- This inappropriateness of exponentially bounded sions. Math. Biosci. 203: 155–170. functions for modeling airborne dispersal processes is Clark, J. S., 1998 Why trees migrate so fast: confronting theory with of importance for risk assessment studies that aim at dispersal biology and the paleorecord. Am. Nat. 152: 204–224. Cousens, R., J. Wallinga and M. Shaw, 2004 Are the spatial pat- long-term, macroscale predictions. The dispersal poten- terns of weeds scale-invariant? Oikos 107: 251–264. tial is a key uncertainty in predicting the risk of an air- Darwin, C., 1859 The Origin of Species. John Murray, London. borne disease (Yang 2006) and the same should apply Davies, S., A. White and A. Lowe, 2004 An investigation into ef- fects of long-distance seed dispersal on organelle population ge- to the spread of pollen of genetically modified or netic structure and colonization rate: a model analysis. Heredity droplet-borne diseases such as foot-and-mouth disease. The 93: 566–576. use of exponentially bounded functions for predicting Devaux, C., C. Lavigne,H.Falentin-Guyomarc’h,S.Vautrin,J. Lecomte et al., 2005 High diversity of oilseed rape pollen dispersal of organisms that in fact follows a small- clouds over an agro- indicates long-distance dispersal. exponent power law in this context is inappropriate Mol. Ecol. 14: 2269–2280. because such models would tend to seriously underes- Feller, W., 1968 An Introduction to Probability Theory and Its Applica- tions, Ed. 3, pp. 295–298. Wiley, New York. timate risks. Ferrandino, F. J., 1993 Dispersive epidemic waves: I. Focus expan- The extensive simulations described here show a sion within a linear planting. Phytopathology 83: 795–802. characteristic population genetic pattern generated by Fitt, B. D. L., P. H. Gregory,A.D.Todd,H.A.McCartney and O. C. MacDonald, 1987 Spore dispersal and plant disease gra- LDD in a homogeneous landscape. This is a simplifica- dients: a comparison between two empirical models. J. Phytopa- tion of real conditions but is partly met in the modern thol. 118: 227–242. type of agriculture that favors monoculture. However, Gittenberger, E., D. S. J. Groenenberg,B.A.S.Kokshoorn and reece fragmented or barriers like mountains or R. C. P , 2006 Molecular traits from hitch-hiking snails. Nature 439: 409. may influence population genetic structure. Extensions Grassberger, P., 1983 Generalized dimensions of strange attrac- of the present model to accommodate for inhomoge- tors. Phys. Lett. A 97: 227–230. alley artley allimanis unin neous landscapes will be necessary to investigate this in H , J. M., S. H ,A.S.K ,W.E.K ,J.J. Lennon et al., 2004 Uses and abuses of fractal methodology detail. Simulations using long-range exponential func- in ecology. Ecol. Lett. 7: 254–271. tions have shown such LDD resulting in faster bridging Hastings, H. M., 1993 Fractals: A User’s Guide for the Natural Sciences. Oxford University Press, London/New York/Oxford. between niches over unsuitable habitats and greater ovmøller ustesen rown e orre H , M. S., A. F. J and J. K. M. B , genetic diversity in remote subpopulations (L C 2002 Clonality and long-distance migration of Puccinia striifor- et al. 1997; Davies et al. 2004; Bialozyt et al. 2006). It is mis f.sp. tritici in north-west Europe. Plant Pathol. 51: 24–32. to be expected that fat-tailed LDD will result in an even Klein, E. K., C. Lavigne,H.Picault,M.Renard and P.-H. Gouyon, 2006 Pollen dispersal of oilseed rape: estimation of the dis- faster niche colonization and a further increase in ge- persal function and effects of field dimension. J. Appl. Ecol. netic diversity of remote subpopulations. Predictions on 43: 141–151. the presence of fractal structure are more difficult as Kot, M., M. A. Lewis and P. van den Driessche, 1996 Dispersal data and the spread of invading organisms. Ecology 77: 2027– this may depend on the niche distribution. 2042. unin We thank two helpful anonymous reviewers. This research was K , W. E., 1998 Extrapolating species abundance across spatial supported by the Biotechnology and Biological Sciences Research scales. Science 281: 1513–1515. Le Corre, V., N. Machon,R.J.Petit and A. Kremer, 1997 Col- Council and was part of the BioExploit project in the European Union onization with long-distance seed dispersal and genetic structure Framework 6 Programme. of maternally inherited genes in forest trees: a simulation study. Genet. Res. 69: 117–125. Lett, C., and H. Østerga˚rd, 2000 A stochastic model simulating LITERATURE CITED the spatiotemporal dynamics of yellow rust on wheat. Acta Phy- Austerlitz, F., and P. H. Garnier-Ge´re´, 2003 Modelling the im- topathol. Entomol. Hung. 35: 287–293. pact of on genetic diversity and differentiation of Male´cot, G., 1969 The Mathematics of Heredity. W. H. Freeman, San forest trees: interaction of life cycle, pollen flow and seed long- Francisco. distance dispersal. Heredity 90: 282–290. McCartney, H. A., 1987 Deposition of Erysiphe graminis conidia on Austerlitz, F., C. W. Dick,C.Dutech,E.K.Klein,S.Oddou- a barley crop. 2. Consequences for spore dispersal. J. Phytopa- Muratorio et al., 2004 Using genetic markers to estimate thol. 118: 258–264. the pollen dispersal curve. Mol. Ecol. 13: 937–954. McCartney, H. A., and A. Bainbridge, 1987 Deposition of Erysiphe Aylor, D. E., 2003 Spread of plant disease on a continental scale: graminis conidia on a barley crop. 1. Sedimentation and impac- role of aerial dispersal of pathogens. Ecology 84: 1989–1997. tion. J. Phytopathol. 118: 243–257. Bialozyt, R., B. Ziegenhagen and R. J. Petit, 2006 Contrasting ef- Nathan, R., G. Perry,J.T.Cronin,A.E.Strand and M. L. Cain, fects of long distance seed dispersal on genetic diversity during 2003 Methods for estimating long-distance dispersal. Oikos range expansion. J. Evol. Biol. 19: 12–20. 103: 261–273. Brown, J. K. M., and M. S. Hovmøller, 2002 Aerial dispersal of Nathan, R., N. Sapir,A.Trakhtenbrot,G.G.Katul,G.Bohrer pathogens on the global and continental scales and its impact et al., 2005 Long-distance biological transport processes through on plant disease. Science 297: 537–541. the air: Can nature’s complexity be unfolded in silico? Divers. Bullock, J. M., and R. T. Clarke, 2000 Long distance seed dispersal Distrib. 11: 131–137. by wind: measuring and modelling the tail of the curve. Oecolo- Nei, M., 1973 Analysis of gene diversity in subdivided populations. gia 124: 506–521. Proc. Natl. Acad. Sci. USA 70: 3321–3323. 448 L. U. Wingen, J. K. M. Brown and M. W. Shaw

Nichols, R. A., and G. M. Hewitt, 1994 The genetic consequences Shaw, M. W., 1995 Simulation of population expansion and spatial of long distance dispersal during colonization. Heredity 72: 312– pattern when individual dispersal distributions do not decline ex- 317. ponentially with distance. Proc. R. Soc. Lond. Ser. B 259: 243–248. O’Hara, R. B., and J. K. M. Brown, 1998 Movement of barley pow- Shaw, M. W., 1996 Simulating dispersal of fungal spores by wind, dery mildew within field plots. Plant Pathol. 47: 394–400. and the resulting patterns. Asp. Appl. Biol. 46: 165–172. Ostling, A., J. Harte,J.Green and A. Kinzig, 2003 A - Shaw, M. W., T. D. Harwood,M.J.Wilkinson and L. Elliott, level fractal property produces power-law species-area relation- 2006 Assembling spatially explicit landscape models of pollen ships. Oikos 103: 218–224. and spore dispersal by wind for risk assessment. Proc. R. Soc. Paradis, E., S. R. Baillie and W. J. Sutherland, 2002 Modeling B 273: 1705–1713. Shlesinger, M. F., 1996 Le´vy walks versus Le´vy flights, pp. 279–283 large-scale dispersal distances. Ecol. Modell. 151: 279–292. tanley strowsky Pedgrey, D. E., 1986 Long distance transport of spores, pp. 346–365 in On Growth and Form,editedbyH.E.S and N. O . Nijhoff, Dordrecht, The Netherlands. in Plant Disease Epidemiology, Population Dynamics and Management, impson eonard ry S , E. H., 1949 Measurement of species diversity. Nature 163: edited by K. J. L and W. E. F . Macmillan, New York/ 688. London. tockmarr ress eukolsky etterling lannery S , A., 2002 The distribution of particles in the plane dis- P , W. H., S. A. T ,W.T.V and B. P. F , persed by a simple 3-dimensional diffusion process. J. Math. Biol. 1996 Numerical Recipes in C, Ed. 2, pp. 274–286. Cambridge Uni- 45: 461–469. versity Press, Cambridge/London/New York. Wright, S., 1951 The genetical structure of populations. Ann. Eu- RDevelopment Core Team, 2004 R: A Language and Environment for gen. 15: 323–354. Statistical Computing. R Foundation for Statistical Computing, Yang, X. B., 2006 Framework development in plant disease risk as- Vienna, Austria. sessment and its application. Eur. J. Plant Pathol. 115: 25–34. Sackett, K. E., and C. C. Mundt, 2005 Primary disease gradients of wheat stripe rust in large field plots. Phytopathology 95: 983–991. Communicating editor: M. Veuille