Copyright Ó 2007 by the Genetics Society of America
DOI: 10.1534/genetics.107.077206
The Population 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 spores 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 powerlaw 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.
T
HE importance of long-distance dispersal (LDD)
for the distribution and evolution of organisms
has long been recognized and was acknowledged as
early as 1859 by Darwin (Darwin 1859). Until recently,
however, population genetic and evolutionary studies
have concentrated mainly on short-distance dispersal
that is easier to measure but nonetheless has important
consequences for local population dynamics. Owing to
better methodology for assessing LDD and increased
awareness of its importance, interest in it has risen
again in the last 15 years (Nathan et al. 2003).
LDD plays an important role in colonization of
islands (Cain et al. 2000 and references therein;
Gittenberger et al. 2006), in range expansion (Cain
et al. 1998; Clark 1998), and in the rates of population
expansion and spread of epidemics (Shaw 1995; Kot
et al. 1996). However, LDD is rare and difficult to analyze
in detail in the field. Modeling of LDD has thus become
an instrument to investigate its importance in evolutionary and ecological processes. One example is the
range expansion of oak trees to the north during the
postglacial recolonization of Europe. LDD plays an
important role in explaining the speed of the expansion
(Le Corre et al. 1997; Austerlitz and Garnier-Géré
1
Corresponding author: Department of Disease and Stress Biology, John
Innes Centre, Colney, Norwich NR4 7UH, United Kingdom.
E-mail: [email protected]
Genetics 177: 435–448 (September 2007)
2003; Davies et al. 2004; Bialozyt et al. 2006). Computer models of the process explained the patchy
genetic patterns, observed in modern oak populations,
by LDD founder events (Davies et al. 2004; Bialozyt
et al. 2006).
Dispersal by wind is a major mechanism of LDD.
Fungal spores causing severe agricultural diseases are
dispersed in rare events over hundreds or even thousands of kilometers (Brown and Hovmøller 2002).
Transport through the air is profoundly affected by
turbulence over a wide range of spatial scales (Aylor
2003; Nathan et al. 2005).
There is an ongoing debate about what kind of
dispersal kernel, the function that describes the probability that a propagule will be deposited at a given
distance, is best suited to describe LDD. Several studies
have modeled LDD of insects or seeds by a mixture of
two exponential dispersal distributions, one with a short
median dispersal distance and one with a very long one
(e.g., Nichols and Hewitt 1994; Bialozyt et al. 2006).
Inferring true dispersal curves from small, winddispersed biological objects like spores or pollen is difficult. Measured dispersal distributions are frequently
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
et al. 1996). Many pollen dispersal data are best fitted
with an inverse power-law function (Bullock and Clarke
436
L. U. Wingen, J. K. M. Brown and M. W. Shaw
2000; Austerlitz et al. 2004; Devaux et al. 2005; Klein
et al. 2006; Shaw et al. 2006). For fungal spores, the
question of the best-fitting dispersal function is hindered
by the necessity of large experimental plots free from too
much background infection. A recent study addressing
the above problems showed that dispersal of the wheat
stripe or yellow rust fungus (Puccinia striiformis) fitted a
power-law model well if enough sufficiently distant spore
traps were used (Sackett and Mundt 2005). Moreover,
although several physical processes underlie wind dispersal, theoretical arguments strongly propose that LDD
of small objects can be modeled by a single function that
will have inverse power-law behavior in the tails (Shaw
1995; Kot et al. 1996; Stockmarr 2002; Aylor 2003;
Shaw et al. 2006). Uplift is the most important factor for
heavier propagules but many factors are equally important for smaller objects such as spores (Nathan et al.
2005). A simplification of the stochastic dispersal process
by a single negative power-law dispersal function is a
useful basis for theoretical modeling.
Recent simulation studies have addressed either the
influence of LDD, modeled as a dual exponential function, on population genetic structure (Bialozyt et al.
2006) or the influence of power-law dispersal function
on spatial distribution of species (Cannas et al. 2006).
This article investigates the influence of LDD, modeled as a negative power law, on population genetic
structure of populations in quasi-equilibrium. Inverse
power-law functions with exponents in the range of 1 ,
b # 2 were used as dispersal functions to simulate fattailed dispersal. The resulting population structures
were compared to those generated by a negative exponential dispersal function or a global dispersal (uniform
random) function. Widely used statistics from ecology
and population genetics were applied to the resulting
populations. Some of them were more suitable than
others to describe the population structures and to distinguish the outcomes of different modes of dispersal.
The simulations reported here used an arena several
orders of magnitude larger than the median dispersal
distance of the dispersal function. We used a novel simulation strategy that allowed us to investigate a range of
spatial scales covering nine orders of magnitude. This
usage of a large arena may be of paramount importance
for a thorough investigation of the consequences of
LDD as it is hypothesized that the population structure
of organisms with small wind-dispersed propagules is influenced over scales ranging from centimeters to several
hundred or even thousands of kilometers (Pedgrey 1986;
O’hara and Brown 1998; Brown and Hovmøller 2002).
genetic structure of populations with a roughly stable population size. This scenario corresponds to a natural population
that is restricted in growth, e.g., by limited space or limited
nutrients. The simulation arena used was very large and so the
occupying population should be very large as well. As computer memory was inevitably limited, only sample lineages
were simulated. A large population was assumed to be present
in the background of these individuals, competing for resources and thus limiting the expansion of the simulated
individuals. A birth process with Poisson-distributed progeny
number of one individual per parent and a fixed death age of
one generation were used to simulate these sample lineages in
a fluctuating, nonexpanding population.
General model settings: Simulations were initiated with 30,000
individuals, initially all of different genotypes, each represented by a 32-bit number, in effect 32 biallelic loci. The initial
individuals were placed randomly in a simulation arena of
108 3 108 square units in size, with 1 unit corresponding to the
closest distance allowed between two individuals. Individuals
gave birth to offspring at the beginning of each time step,
which were immediately dispersed according to the chosen
dispersal function. The genotype of a new individual was either
the same as that of the parent or mutated by conversion of one
random bit of the 32-bit genotype. Mutation took place at random
with a frequency set by the mutation rate m ¼ 104. Individuals
died after 1 time step and thus had the chance to produce
progeny only once in their lifetime. Offspring were not placed
outside of the simulation arena or closer to other offspring than
the minimal interaction distance. If the simulation generated
such an event, a new location was calculated until a legitimate one
was found. Individuals possibly adjacent to a given point were
found quickly using the indexing algorithm in Shaw (1996). The
simulations were assumed to be a part of a huge population of
uniform density. All ‘‘background’’ lineages were assumed to
disperse in the same way as the simulated lineages and thus
result in a similar population genetic structure. The main
simulations were aimed to run for 50,000 generations.
Simulations that were used mainly to calculate the fractal
dimensions were run for 10,000 generations only.
Dispersal functions: The dispersal of the spores was modeled
by an inverse power-law probability density function with the
spore concentration t(r j u) at distance r from the source along
a given bearing u given by
tðr j uÞ }
ð1Þ
with b . 1 (Shaw et al. 2006). Of special interest were values
of b # 2. Theoretical and experimental results suggest this is
the relevant range in wind dispersal of small particles like
fungal spores and pollen (Mccartney 1987; Mccartney and
Bainbridge 1987; Ferrandino 1993; Sackett and Mundt
2005; Klein et al. 2006; Shaw et al. 2006). If not stated
otherwise, simulations used values of 1.2 # b # 4. The main
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 wellknown Cauchy distribution. Quasi-random power-law variates
were generated as described in Shaw (1996). Two rational
function approximations were derived using the algorithm in
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
Computer simulations: An individual-based, spatially explicit model of haploid individuals without sex in a continuous
habitat was used, developing the models of Shaw (1995,
1996). The simulation aimed at investigating the spatial
1
1 1 rb
loge ðmedianðbÞÞ ¼
For b $ 2,
0:2816583 0:4939602b 1 0:3358465b 2
:
1 1:7971499b 1 0:8078619b 2
ð2Þ
Population Structure Due to Different Types of Dispersal
loge ðmedianðbÞÞ ¼
1:7316499 1:1767354b 1 1:1298891b 2
:
1 1 2:0186175b 2:2337080b 2
ð3Þ
For comparison, simulations were also done following the
commonly used dispersal model of an exponential decline of
spore concentration with a characteristic scale of k:
tðr j uÞ } e r =k :
ð4Þ
Simulations with global or uniform random dispersal were
performed for further comparison, in which offspring were
placed randomly in the arena, with both x and y coordinates
drawn from a uniform distribution. Simulations were repeated
20 times for the negative exponential dispersal and 12 times
for other dispersal functions.
Parameter settings: Distances in our model were chosen to fit
experimental data, especially on wind dispersal of powdery
mildew (Blumeria graminis) and yellow rust (P. striiformis) of
cereals. The minimal interaction distance between individuals, 1 unit, corresponding to the minimum distance between
two distinct rust or mildew lesions on one leaf, was assumed for
the sake of simplicity to be 10 cm. Other authors have assumed
it of similar magnitude (2 cm was used as the size of a single
yellow rust lesion by Lett and Østergård 2000). The median
dispersal distance was set to 30 units for the exponentially
bounded function. The power-law functions were adjusted to
approximate the same median value, corresponding to 3 m in
real-world dimensions. The median dispersal for powdery
mildew conidiospores is estimated to be 3.1 m for field disease
gradients (half the distance of the exponential model; Fitt
et al. 1987) or 1 m to several meters (modeled from deposition
velocities and impaction efficiency, with 50% deposition rate;
Mccartney 1987). Modeled estimates for wheat yellow rust
are 2.8 m–6.5 m (calculated from exponential fits to the three
largest data sets for upwind dispersal in Sackett and Mundt
2005). Estimates by O’hara and Brown (1998) are similar.
The median dispersal distance setting was thus in concordance with field data. The extension of our simulation arena
of 108 units can then be regarded as a distance of 104 km, e.g.,
spanning further than the whole of Europe.
Software and hardware: Simulation software was written in
Kylix (Borland Delphi for Linux, version 14.5, Open Edition)
using the Free Pascal Compiler (version 1.9.8). Simulations
were run on a GNU/Linux platform with four 2.8-GHz
processors. Statistical analysis of populations and graphical
output were performed using the free R software suite (version
2.1.0; R Development Core Team 2004).
Characterization of populations: Statistical characterization
of populations was performed at different time steps. Results
are shown for generation 6000 and in some cases for generation 40,000. If not stated otherwise, results from later generations did not notably differ from those from generation 6000.
Results of repeated runs were summarized as box plots.
The spatial genetic structure of the simulated lineages was
analyzed using graphs of genetic dissimilarity against genetic
distance. For this, 20,000 individuals were drawn randomly
from the population and grouped into pairs. The geographic
distance and the genetic dissimilarity (1, dissimilar; 0, similar)
of each pair were determined. The distance range was divided
into 100 intervals and the percentage of dissimilar pairs in
each interval was plotted against the logarithm of the midpoint of the distance interval.
Spatial statistics: Several statistics were calculated at different
spatial scales. These were fractal dimensions D0 and D2, conditional incidence IC, conditional Simpson incidence IC2, and
a measure for population subdivision GST. Analysis was done
for a square area a little smaller than the simulation arena with
437
side length of 226 (6.7 3 107) units, randomly placed within
the simulation arena. The analysis area was divided into
smaller and smaller squares by repeated equal divisions into
four. The smallest square had a side length of 2 units. Each step
of this subdivision formed one scale of analysis. The units of
analysis were the squares of subdivision (‘‘boxes’’), which can
also be regarded as subpopulations. If boxes contained one or
more individuals they were counted as ‘‘occupied’’ and were
analyzed, and if they had none they were counted as empty.
A fractal distribution of individuals is reflected by a relationship of the type
D0
1
N ðsÞ }
s
ð5Þ
with N(s) the number of boxes needed to cover all individuals
at the scale s, the side length of the boxes (Hastings 1993).
The exponent D0 is the box-counting dimension. By introducing a constant k0 (5) can be linearized to
logðN ðsÞÞ ¼ D0 log
1
1 logðk0 Þ:
s
ð6Þ
A practical approach to determine D0 was to cover the arena
of the simulated populations with grids of different box sizes
and do box counts for the different scales (Halley et al. 2004),
counting the number of boxes, Nj(s), needed to cover individuals of a given genotype j and at each scale (Figure 1). This
was done for all common genotypes of frequency of at least 10,
partly to reduce the computational demands and partly because rare genotypes will most likely not be detected in a
population study. Moreover, rare genotypes contribute little to
calculation of D0 as their box-occupation pattern is inevitably
similar at most scales. D0 was then estimated by the negative
slope of all log(Nj(s)) on log(s) by applying a regression analysis to the linear region of the graph ( judged by eye).
The correlation dimension D2 is related to D0 and either
equal to or smaller than it (Grassberger 1983). It detects a
power-law relationship between the concentration of individuals in the boxes and the scale. The concentration is measured
by the Simpson index, IS, calculated from the proportion
P ðsÞ of2
individuals that fall in each box, pi, as IS ðsÞ ¼ Ni¼1
ðpi Þ
(Simpson 1949), with N(s) the number of boxes at that scale.
The relationship is thus
D2
1
IS ðsÞ}
:
s
ð7Þ
By introducing a constant k2 (7) can be linearized by analogy
to (6) as
1
ð8Þ
logðIS ðsÞÞ ¼ D2 log
1 logðk2 Þ:
s
D2 was determined from the linear region of the graph of
logðISj ðsÞÞ against log(s) for all common genotypes j (Halley
et al. 2004).
The conditional incidence IC is a measure of self-similarity,
effectively the derivative of D0 (Shaw 1995; Shaw et al. 2006).
IC is calculated as the ratio of two incidences at neighboring
scales,
IC ¼
N ðsÞ=Ns
I ðsÞ
¼
;
N ðs 1Þ=Ns1 I ðs 1Þ
ð9Þ
with I(s) the proportion of occupied boxes N(s) among total
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
Figure 1.—Example of the box-counting procedure. (A)
The big squares represent the simulation arena with grids
of different box sizes superimposed. The arena is populated
by individuals of two genotypes (solid and shaded dots). A box
count for each genotype and each box-side length, or scale, is
performed. The box count is the number of boxes containing
one or more individuals of a given genotype. In the above example, all boxes that contribute to the box counts for the
‘‘solid’’ individuals are shaded. Scales and counts for those individuals are listed under the arenas and used for the plot in
B. (B) The results from the box counting are plotted as log
(box number) against log(scale). If the resulting graph, or
parts of it, shows a linear region with a measurable gradient,
the negative of this gradient is the box-counting dimension D0
of that genotype over the extent of the linear region. D0 is an
indication of a fractal distribution of that genotype.
distribution. For constant IC and each scale being half the
length of the next larger one (which results in a subdivision
into quarters), the relation between D0 and IC is thus
D0 ¼ 2 1 log2 ðIC Þ
ð10Þ
(Shaw 1995).
The conditional Simpson incidence IC2 was calculated in a
similar way but using the Simpson index divided by the total
number of boxes as a measure of incidence at a scale:
P
N ðsÞ 2
pi =Ns
i
IS ðsÞ=Ns
IC2 ¼ PN ðs1Þ ¼
:
ð11Þ
2
I
ðs
S 1Þ=Ns1
p =Ns1
i
i
IC and IC2 were calculated separately for each common
genotype at each scale.
A common way of summarizing population genetic structure over a wide range of spatial scales is to use F-statistics
(Wright 1951), especially the coefficient of inbreeding, FST.
GST, the coefficient of gene differentiation, is used as an analog
of FST (Nei 1973) and often used for haploid organisms and
was applied here by regarding the occupied boxes as subpopulations at different scales. The GST values estimated from
the simulated lineages are proportional to those that would be
found by unbiased sampling from the whole populations.
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 descendant 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 between 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
12,900 for PL r2.0, 40,700 6 12,800 for PL r2.5, 42,400 6
13,500 for exponential, and 40,900 6 15,000 for global
dispersal (mean 6 standard deviation); the differences
of the means were statistically significant. These numbers were products of large numbers of calls of the
Poisson generator. Since the Poisson generator and the
dispersal module were tested independently and had no
shared variables or parameters, variations in population
size must have been due to slight remaining long-range
correlations in the underlying random-number generator, coupled with differences in the number of calls
made to it. The generator, Knuth’s subtractive method
to generate uniform random deviates (Press et al. 1996),
is widely recommended and has passed all the usual
tests. However, no generator is truly random and there
will always be some remaining signatures of the underlying deterministic nature of the algorithm. Since
the imbalance between large and small random Poisson
deviates was of the order of only 1 in 105, we believe the
general patterns in the data were robust. This belief was
supported by the observation that in general results
from simulations that went extinct were very similar to
the long-run results. Exceptions were runs where the
population size dropped dramatically under 10,000 and
recovered, causing a dramatic drop in genotype number
to near uniformity.
Initially, all 30,000 individuals were of different genotypes. Most of the genotypes died out within the first
4000 generations in a process similar to exponential
decay as is expected for a Poisson birth process with
constant death rate. Without mutation, all genotypes
but one died out (data not shown). With mutation, the
number of genotypes stabilized after 4000 generations
at a mean value of 62 6 20 (Figure 2), indicating an
equilibrium between mutation and drift. In addition to
similar genotype numbers, the distribution of genotype
Population Structure Due to Different Types of Dispersal
Figure 2.—Decrease of genotype number with time. The
logarithm of genotype number of inverse power-law ½PL
r1.5 (black), PL r2.0 (dark shading), and PL r2.5 (medium
shading) and exponentially (light shading) and globally
(very light shading) dispersed lineages is plotted against time.
Box plots correspond to the statistics of 20 (exponential) or
12 replicates (all others).
frequencies was also similar between time steps and
dispersal functions.
Spatial patterns: The different dispersal functions
generated different spatial population patterns (Figure 3).
Differences became more prominent over time. Initially
all individuals were randomly distributed. In the case
of global dispersal, that distribution persisted (not
shown). For the other dispersal functions, the combination of dispersal and the Poisson birth process led
to an aggregation of individuals in clusters. Individuals
of the same genotype tended to cluster together because
the dispersal patterns were inversely related to distance.
This clustering process was strongest for the exponential simulations, in which tight clusters arose over time,
of which most died out (Figure 3, bottom row) and only
one cluster survived in the analyzed simulations after
40,000 generations. The exponential dispersal pattern
did not enable new clusters to form as the maximum
distance over which dispersal was at all probable was in
the same range as the spatial size of an individual cluster.
The structures that arose from PL r2.5 and PL r2.0
dispersal also led to a few distinct clusters, but the process of reducing the number of clusters was slower for
PL r2.5 and even slower for the PL r2.0 dispersal. The
resulting clusters were less dense; the PL r2.5 dispersed
lineages were clustered more densely than the PL r2.0
dispersed ones (Figure 3, middle rows). LDD was sufficiently frequent with PL r1.5 dispersal to counteract
the concentrating effect of the Poisson birth process
(Figure 3, top row). After an initial loss of single individuals and small clusters within the first 100 generations, the lineages remained scattered in clusters of a
wide range of sizes over the entire arena. The number
and sizes of clusters appeared stable in that state over
the whole duration of the simulations (Figure 3, top
439
row, different time steps). It was hard to define a cluster
unambiguously with PL r1.5 dispersal, so it was not possible 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 depended strongly on the probability of LDD and was
much greater for smaller b. The difference between dispersal 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
radius.
Spatial genetic structure: Population genetic theory
states that limited dispersal results in individuals located
close to each other being genetically more similar than
individuals further apart (Malécot 1969). However,
many of the presented simulations were using fat-tailed
LDD. The relationship of genetic dissimilarity and geographic distance at quasi-equilibrium was thus of interest
and investigated as the percentage of genetic dissimilar
pairs found at different distances (Figure 4). Individuals
were initially genetically dissimilar and were distributed
randomly over the whole arena with wide spacing between them. Over time, genetically similar individuals
appeared as particular lineages grew by chance. In lineages dispersing with an exponential dispersal function
these individuals were found near one another. In these
lineages genetically dissimilar individuals appeared near
one another as well, as mutants arose within lineages
of spatially adjacent individuals. The graphs of genetic
dissimilarity reflect the same clustering tendencies of
genotypes as discussed above for the whole population
(Figure 3). The strong clustering tendency of the
exponential dispersal resulted in the dissimilarity graph
having a short extension on the log(geographic distance) axis (Figure 4). The extension of the graphs was
longer for the power-law dispersal functions, reaching
higher geographic distances. This trend increased with
decreasing b. The extremely fat-tailed distribution PL
r1.5 dispersed sufficient individuals to distances comparable to the arena size to cause pairs of similar and
dissimilar individuals to be found at far distances at all
time steps. With all dispersal functions but the global
dispersal an increase of genetic dissimilarity with geographic distance was found (Figure 4).
Fractal dimensions D0 and D2: Many natural objects
have relevant features on a variety of different scales. If
their properties are similar across a wide range of scales,
they are self-similar over that range and have a fractal
structure. Natural objects are often not ideal fractals,
but are sufficiently fractal that the tools of fractal geometry, including characterization by noninteger dimensions, can be used to describe them. Fractal measures,
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, PL r2.0,
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, 100 for PL r2.0, 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
1998) and power laws, which are closely connected with
fractals, are a major descriptive tool in ecology (Halley
et al. 2004).
The most basic fractal dimension is the box-counting
dimension (Equation 6). This box count was applied
separately to individuals of the same genotype in the
simulated samples at different scales and summed up in
the graph of log(D0) against log(s) (Figure 5). For the
exponentially dispersed lineages the maximum scale,
the point where the regression line meets the log(s)
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, PL r2.0, PL r2.5, and exponentially dispersed lineages after 6000 (top row) and 40,000 (bottom row) generations are shown. Geographic distance scale is given as log10 (1 ¼ 101, 2 ¼ 102, . . . ).
for the inverse power-law dispersed lineages. This simply
reflects the limited spatial extent of any genotype with
exponential dispersal. In PL r2.5 dispersed lineages, the
maximum spatial scale was 105, in PL r2.0 it was 106,
while in PL r1.5 dispersed lineages, the maximum scale
was larger than the simulation arena (Figure 5). This
was confirmed with a limited number of simulations in a
larger arena where still the line did not meet the log(s)
axis (data not shown).
A regression analysis for the box-counting graphs of
the exponentially dispersed lineages produced a coefficient of determination of R 2 ¼ 0.87. The linear
region stretched over a spatial range of about two orders
of magnitude only, which is too short to merit description as a truly fractal structure (Halley et al. 2004).
By contrast, broader linear regions, spanning three orders of magnitude and three to four orders of magnitude, were found in PL r2.5 and PL r2.0 dispersed
lineages, indicating fractal spatial patterns over several
orders of magnitude with fractal dimensions of D0 ¼
0.994 6 0.008 (R 2 ¼ 0.95) and D0 ¼ 0.760 6 0.006 (R 2 ¼
0.93), respectively. As ranges were limited these can be
classified only as a quasi-fractal spatial pattern. The regression analysis of the box-counting curve of PL r1.5
dispersed lineages found a poorer fit with R 2 ¼ 0.85 but
the linear region stretched over a much wider scale, six
orders of magnitude. Hence PL r1.5 populations had a
more extended fractal structure than PL r2.0 populations but the pattern seemed to be less homogeneous.
The fractal dimension was D0 ¼ 0.439 6 0.004 (Figure 5).
However, the box-counting dimension does not consider the number of individuals in each box, which can
vary enormously. To capture this aspect of population
structure, the correlation dimension D2 was applied
(Equation 8). This uses the Simpson index (Simpson
1949) that quantifies the evenness of occurrence between subpopulations (boxes).
In PL r1.5 dispersed lineages, a fractal dimension of
D2 ¼ 0.534 6 0.004 was calculated (R 2 ¼ 0.90). The
linear region stretched over nearly six orders of magni-
tude (Figure 5). The distribution of genotypes therefore
shows a very strongly fractal pattern, over most of the
observed area. In PL r2.0 dispersed lineages, the fractal
pattern is a little weaker, D2 ¼ 0.777 6 0.006 (R 2 ¼ 0.92),
and the linear region stretched over four orders of magnitude, which, though less than for PL r1.5, is still quite
large by comparison with reports in natural ecological
systems (Halley et al. 2004). Results for the PL r2.5
dispersed lineages indicate an even weaker fractal
structure with D2 ¼ 1.008 6 0.07 (R2 ¼ 0.94). The linear
region of the log(Simpson index) against the log(s) plot
for the exponentially dispersed lineages stretched over
only two orders of magnitude and so did not indicate a
fractal genotype distribution. As expected, globally dispersed lineages did not show patterns of any kind as D0
and D2 were both zero.
The fractal dimensions were determined for five
further power-law dispersal functions (1.2 # b # 4) at
generation 6000. Linear relationships between the fractal dimensions and the inverse of the power-law exponent were found (Figure 6). The regression equations of
fractal dimension on 1/b were D0 ¼ 2:07ð1=bÞ 1 1:82
(R 2 ¼ 0.99) and D2 ¼ 1:94ð1=bÞ 1 1:81 (R 2 ¼ 0.99).
Conditional incidence (IC) and conditional Simpson incidence (IC2): The self-similarity measures were applied to
individuals of the same genotypes of the simulated lineages. Stable patterns of IC and IC2, respectively, against
log(s) were found, which were characteristic of the
dispersal function (Figure 7). As found by Shaw (1995)
the IC values describing exponentially dispersed lineages were much larger than those in PL r2.0 dispersed
lineages.
The IC graphs of the exponentially dispersed lineages
were characterized by a sharp rise from 0.25, which is the
minimum possible IC value for a grid based on a division
of each square into four smaller squares, to a maximum
value around 0.75 over one to two orders of magnitude
followed by a little less steep decline (Figure 7). The fattailed power-law dispersed lineages showed flatter IC
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 dimension D0 (left column) and correlation dimension D2 (right column)
of clonal descendants of inverse
power-law (PL r1.5, PL r2.0, and
PL r2.5) and exponentially and
globally dispersed lineages at generation 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 correspond to the statistics of 20 (exponential) or 12 replicates (all
others).
Population Structure Due to Different Types of Dispersal
Figure 6.—Relationship between fractal dimensions and
1/b (b: power-law exponent). Data points were plotted as solid
circles (D0) or open squares (D2) and regression lines were
fitted to the two fractal dimension types (solid line, D0; dashed
line, D2).
two and three orders of magnitude for PL r2.5 and PL
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
fat-tailed inverse power-law dispersal had a much larger
extent than that produced by exponential dispersal.
However, IC graphs for PL r4.0 dispersed lineages looked
identical to those for exponential dispersed lineages
and thus had the same spatial extent (graphs not shown).
IC graphs for globally dispersed lineages were, as expected, uninformative.
For the very fat-tailed inverse power-law dispersal
function PL r1.5, the rise of the IC against the log(s)
curve was very shallow over about two to three orders of
magnitude. A flat IC curve signifies a self-similar structure, so the underlying genetic structure of PL r1.5
lineages was close to self-similar over a limited but substantial range.
The results from the IC2 analysis were in most respects
similar to those from the IC analysis. IC2 against log(s)
graphs had the same modes as the IC graphs. The IC2
graphs of the global and exponential dispersal were
nearly identical to the IC graphs. However, the two extreme fat-tailed dispersal functions under investigation
resulted in the IC2 graphs having a plateau, reflecting a
self-similar structure in the lineages. For PL r2.0 the
plateau stretched about two orders of magnitude. For
PL r1.5 the plateau stretched to all scales above 102.
Although IC2 analysis results were very similar to the IC
results, the IC2 was more sensitive to the self-similar
structure present in the lineages.
Gene differentiation GST values can be interpreted
as characterizing the population structure by the proportion of total gene diversity between subpopulations.
High GST values near 1 were found for all dispersal
functions at small scales (Figure 8). This means that
almost all the variation in the populations occurred
between subpopulations at small scales. Subpopulations
443
contained only one main genotype. In the PL r2.0 and
PL r1.5 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 exponentially 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 genotypes present in the population and the location of
population clusters. GST graphs from the PL r2.5 dispersed lineages showed an intermediate behavior between PL r1.5 and exponential GST graph trends. In the
case of global dispersal, GST values were either 1 for
small and medium scales where only single individuals
were present per box or 0 at large scales, revealing perfect mixing at those scales. A transition phase with
intermediate GST values is very short.
DISCUSSION
Several experiments and theoretical arguments suggest that LDD dispersal of small propagules by wind is
best described by inverse power-law dispersal functions.
Here, the importance of such LDD dispersal events on
population genetic structure in quasi-equilibrium is investigated by computer simulation of lineages of clonal
haploid organisms, which were dispersed following
inverse power-law dispersal functions with an exponent
1 , b # 4. Our most important finding is a fractal distribution of genotypes in the resulting populations for
small b, which is best detected by the D2 fractal dimension, applying the Simpson index to subpopulations
at different scales (Figure 5). In contrast, if dispersal
is modeled by an exponentially bounded function, no
fractal population genetic structure is produced. The D2
statistic therefore is well suited to differentiate populations dispersed by inverse power-law functions from
those dispersed by exponentially bounded functions.
These results corroborate results from studies of focal
expansion with fat-tailed dispersal, which showed that
LDD produces a patchy, fractal-like, spatial population
structure very different from the structure of a homogeneous front found with exponentially bounded dispersal (Shaw 1995; Lett and Østergård 2000; Cannas
et al. 2006). With the present quasi-equilibrium setting we
were able to quantify differences in fractal structure.
It was furthermore possible to find a simple relationship between the fractal dimension of the population
structure and the migration rate (Figure 6). The linear
relationship between the D2 fractal dimension and the
inverse power-law exponent was D2 ¼ 1.8 2.0 3 1/b.
If the necessary data were available, it should be possible
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 powerlaw (PL r1.5, PL r2.0, and PL r2.5) and
exponentially and globally dispersed
lineages at generation 6000 plotted
against log10 of scale. Box plots correspond to the statistics of 20 (exponential) or 12 replicates (all others).
exponent, from the fractal structure observed in field
studies. The factor of 2 appearing in this equation suggests that the relationship has a simple mathematical
explanation, but we cannot deduce it.
The power-law dispersal distribution used for this
study is in essence very similar to Lévy-flight process
(Shlesinger 1996). The resulting genotype distribution should have a Lévy-stable distribution (scale free)
Population Structure Due to Different Types of Dispersal
445
Figure 8.—GST at different scales. GST of inverse power-law (PL r1.5, PL r2.0, and PL r2.5)
and exponentially dispersed lineages at generation 6000 plotted against log10 of scale is shown.
Box plots correspond to the statistics of 20 (exponential) or 12 replicates (all others).
in good agreement with the fractal population structures we report here. The difference between power-law
and exponentially dispersed populations is also demonstrated by self-similarity statistics IC and IC2 (Figure 7),
which are effectively derivatives of D0 and D2, respectively. The IC statistic was suggested by Shaw (1995) and
a very similar statistic but for the factor of subdivision,
the species-level self-similarity parameter a, was recently
suggested by Ostling et al. (2003) as helpful for
characterizing species distribution in ecology. In particular, if IC2 is applied to PL r1.5 dispersed populations, it
is constant over several orders of magnitude of scale and
reveals the highly self-similar structure of those populations (Figure 7). IC2 values for exponentially dispersed
populations are very different in showing a distinctive
peak and no self-similar structure is detected. The selfsimilarity statistics can pick up other patterns besides
self-similarity in the spatial structures. Peaks are a sign of
visual patchiness, they are caused by areas with more
occupied boxes, and the scale immediately greater than
the peak is the scale of greatest patchiness (Cousens
et al. 2004). IC2 may be the most suitable tool if applied
to noisy field data to detect fractal structures or selfsimilarity. The expected graphs may indicate differences stronger than the fractal dimension analysis. In
theory, IC2 values are interchangeable with the fractal
dimension by the relation D2 ¼ 2 1 log2(IC2). However,
it seems difficult to estimate IC2 from the graphs as
plateaus are short for all values of b $ 2. From the PL r1.5
simulations, the value for IC2 can be estimated as 0.36
(Figure 7), implying D2 ¼ 0.52, which is near the value of
0.54 from D2 statistics. However, it seems preferable to
use the IC2 graphs for detecting self-similarity only and
determine the correlation dimension in case self-similarity was found. To test the distinctness of the selfsimilarity statistics, some preliminary simulations with
different median dispersal distances were performed.
The mode of the IC or IC2 curves moved to larger scales
for higher median dispersal distances. However, the
shape of the graph was not changed as long as the mode
still lay within the arena. An exponentially bounded
distribution induced a different pattern from that of
an inverse power-law distribution with b # 2. Even if the
median dispersal distance of the exponential function
was very long range the IC graph still showed a distinctive peak, underlining the potential of this statistic to
discriminate between fat-tailed LDD and exponential
dispersal.
Applying our statistics to field data in the future will
show how robust they are to noise and sparsity of data at
larger scales. Data at many different scales are necessary
for using the new statistics and to detect the different
structures we predict. We therefore suggest that sampling schemes for population genetic analyses in which
the nature of dispersal between subpopulations may be
significant should routinely include as wide a range of
spatial scales as possible. Sampling on a wider range of
spatial scale does not necessarily mean that more samples are needed but that the sampling scheme has to be
optimized for this task.
446
L. U. Wingen, J. K. M. Brown and M. W. Shaw
With all dispersal functions except global dispersal,
an increased genetic dissimilarity with geographic distance was found (Figure 4). Some population studies
include the estimation of mean dispersal distances from
the genetic structure of the population. This strategy is
problematic for power-law dispersal with b # 2 because
mean and variance are not defined for these dispersal
functions. As this is a substantial problem it cannot be
dealt with in the frame of this work.
In contrast to the fractal or self-similar measures, the
common population genetic statistic GST is not very useful for differentiating between different dispersal types
(Figure 8). Patterns of change were rather similar for all
dispersal functions. To apply the statistic to field data to
determine the underlying dispersal function would be
difficult, as clear differences are small and apparent
only at very long scales, at distances from the source
where data will normally be very scarce.
The emergence of fractal or scale-invariant patterns
in simulations of power-law dispersed biological objects
was shown previously by Cannas et al. (2006). They
investigated the expansion of an invading species from a
central focus, such as a newly established weedy species.
Using exponents of 2 , b # 3, the resulting spatial
structure showed a fractal character, but only at some
times and at the border of the main patch. In contrast,
our study showed a fractal pattern of genotype distribution developing from an initially uniform random distribution of genotypes for a population with power-law
dispersal and stable size. Once developed, the fractal
pattern remained present during the whole remaining
time of simulation because mutation in our model acts
as a continual source of new invasions superimposed on
the existing population. Moreover, fractal patterns appeared most strongly with small power law coefficients,
of magnitude ,2.
Fat-tailed dispersal of haploid individuals results in a
fractal distribution of haploid genotypes over several
scales. However, true nonsexual propagation is not very
frequent in nature. Even organisms that use clonal reproduction for the rapid production of many descendants generally have occasional sexual reproduction,
such as many ascomycete and basidiomycete fungi.
The sexual phase leads to an uncoupling of genes and
loss of linkage disequilibrium. We predict, therefore,
that our main conclusions will apply equally to alleles of
a single gene in a sexual population. In particular, we
predict that power-law dispersal will result in a fractal
distribution of alleles of a gene in a sexual population,
just as it does of genotypes in a clonal population.
This study investigates LDD in a simulation arena that
is large enough to reflect the dispersal of wind-blown
small particles like fungal spores at scales ranging
from leaves to continents. Such LDD is observed in nature, e.g., the spread of recently mutated genotypes of
P. striiformis (wheat yellow rust) from the United Kingdom to Denmark, France, and Germany (Brown and
Hovmøller 2002; Hovmøller et al. 2002) or the
regular reinfection of wheat-growing areas in northern
China with wheat yellow rust from more southerly
regions (Brown and Hovmøller 2002). The ability of
the PL r1.5 dispersal to generate a wide-ranging distribution of an individual genotype over the whole arena
was clear. The observations of LDD of wheat yellow rust
are so far in accordance with those of our model. Our
study is consistent with the presence of fractal structure
in natural rust or mildew populations but more data on
several scales than are usually collected for a population
genetic study for confirmation would be needed.
The fractal analysis of the PL r1.5 dispersed populations does not determine a maximum scale of dispersal
as there may be none with this kind of dispersal function. Realistically, the range of dispersal will be finite,
because of processes such as deposition or death, but
nevertheless very large (Shaw et al. 2006). This was not
considered in the present study as the population genetic structure will be dominated by the effective dispersal function at scales well below the arena size.
For larger wind-dispersed particles like seeds, a powerlaw dispersal function with b . 2 will apply (Cannas et al.
2006). However, inverse power-law distribution functions may also be useful to model nonwind LDD, as,
e.g., the dispersal of small birds, tits, which followed
an inverse-square (Cauchy) law (Paradis et al. 2002).
Larger negative power-law exponents than b . 4 may
not be very useful for simulations of biological processes
because the spatial pattern produced cannot be distinguished from that arising from a negative exponential
and thus the less computationally expensive model of
negative exponential dispersal can be chosen.
From the present study it appears to be inappropriate
ever to use exponentially bounded dispersal functions
with a scale parameter much smaller than the size of the
available arena to model biological dispersal processes.
With exponential dispersal alone a lineage of a dispersing clonal organism (or, we propose, an allele in a sexual
organism) will inevitably stay in an area a few orders of
magnitude larger than the median dispersal distance
(Figure 3). In a nongrowing population this leads to
a concentration of each genotype in a single location.
Such populations would be much more prone to extinction in the case of changes in the environment. If
the only means of dispersal is an exponentially bounded
process, any species is in fact evolving as strongly isolated populations, even if it occupies a large homogeneous, suitable habitat. It therefore seems likely that the
majority of organisms have a LDD means of dispersal,
which will have allowed their ancestors to avoid restriction to a limited area. This argument may seem to depend
on the short median dispersal distances we assumed.
However, long-distance (e.g., 105 km) exponential dispersal is inconsistent with all observations of dispersal of
spores, pollen, or seeds. If a mixture of two exponential
functions is used as in Nichols and Hewitt (1994), the
Population Structure Due to Different Types of Dispersal
model still exhibits a distinct scale, albeit the larger one,
with quite distinct isolated patches evolving independently. Assuming a high proportion of long-distance
exponential dispersal in the mixture conflicts with the
lack of such observations in field data, as stated before.
This inappropriateness of exponentially bounded
functions for modeling airborne dispersal processes is
of importance for risk assessment studies that aim at
long-term, macroscale predictions. The dispersal potential is a key uncertainty in predicting the risk of an airborne disease (Yang 2006) and the same should apply
to the spread of pollen of genetically modified plants or
droplet-borne diseases such as foot-and-mouth disease. The
use of exponentially bounded functions for predicting
dispersal of organisms that in fact follows a smallexponent power law in this context is inappropriate
because such models would tend to seriously underestimate risks.
The extensive simulations described here show a
characteristic population genetic pattern generated by
LDD in a homogeneous landscape. This is a simplification of real conditions but is partly met in the modern
type of agriculture that favors monoculture. However,
fragmented habitats or barriers like mountains or oceans
may influence population genetic structure. Extensions
of the present model to accommodate for inhomogeneous landscapes will be necessary to investigate this in
detail. Simulations using long-range exponential functions have shown such LDD resulting in faster bridging
between niches over unsuitable habitats and greater
genetic diversity in remote subpopulations (Le Corre
et al. 1997; Davies et al. 2004; Bialozyt et al. 2006). It is
to be expected that fat-tailed LDD will result in an even
faster niche colonization and a further increase in genetic diversity of remote subpopulations. Predictions on
the presence of fractal structure are more difficult as
this may depend on the niche distribution.
We thank two helpful anonymous reviewers. This research was
supported by the Biotechnology and Biological Sciences Research
Council and was part of the BioExploit project in the European Union
Framework 6 Programme.
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