Journal of
Ecology 2004
92, 747– 757
How far can a hawk’s beard fly? Measuring and modelling
the dispersal of Crepis praemorsa
Blackwell Publishing, Ltd.
OLAV SKARPAAS*‡, ODD E. STABBETORP†, INGEBORG RØNNING* and
THOMAS O. SVENNUNGSEN*
*Department of Biology, University of Oslo, PO Box 1050 Blindern, N-0316 Oslo, Norway, †The Norwegian Institute
of Nature Research, Dronningens gt. 13, PO Box 736 Sentrum, N-0105 Oslo, Norway, and ‡Department of Biology,
The Pennsylvania State University, 208 Mueller Laboratory, University Park, PA 16802, USA
Summary
1 We measured and modelled the dispersal of a wind-dispersed herb, the leafless
hawk’s beard (Crepis praemorsa, Asteraceae), using a combination of measurement
techniques, selected empirical models and mechanistic models originally developed for
trees.
2 Dispersal was measured by releasing individual seeds and by placing seed traps
around an experimentally created point source of seeding plants. Dispersal distances
varied considerably between the experiments. In the seed releases, dispersal distances
were positively related to horizontal wind speed (linear regression, P < 0.001) and,
under favourable conditions, many seeds dispersed over several metres, with a few at
> 30 m.
3 Four empirical models (the inverse power (IP), the negative exponential (NE),
Bullock & Clarke’s mixed model (MIX) and Clark et al.’s 2Dt model) were fitted to the
data. IP and NE models often failed to accommodate the shape of the empirical
distribution of dispersal distances, and the MIX model, although extremely flexible,
tended to overfit the data. The 2Dt model was, however, flexible and realistic in each
experiment. Nevertheless, the parameter values for all of the empirical models varied
dramatically between experiments; no set of parameter values predicted the observed
dispersal distances under all conditions.
4 Two mechanistic models (Greene & Johnson’s analytical model (GJ) and Nathan
et al.’s individual-based simulation model (NSN)), originally developed for trees, were
parameterized using independent data and parameter values from the literature.
Although the NSN model provided a poor fit for the seed trap experiment, it performed
almost as well as the best empirical models in the seed release experiments. Its predictions were further improved by including convection, and predicted dispersal > 30 m
corresponded closely with our observations. The prediction of low (< 1%) dispersal
> 200 m requires further validation.
5 We conclude that dispersal models for wind-dispersed trees can be adapted for herbs
with different dimensions and diaspore characteristics. Mechanistic simulations are
superior because they are robust to environmental heterogeneity, as well as being
informative in terms of understanding and predicting the effects of species characteristics and ecological factors on dispersal distances. Future empirical studies are needed
at a wide range of environmental conditions, with careful measurement of conditions
such as the strength and variability of horizontal and vertical wind speeds.
Key-words: Asteraceae, dispersal experiments, dispersal models, leafless hawk’s beard
(Crepis praemorsa), wind dispersal
Journal of Ecology (2004) 92, 747–757
© 2004 British
Ecological Society
Present address and correspondence: Olav Skarpaas, Department of Biology, The Pennsylvania State University, 208 Mueller
Laboratory, University Park, PA 16802, USA (tel. + 1 814 865 7912; fax + 1 814 865 9131; e-mail [email protected]).
748
O. Skarpaas et al.
© 2004 British
Ecological Society,
Journal of Ecology,
92, 747–757
Introduction
With the rise of spatial ecology ( Tilman & Kareiva
1997), it has become clear that dispersal is extremely
important in many biological contexts. The shape of
the dispersal kernel is a critical determinant of spatial
spread (e.g. Kot et al. 1996; Neubert & Caswell 2000;
Clark et al. 2001), and it affects interactions (Laterra &
Solbrig 2001) and large-scale biogeographical patterns
(Cain et al. 1998). Still, there is a lack of studies combining
empirical measurements and mathematical modelling
of dispersal for plants. This is perhaps the greatest
obstacle to progress in plant metapopulation biology
(Husband & Barrett 1996).
Wind dispersal is a well-studied mode of dispersal
in plants, both in terms of empirical and theoretical
studies (e.g. Nathan & Muller-Landau 2000; Greene &
Calogeropoulos 2002). However, because of economic
interests in forestry, much more attention has been given
to the spatial redistribution of wind-dispersed trees than
to herbs and shrubs (Greene & Calogeropoulos 2002).
In this paper we measure dispersal and compare ‘classical’ and recent promising empirical and mechanistic
models for the leafless hawk’s beard (Crepis praemorsa,
Asteraceae). This perennial herb inhabits meadows in a
fragmented and changing agricultural landscape where
dispersal is critical for regional persistence (Skarpaas 2003).
A number of dispersal models have been developed
(e.g. Turchin 1998; Nathan & Muller-Landau 2000;
Greene & Calogeropoulos 2002; Levin et al. 2003), but
the empirical data to test these models are still lacking
for many species, including C. praemorsa. Dispersal
distances and seed shadows have previously been measured using either of two different general approaches
(seed traps in the vicinity of reproductive individuals or
following individual seeds), but we used both approaches,
as they have complementary strengths and weaknesses.
To obtain a large sample size when following individual seeds is time-consuming and seed shadows have
therefore often been measured using seed traps. However, a generic problem with such experiments has been
the failure to measure long-distance dispersal. While
some contend that this is because most seeds do not
disperse very far (Cain et al. 2000), it may be because we
have not looked carefully enough (Greene & Calogeropoulos 2002). The area over which seeds are spread
from a seed source increases with increasing distance
from the source by a factor 2πr (unless there is a strong
directional bias), but the seed trap area is often held
constant. This problem can be alleviated by maintaining the proportion of the circumference sampled at
increasing distances ( Bullock & Clarke 2000), but this
approach becomes practically impossible as distance
from the source and, hence, area to be sampled continues to increase.
The problem of sampling biased towards shorter
distances is reduced by following individual seeds. This
also allows wind speeds and other factors (e.g. release
height) of presumed importance to dispersal distance to
be measured independently for each seed. This facilitates
statistical testing of the effects on dispersal distance of
different factors, such as wind speed and release height.
Mathematical models of seed dispersal have been
developed along two main lines (Nathan & Muller-Landau
2000): (i) empirical (or phenomenological) models and
(ii) mechanistic models. Empirical models ignore
dispersal mechanisms, they are simply functions fitted
to observed seed shadows. Mechanistic models on
the other hand, are formulated with the dispersal
mechanisms in mind, and can be parameterized using
independent data on the dispersal vector and medium
(e.g. wind velocities or animal movement; Turchin 1998).
Two much-used empirical models, the inverse power
model and the negative exponential model, are both
mathematically simple, but in most cases neither of
them fit the shape of empirical dispersal kernels at all
distances (Clark et al. 1999; Bullock & Clarke 2000).
Clark et al. (1999) and Bullock & Clarke (2000) proposed
two alternative models, the 2Dt (two-dimensional
student’s t) and the mixed model (a mix of the negative
exponential and inverse power), respectively, that seem
sufficiently flexible to fit empirical dispersal kernels
both near and far from the source.
However, when dispersal distances depend strongly
on factors that are variable in space and time, there is a
limit to the validity of empirical models. Strictly speaking,
empirical models cannot be used in other situations
than the ones they were parameterized for. Dispersal
studies of plumed seeds suggest that there is considerable variation in dispersal distances depending on, among
other things, seed weight, diaspore morphology,
horizontal wind speed, updrafts and turbulence (e.g.
Burrows 1973; Okubo & Levin 1989; Greene & Johnson
1992a; Andersen 1993; Soons 2003; Tackenberg 2003).
To account for the effects of such factors, mechanistic
models are needed.
Most mechanistic models are analytically tractable
and give completely specified dispersal curves (probability density functions), but often at the cost of
making unrealistic assumptions. In reviewing a number
of mechanistic models, Nathan et al. (2001, p. 376)
concluded that because of problems with model structure and assumptions, ‘analytical models are unlikely
to accomplish the objectives of gaining better understanding and predictive ability’ (see also Bullock &
Clarke 2000). As an alternative, Nathan et al. (2001)
proposed a simulation approach in which model tractability is sacrificed to avoid unrealistic assumptions.
The dispersal curve is defined as a function of random
variables with known distributions from which random
values are drawn to produce an estimate of the total
dispersal curve by simulation.
The mechanistic simulation approach has recently
been used with success for a variety of different species,
including herbs as well as trees (Nathan et al. 2001;
Soons 2003; Tackenberg 2003). The empirical 2Dt
model has been tested and found useful for a number of
tree species (Clark et al. 1999), and the mixed model has
749
Wind dispersal of
Crepis praemorsa
been applied to small-seeded shrubs (Bullock & Clarke
2000), but as far as we know these empirical models remain
to be tested for herbs with plumed seeds. In this study
we test these models for Crepis praemorsa and contrast
them with ‘classical’ dispersal models, using data from
seed release experiments and a seed trap experiment.
Materials and methods
 
The leafless hawk’s beard (Crepis praemorsa (L.) Tausch)
is a perennial herb confined to natural and semi-natural
grasslands in central and northern Eurasia (Hultén &
Fries 1986; Elven 1994). It is rare and declining in several countries in north-western Europe (Stoltze & Pihl
1998; DN 1999; Rassi et al. 2001), presumably because
of habitat loss as a result of landscape changes following the industrialization of agriculture and cessation of
traditional land use (Stoltze & Pihl 1998; Rassi et al.
2001). In the core areas for this species in SE Norway,
its primary habitats in traditional agricultural landscapes are highly fragmented and continuously changing (Framstad & Lid 1998; Norderhaug 1999). In this
setting, dispersal is critical for regional persistence.
Every year individual plants produce a basal leaf
rosette in which most of the biomass is concentrated.
Large rosettes reproduce both sexually and asexually
(Kemppainen et al. 1991), but long-distance dispersal
and colonization can only be achieved by seed. Sexually reproducing plants normally develop one upright
leafless flowering stem with several light yellow flower
heads. All of the flowers are ligulate and produce monomorphic achenes (in contrast to some other composites).
The achenes are small (about 4 mm long) and light and
carry a ring of pappus hairs. In the following we use the
term ‘seed’ to refer to the achene-pappus unit, unless
reference is made to a specific part of the diaspore.
 
© 2004 British
Ecological Society,
Journal of Ecology,
92, 747–757
Dispersal distances were measured by carrying out two
single seed release experiments and a seed trap experiment
in three different locations in SE Norway in July 2001.
Seeds for single seed releases were selected at random
from a pool of seeds collected from 10 individuals at
Solheim, Ringsaker. The seed releases were carried out
at a coastal meadow at Feskjær ( Tønsberg municipality, Vestfold) and a mowed lawn at Helgøya (Ringsaker
municipality, Hedmark). We refer to these experiments
as seed release 1 and 2, respectively. Both sites were
selected because they were flat and open areas with few
objects that disturbed the wind patterns. Seed release 1
was carried out in the afternoon (17.00–18.00) during
a period with a light afternoon breeze, whereas seed
release 2 was carried out during a calm period in the
morning (10.00 –12.00). The numbers of seeds released
in the two experiments were 52 and 50, respectively.
The diaspores were released manually one by one from
normal plant height (38 – 61 cm in release 1, 50 cm in
release 2) at the same point. The achenes were held
between fingers with the pappus hairs up, released at
random (intervals of 30 seconds to 2 minutes), and
followed until landed or lost from sight at the end of
the distance measuring tape (30 m). For every seed, we
recorded height of release, wind speed and distance
travelled. Wind speed was measured at the moment of
release using a Young 05103 propeller-type anemometer
mounted 2 m above ground.
In the seed trap experiment, we used a similar
approach to Bullock & Clarke (2000) in that we created
a point seed source and placed seed traps in sectors (a
constant proportion of each annulus) at increasing
distance intervals from the source. However, we used
adhesive tape rather than pots as it was then easier to
cover a larger total trap area: this approach was tested
in a pilot study for Tussilago farfara (Skarpaas &
Stabbetorp 2003). In the present study, 101 flowering
plants of C. praemorsa were collected from the largest
known population at Nes, Ringsaker, at the time of
seed ripening. The plants were excavated with the topsoil and root system as intact as possible, and placed in
two 40 × 70 cm plastic boxes located in the centre of a
recently cut hay field at the organic farm Alm Østre
(Stange municipality, Hedmark). The surrounding
field was divided into 32 sectors of equal size (11.25°)
delineated by cords strung between metal poles in the
ground and numbered clockwise from the north. Seed
traps consisting of 10-cm-wide poly ethylene adhesive
tape (Stockvis tapes) were placed horizontally around
the entire circumference at 0.7, 1.0, 1.3, 1.8, 2.4, 3.2,
4.3, 5.7, 7.5 and 10 m from the centre of the source.
Thus, sampling intensity at each distance was maintained by increasing the size of the traps (i.e. the length
of the adhesive tape). Additional traps were placed
along the direction of dominant local wind directions
at this time of year (T. Sund, personal communication),
with tape laid at 13, 17, 22, 28, 35 and 46 m in six
sectors (3, 7, 15, 19, 23 and 31), and at 52, 62, 73, 85 and
100 m in the diametrically opposite sectors 3 and 19.
The total trap area was 57.4 m2, compared with 16.3 m2
in Bullock & Clarke (2000).
The experiment was terminated after 9 days, when
most of the seeds had been dispersed. Unfortunately,
because seed ripening happened suddenly, and earlier
than previous years, and the dispersal period was shorter
than expected, the anemometer was not in place during
the experiment. Wind measurements were therefore
taken approximately 100 m away from the transplanted
plants during the two following weeks at the same
height as in the seed releases. Measurements were made
every hour and stored in a data logger.
The total number of seeds released (Q) in the
seed trap experiment was estimated by counting seed
attachment points in 50 flower heads (5 from each of 10
plants), the number of flower heads per plant and the
proportion of flower heads that had released the seeds
by the termination of the experiment.
750
O. Skarpaas et al.
Table 1 Empirical and mechanistic dispersal models tested for Crepis praemorsa. Symbols are defined in Table 2. Note that the
formulation of the dispersal kernels in this table differs from formulations in the references. Because the dispersal kernel is
considered separately from Q (equation 1), our a’s are not directly comparable with the corresponding parameters in Bullock &
Clarke (2000). Greene & Johnson’s (1989) model (GJ) is expressed in terms of seed density (obtained by dividing Greene &
Johnson’s equation 5 by 2πr). The formulation of the NSN model was obtained by combining Nathan et al.’s equations 10 and 11
Model
Dispersal kernel f(r)
Empirical
IP
NE
MIX
a1 r
2Dt
References
− b1
a2 exp( −b2r )
a3 exp( −b3 r ) + a4 r
Bullock & Clarke (2000)
Clark et al. (1999), Bullock & Clarke (2000)
Bullock & Clarke (2000)
− b4
p

r2 
π u 1 + 
u 

Clark et al. (1999)
p+1
Mechanistic
GJ
NSN
  ln(rF /HU )  
g

exp − 
2 π r σu 2 π
  σu 2  


H − d 
um  [ H − d ] ln 
 + z0 
z0 
g (r )



,r=
A
z − d 
F ln  m

 z0 
1
 
In this paper we express each dispersal model as a onedimensional seed shadow (s) (seeds m−2):
s = Q f (r)
eqn 1
where Q is the numbers of seeds released, f(r) is the
dispersal kernel (m−2) and r is the radial distance (m)
from the source. The dispersal kernel is the core of
any dispersal model. It can take many forms (e.g. Clark
et al. 1999; Nathan & Muller-Landau 2000; Greene &
Calogeropoulos 2002). The models considered in this
paper are listed in Table 1, and the symbols are defined
in Table 2.
 
We tested four empirical dispersal models for Crepis
praemorsa (Table 1): (i) the inverse power model (IP);
(ii) the negative exponential model (NE); (iii) the mixed
model (MIX); and (iv) the 2Dt model. These models
are developed and described in detail in the literature.
We follow the accounts of Clark et al. (1999) and
Bullock & Clarke (2000), but modify the notation to
make it consistent.
Following Bullock & Clarke (2000) we fitted the
empirical models expressed in terms of counts rather
than densities. The expected seed count (c) is taken to be:
c = As = AQ f (r)
© 2004 British
Ecological Society,
Journal of Ecology,
92, 747–757
Greene & Johnson (1989)
2
eqn 2
where A is the trap area and f is evaluated at r, the midpoint of the trap. Using the respective dispersal kernels
for NE and IP (Table 1) in equation 2, these models
were fitted to data from the dispersal experiments by
Nathan et al. (2001)
generalized linear modelling via maximum likelihood
(the function ‘glm’ in R; The R Development Core
Team 2003) using a log link function and assuming a
Poisson error structure. The product AQ was treated as
an offset variable (i.e. no coefficient estimated). In the
seed trap experiment c is the total seed count in the
traps at distance r, A is the total seed trap area at this
distance and Q is the estimated number of seeds
released by the source plants. To make the observations
comparable across the three experiments we used the
distances of the seed traps as midpoints of histogram
classes for the seed release experiments (we also added
several classes below 0.7 m). Thus, in the seed release
experiments c for a given distance ri is the total count
of seeds falling between rmin = ri − (ri − ri−1)/2 and rmax =
2
2
ri + (ri+1 − ri)/2, and A = 2π( rmax − rmin).
To fit the MIX and 2Dt models we used numerical
maximum likelihood estimation assuming a Poisson
error distribution for seed counts (e.g. Clark et al. 1999).
The Poisson likelihood of the model, given the data and
parameters is:
n
L=
∏
i =1
c
çi i exp ( −çi )
ci !
eqn 3
where ci is the observed seed count in seed trap (or area)
i and çi is the estimated seed count using equation 2.
The negative log-likelihood (–ln L) was numerically
minimized using simulated annealing by the function
‘optim’ in R (The R Development Core Team 2003).
The values for p and u at the minimum of this function
are the maximum likelihood parameter estimates.
 
Two different mechanistic models were considered for
Crepis praemorsa (Table 1): (i) Greene & Johnson’s
751
Wind dispersal of
Crepis praemorsa
Table 2 Definitions of symbols for variables and parameters
of the dispersal models in Table 1
Symbol
Definition (unit)
s
c
r
f (r)
ai
Seed shadow (seeds m−2)
Seed count (seeds)
Radial distance from seed source (m)
Dispersal kernel (m−2)
Scaling parameter in the IP,
NE and MIX models
Curvature parameter in the IP,
NE and MIX models
Parameter in the 2Dt model (m2)
Parameter in the 2Dt model
Seed falling velocity (m second−1)
Seed release height (m)
Displacement height (m)
Roughness length (m)
Measurement height for wind speed (m)
Horizontal wind speed (m second−1)
at measurement height
Horizontal wind speed (m second−1)
between H and d
Geometric mean horizontal wind speed
(m second−1) between H and d
Variance of ln U (m second−1)
Vertical wind speed (m second−1)
Frequency distribution
Area of seed trap or ground (m2)
bi
u
p
F
H
d
z0
zm
Um
U
Ug
σU
W
g(r)
A
differential equation (GJ, Greene & Johnson 1989);
and (ii) Nathan et al.’s mechanistic simulation model
(NSN, Nathan et al. 2001). Both of these models are
derived from the ballistic equation (Greene & Johnson
1989; Nathan et al. 2001):
r=
© 2004 British
Ecological Society,
Journal of Ecology,
92, 747–757
HU
,
F
eqn 4
where H is release height, U is horizontal wind speed
and F is falling velocity. The GJ model is derived
assuming that U is lognormal, seeds detach randomly
with respect to wind velocity, and F and H are constant.
This leads to a relatively simple dispersal kernel (although
it is complex compared with the empirical models;
Table 1). In the NSN model these simplifying assumptions are relaxed. In this model H, F and U are treated
as random variables (see below).
Falling velocity was measured for 40 seeds collected
from eight randomly selected individuals (five seeds
each) in natural populations in Ringsaker (Hedmark)
and Nes (Akershus). The descent of seeds was timed
in an airtight tube of known length (Sheldon & Burrows
1973; Andersen 1992). We used a digital stopwatch
and a 1-m-long cardboard poster tube closed at the
bottom by a glass jar and at the top by the upper
half of a plastic bottle. This simple design allowed the
seeds to accelerate to the terminal falling velocity before
the timing of the descent through the tube.
We tested two versions of the mechanistic models, with
and without vertical wind. Vertical wind was incorporated in the models by replacing the falling velocity F
with realized falling velocity FW = F –W, where W is
vertical wind speed. The models with vertical wind
included are denoted GJW and NSN W. We made no
measurements of vertical wind, but assumed the same
distribution as Nathan et al. (2001, Nir’Ezyon site).
Values for the other parameters were taken from the
dispersal experiments. In the seed trap experiment mean
release height H was calculated as the mean height of
the inflorescence midpoint of 10 plants.
For the GJ model wind horizontal velocities at 2 m
were transformed to wind velocities between H and the
ground using the formula:
U =
ln [( H − d )/ z0 ] − 1
Um
ln [( z m − d )/ z0 ]
eqn 5
where Um is the wind speed at measurement height zm
(2 m) and z0 and d are two parameters shaping the
wind profile above rough surfaces, such as vegetation:
z0 (roughness length) is the scale of the total magnitude
of the shear forces acting on the surface, whereas d (displacement length) scales the distribution of these forces
in the surface canopy (Wieringa 1993). Together, z0 and
d shift the origin of the logarithmic wind profile from
the ground level to z0 + d, i.e. horizontal wind speed is
0 at the height z0 + d. Equation 5 is a modified version
of Greene & Johnson’s (1989) equation 7, which includes
z0 but not d. This may be appropriate for trees, but d
must be taken into account when wind profiles are
modelled close above the vegetation surface (Wieringa
1993). The values for the roughness parameters were
obtained from Wieringa’s (1993) compilation of representative values for long grass (seed release 1), short grass
(seed release 2) and stubble (seed trap experiment).
Note that the logarithmic wind profile is an integral
part of the NSN model, and hence the measured wind
speeds can be used directly without transformation in
this model.
The assumed distributions of parameters were tested
for the empirical data using the Kolmogorov-Smirnov
test (Sokal & Rohlf 1995) as implemented in S-plus
(Venables & Ripley 1997). The horizontal wind speed
distributions were significantly different from lognormal
in seed release experiment 1 and the seed trap experiment,
but not significantly different from normal in any of the
experiments. We therefore assumed a normal distribution of wind speeds in the simulation of the NSN models.
We used 100 000 simulations of the NSN models to
estimate the seed shadow in each of the three dispersal
experiments. The simulations were carried out using
scripts written specifically for this purpose in R (The R
Development Core Team 2003). For every single seed
random values for the parameters were drawn from the
specified probability distributions corresponding to
empirical distributions or obtained from the literature
(see Results). Some of the parameter distributions were
sufficiently wide to produce unrealistic values, such as
negative horizontal wind speed and negative realized
falling velocity, both of which give negative dispersal
distances. Therefore, we imposed the following constraints: U > 0, H > d and F > W. In the simulations we
752
O. Skarpaas et al.
implemented the constraints (as did Nathan et al.
2001) by drawing new values for the parameters (both
parameters in the latter case) until the inequalities were
satisfied.
 
The models were evaluated using the likelihood as
an indicator of overall fit (Clark et al. 1999). For all
models we calculated the Poisson likelihood L for seed
counts (equation 3), and compared the models using
–ln L. The model with the lowest –ln L is the best in
terms of overall fit. This measure conceals variability in
the fit for different parts of the seed shadow (e.g. shortdistance vs. long-distance dispersal). Therefore, we also
evaluated the models graphically.
Results
 
Observed dispersal distances in the three dispersal
experiments ranged from 3 cm (the minimum in seed
release 2) to > 30 m in seed release 1. The two seeds that
dispersed > 30 m were caught by updrafts and passed
out of sight several metres above the ground at the end
of the measuring tape (30 m). Dispersal distances were
highly right skewed in all three experiments, but the
median, mean and extreme values differed considerably
among the experiments ( Table 3), being consistently
highest in seed release 1, lowest in seed release 2 and
intermediate, albeit closer to release 2, in the seed trap
experiment. The distributions of horizontal wind speeds
were again higher in release 1 than release 2, with seed
trap values similar to release 1 (Table 3).
In the seed release experiments, dispersal distance
(ln-transformed) was positively related to horizontal
wind speed (linear regression, ln r = − 1.055 + 1.746U,
P < 0.001 for the wind coefficient) when the data from
the two experiments were pooled. When analysed separately, the effect of horizontal wind speed was significant in release 2 (ln r = − 1.297 + 1.723U, P < 0.001)
and positive, but not statistically significant, in release
Table 3 Summary of dispersal distances and wind speeds in
the three dispersal experiments for Crepis praemorsa
Release 1
© 2004 British
Ecological Society,
Journal of Ecology,
92, 747–757
Release 2
Trap
Dispersal distances r (m)
Minimum
0.65
Median
1.99
Mean
3.51
Maximum
> 30.00
0.03
0.57
0.86
3.81
0.70*
0.70
1.03
4.30
Wind speeds U (m second−1)
Minimum
0.38
Median
0.93
Mean
0.95
Maximum
1.74
0.01
0.40
0.41
1.49
0.09
0.88
0.90
2.04
*No seed traps were located closer to the source than 0.7 m.
1 (ln r = − 0.129 + 0.261U +1.208H, P = 0.310 and
0.456 for U and H, respectively).
In the seed trap experiment the estimated mean seed
production per flower head was 24.32 (SD = 4.49, n =
50) and the mean number of empty flower heads per
plant was 7.30 (SD = 4.24, n = 10). This gives an estimated total of 17 931 (SD = 11 106) seeds released
during the experiment for the 101 plants but, despite
the extensive sampling effort, only 168 seeds were caught
in the traps. A large number of seeds were observed
among the rosettes of the source plants, i.e. within
0.7 m of their centre.
 
The parameters of the fitted empirical models differed
greatly among dispersal experiments (Table 4), reflecting the differences in observed dispersal distances
(Table 3). All of the models fit the observations fairly
well within a certain range in all of the dispersal experiments (Fig. 1). The IP model tended to over-predict
seed densities at short and long distances and underpredict densities at intermediate distances. The NE
model captured the shape of the observed curve over
a greater range, but tended to under-predict longdistance dispersal. The MIX model was the most flexible
of the empirical models, following the data points closely
in all three studies. This resulted in over-fitted curves
with a tendency to predict fat tails (Fig. 1). The 2Dt
was almost as flexible as the MIX model but avoided
overfitting. It was intermediate in shape between the IP
and NE models in the seed release experiments and
approached the IP model in the seed trap experiment.
 
There were considerable differences among the mechanistic models (parameters summarized in Table 5),
and between the observed and predicted dispersal
curves in the different dispersal experiments (Fig. 2).
In seed release 1 the GJ and GJW models correctly
predicted low seed densities at short distances, but failed
Table 4 Estimates of empirical dispersal model parameters
for the three dispersal experiments for Crepis praemorsa. See
Tables 1 and 2 for model formulations and notation
Model
Parameter
Release 1
Release 2
Trap
IP
a1
b1
0.009
1.795
0.010
2.114
0.002
3.434
NE
a2
b2
0.022
0.540
0.398
2.280
0.024
2.418
MIX
a3
b3
a4
b4
0.053
0.832
0.000
0.307
0.372
2.325
0.001
2.297
0.024
2.418
0.024
2.418
2Dt
p
u
0.114
0.797
0.864
0.271
0.718
0.001
753
Wind dispersal of
Crepis praemorsa
© 2004 British
Ecological Society,
Journal of Ecology,
92, 747–757
Fig. 1 Observed and fitted empirical dispersal kernels for
Crepis praemorsa in seed release experiment 1 (upper panel),
seed release experiment 2 (middle panel) and seed trap
experiment (lower panel). Observed values > 0 are indicated
by filled circles, and zeros are indicated by triangles (note that
both axes are on a natural log scale). Fitted empirical models
(lines): IP (dashed), NE (dotted), MIX (dashed-dotted), and
2Dt (solid).
Fig. 2 Observed and predicted dispersal kernels using mechanistic models for Crepis praemorsa in seed release experiment
1 (upper panel), seed release experiment 2 (middle panel) and
seed trap experiment (lower panel). Observed values as in
Fig. 1. Mechanistic models (lines): GJ (dotted), GJW (dasheddotted), NSN (dashed), and NSNW (solid).
754
O. Skarpaas et al.
Table 5 Mean (SD) parameter values used in mechanistic models in the three dispersal experiments for Crepis praemorsa. For
parameters estimated in the laboratory or obtained from the literature, the same values were used in all three experiments. See
Tables 1 and 2 for model formulations and notation
Parameter
Release 1
Release 2
Trap
Distribution
Source
Species
F (m second−1)
H (m)
0.25 (0.07)LN
0.52 (0.05)N
0.25 (0.07)LN
0.50 (0.00)
0.25 (0.07)LN
0.55 (0.15)N
Log-normal
Normal
Laboratory measurement
Field measurement
Ecological
Um (m second−1)
U (m second−1)
W (m second−1)
d (m)
z 0 (m)
2.78 (0.89)N,LN
0.97 (0.31)N,LN
0.10 (0.35)
0.10
0.04
0.75 (0.56)N
0.38 (0.29)N
0.10 (0.35)
0.00
0.01
2.46 (1.08)N
1.54 (0.67)N
0.10 (0.35)
0.10
0.04
Normal
Normal
Normal
Constant
Constant
Field measurement
Field measurement
Nathan et al. (2001)
Wieringa (1993)
Wieringa (1993)
N
Not significantly different from the normal distribution (Kolmogorov-Smirnov goodness-of-fit test, P > 0.05).
Not significantly different from the log-normal distribution (Kolmogorov-Smirnov goodness-of-fit test, P > 0.05).
LN
Table 6 Negative log-likelihood (−ln L) of empirical and
mechanistic models of dispersal in the three dispersal
experiments for Crepis praemorsa. In some cases, likelihoods
were indistinguishable from zero, making −ln L infinite (∞)
Model
Empirical
IP
NE
MIX
2Dt
Mechanistic
GJ
GJW
NSN
NSNW
Release 1
Release 2
Trap
73.10
52.52
26.77
47.04
51.80
27.73
25.85
41.22
19.48
33.52
18.28
19.48
524.90
549.45
∞
30.34
46.56
61.11
37.33
31.55
∞
∞
∞
∞
to predict long-distance dispersal. In seed release 2 both
models under-predicted both short- and long-distance
dispersal. The NSN and NSNW models were closer to
the observed values at all distances from the source in
both seed release experiments. In the seed trap experiment all four of the mechanistic models failed (Fig. 2).
There was an effect of introducing vertical wind in
both the GJ model and the NSN model, but it did not
consistently improve model fit (Fig. 2, Table 6). In the
seed releases, introducing vertical wind improved the
fit of the NSN model, but decreased the fit of the GJ
model. For the seed trap experiment, graphical evaluation (Fig. 2) suggests that the NSNW model did better
than the NSN model, whereas the GJW did worse than
the GJ model.
 
© 2004 British
Ecological Society,
Journal of Ecology,
92, 747–757
In terms of overall fit (–ln L), the MIX model was the
best-fitting model in all three experiments (Table 6).
The NE model fitted reasonably well in seed release 2
and the IP and 2Dt models in the trap experiment. The
2Dt model also appeared to fit the seed release data
(Fig. 1), although the log likelihoods were well above
that of the MIX model (Table 6). Of the mechanistic
models, the NSN models fitted far better than the GJ
models. In the seed trap experiment all of the mechanistic models failed, but in seed release 1 the NSNW
model outperformed all of the empirical models but
the MIX model, and in seed release 2 both NSN models
fitted almost as well as the best-fitting empirical models
(Table 6).
Discussion
The dispersal distances measured for C. praemorsa in
this study are comparable with those of other plants
with plumed seeds (e.g. Willson 1993; Bullock & Clarke
2000). The falling velocity is relatively low compared
with other Asteraceae (Andersen 1992), indicating that
C. praemorsa has a relatively high potential for exploiting favourable wind conditions. In a study of another
composite with a high dispersal capacity, Cirsium
vulgare (Klinkhamer et al. 1988), just over 10% of the
dispersed seeds were caught by updrafts and carried out
of the study area (more than 32 m). In the present study
4% of the seeds were carried away by updrafts in seed
release 1, but no seeds were dispersed that far in release
2, which was carried out on a calm day, with mean wind
speed between H and d only 0.4 m second−1. Detailed
local wind measurements from Nes (10–20 km NW of
the seed trap experiment site) suggest that mean wind
speeds at this height are of the order of 0.5–0.8 m
second −1 in June–July (Utaaker 1963; wind speeds
transformed from measurements at zm = 3 m using
equation 5). Both higher mean wind speed (more than
twice as high) and convection probably played a role in
producing longer dispersal distances in release 1, which
was carried out in the afternoon (around 17.00–18.00,
vs. 10.00–12.00 for release 2), by which time the ground
had warmed up. Timing differences also explain the
greater improvement of adding vertical wind to the NSN
model in release 1 compared with release 2.
The dispersal models tested for C. praemorsa fit the
observations to differing degrees. The empirical models
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Wind dispersal of
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© 2004 British
Ecological Society,
Journal of Ecology,
92, 747–757
are flexible enough for each to give a reasonable fit in
some situations. However, the NE model and the IP
model each have a serious weakness (Clark et al. 1999).
While the NE may fit the observations near the source,
it usually under-predicts long-distance dispersal, i.e. it
is not sufficiently ‘fat-tailed’. The IP model can accommodate a fat tail but does not fit observations near the
source as it tends to infinity at low dispersal distances,
and does not therefore represent a proper density function. The MIX model, produced by combining the NE
and IP models (Bullock & Clarke 2000), is indeed more
flexible, and can accommodate a fat tail and provide
better fit at shorter distances, but it does so at the cost
of an increase in the number of parameters and a possibility of overfitting the data. It is also unfortunate
that the inverse power component that makes the tail
fat also dictates the shape of the curve at short distances
(Fig. 1). This is not a problem in the 2Dt model, which
captured the shape of the dispersal kernel, both near
and far, better than the IP and NE models in all three
experiments. The flexibility of the 2Dt model comes at
the cost of an increase in model complexity, but it does
not increase the number of parameters, and has further
properties that make it preferable to most other empirical models (Clark et al. 1999).
The greatest weakness of the 2Dt model (and the
other empirical models) is that one set of parameter
values is not representative for all conditions, and hence
the model must be reparameterized to reflect realistic
variation in dispersal conditions in new study areas.
In practice, this means measuring dispersal distances
for representative conditions in every study area of
interest. The mechanistic models, on the other hand, are
parameterized using independent data on species characteristics (falling velocity, seed release height), for
which we know the distributions, and ecological conditions (wind, surface roughness), for which data can
be obtained from meteorological stations, simple field
observations and the literature. Of the mechanistic
models tested in this study, the NSN models (Nathan
et al. 2001) performed on a par with the best empirical
models in the seed release experiments and always
outperformed the GJ models (Greene & Johnson
1989).
An underlying assumption in the GJ models is that
horizontal wind speeds are lognormally distributed. In
our experiments the distributions of wind speeds were
closer to normal than lognormal (although not significantly different from lognormal in seed release 1). This
was incorporated in the NSN models simply by changing the distribution from which random horizontal
wind speeds were drawn (with the additional constraint
that U > 0). In the GJ models, on the other hand, the
lognormal distribution is an integral part that cannot
be replaced without making fundamental changes in
the model structure. In an otherwise correct model (i.e.
‘true’ model structure and parameter values), the consequences of assuming a log-normal distribution when
the true distribution is normal are under-prediction of
modal, median and short-distance dispersal distances
and over-prediction of long-distance dispersal. While
the GJ models under-predict short-distance dispersal in
our experiments, they also under-predict long-distance
dispersal, but over-predict median dispersal distance.
Hence, the lognormal assumption is not sufficient to
explain the differences among the models.
A second, and perhaps more important difference
between the GJ and NSN models is that H, F and W
(when applicable) are assumed to be constant in the GJ
model but random variables in the NSN models: in
other words, only the NSN models incorporate the
variance in these quantities. Increasing variance in any of
these parameters flattens the dispersal kernel, increasing both short- and long-distance dispersal. Greene &
Johnson (1989) suggest a way to incorporate these variances in their model (see also Bullock & Clarke 2000),
and this would probably contribute to a better fit, particularly in seed release 1. However, the shape of the
dispersal kernel in the GJ models is still constrained by
the fundamental assumption regarding the distribution of horizontal wind speeds. The simulation framework of the NSN models is much more flexible in terms
of incorporating different probability distributions for
different parameters, and hence more flexible in terms
of the shape of the dispersal kernel.
The effect on model fit of the incorporation of vertical wind in the mechanistic models also depended on
the consideration of variance. Incorporating vertical
wind improved the fit of the NSN model, in which variance was considered in addition to mean, but it did not
improve the fit of the GJ model, in which only the mean
vertical wind speed was incorporated. In the GJW model
the mode of the dispersal kernel was shifted towards
greater dispersal distances, but the range of predicted
distances (the width of the curve) remained the same as
in the GJ model. In contrast, the NSNW model predicts
a wider range of dispersal distances, and the modal dispersal distance is shorter than in the NSN model. This
gives an overall better fit to the data. This is in accordance with several other studies suggesting the importance of considering convection and variability of wind
speeds for proper characterization of dispersal distance
distributions for plumed seeds (Burrows 1973; Soons
2003; Tackenberg 2003)
The predictions of the mechanistic models, in
particular the NSN model, were impressive for the
seed release experiments, but equally disappointing for
the seed trap experiment. Assuming either a threshold
for seed abscission or an area source rather than a
point source would make the models predict longer
dispersal distances (Greene & Johnson 1992b; Greene
& Calogeropoulos 2002). However, in the seed trap
experiment the observed distances were shorter than
predicted, so we have to look for alternative explanations for the failure of the mechanistic models.
Seeds may have been lost because of deteriorating
glue on the tape traps and rain near the end of the
experiment. Furthermore, our measurements of wind
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© 2004 British
Ecological Society,
Journal of Ecology,
92, 747–757
speeds were carried out in the 2 weeks after the experiment, and this period had a mean wind speed of 0.9 m
second−1, which is higher than usual for the area (Utaaker
1963). However, these experimental weaknesses do not
seem to be sufficient to explain the discrepancy as, even
if we assume 90% seed loss and wind conditions as for
seed release 2, i.e. unusually calm for the area, the models
over-predict seed density at most distances.
A more likely explanation for the discrepancy is a
misspecification of the wind profile. To approximate a
point source the plants were placed close together in
the plastic trays (approximately 180 plants m−2). This
may have led to lower wind speeds within the source
(i.e. below the vegetation canopy; Wieringa 1993) and
turbulent flow around the seed source (Okubo & Levin
2001). In either case, the assumption of a logarithmic
wind profile is most likely invalidated. The net consequences of such wind conditions may be reduced
dispersal distances, and perhaps the observed tendency
for most seeds to end up within the source.
This study shows that recent developments in dispersal modelling for wind-dispersed trees and shrubs can
be adapted to a herb where dimensions and diaspore
characteristics differ from the species for which the
models were developed. This applies to empirical as
well as mechanistic models. Although the MIX model
is the most flexible of the empirical models tested in this
study, the 2Dt model seems to be most realistic. In spatial population dynamical modelling dispersal kernels
that are proper density functions, such as the 2Dt, are
easy to work with (Clark et al. 1999). However, the
mechanistic NSN models fit the data almost as well as
the best empirical models despite the fact that they are
parameterized using independent data. In contrast to
the empirical models, the individual-based simulation
approach of Nathan et al. (2001) is robust to measurable environmental heterogeneity in terms of wind
because this variation can be incorporated in model
parameters. The mechanistic approach is also informative in terms of understanding and predicting the
effects of variability in different species characteristics
and ecological factors affecting dispersal distances.
Therefore, the NSN models are to be preferred over the
2Dt as long as their complexity is not an obstacle to the
application.
Under favourable wind conditions, seeds of C.
praemorsa can fly more than 30 m, but beyond this distance we must rely on model predictions. If we accept
the NSNW model, the potential dispersal distance of C.
praemorsa is unbounded. Under the most favourable
conditions studied (release 1) the maximum dispersal
distance predicted for a sample of 100 000 seeds was
141 km, less than 1% of the simulated seeds dispersed
> 200 m and only about 5% dispersed > 30 m (cf. 4%
of the seeds in release 1 dispersed > 30 m). The model
predictions on long-distance dispersal, however, require
empirical validation.
Because of logistic limitations to dispersal measurements, observations of extreme long-distance dispersal
may remain elusive (Nathan et al. 2003). Nevertheless,
there are ways to extend and improve the measurements.
Our results suggest that future studies of this and similar wind-dispersed species need to measure dispersal
under a variety of conditions, and preferably to obtain
measurements of vertical wind speeds. Seed releases
with more seeds on days with updrafts and high wind
speeds may be attempted to increase the chance of
measuring long-distance dispersal. Although observations of extreme distances may be difficult precisely
because they are caused by updrafts or strong winds
(small seeds tend to be lost out of sight under such conditions), increasing the sampling size would probably
give a few more points, at least in the near part of the tail.
Our study shows that seed release studies and seed
traps do not necessarily give similar results. While the
empirical data from the seed releases corresponded
closely to the predictions of the best mechanistic
models, the seed trap data did not, probably due to deviant
wind patterns around the seed source. The chances of
observing long-distance dispersal in seed trap studies
can be improved by increasing the trap area or the seed
source. Increasing the trap area is labourious but technically possible, e.g. using more or bigger sticky traps,
within reasonable limits. Creating larger seed sources
may be easier but point sources consisting of highdensity patches of plants may exacerbate the wind
patterns that violate the assumptions in mechanistic
models and reduce the chances of measuring long-distance
dispersal. This may limit the utility of the seed trap
approach for measuring long-distance dispersal of less
fecund species. Following individual seeds directly or by
markers (stable isotype, genetic or other; Nathan et al.
2003) may be more useful for these species. Closely
linked observations of individual propagules and
important characteristics of the dispersal mechanism,
such as the strength and variability of horizontal and
vertical wind speeds, is a powerful approach to a mechanistic understanding of dispersal patterns (Levin et al.
2003; Nathan et al. 2003).
Acknowledgements
We would like to thank A. Bruserud and R. Haugan for
helping to find populations of C. praemorsa at Ringsaker,
T. Sund for providing a field for the seed trap experiment,
A. Schjolden for assisting in seed releases and J.M.
Bullock, J.M. van Groenendael, R.A. Ims, I. Nordal and
an anonymous reviewer for giving helpful comments on
the manuscript. The work was supported financially by
the Norwegian Research Council (grant no. 134797/410).
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Received 20 November 2003
revision accepted 28 April 2004
Handling Editor: Mark Williamson