How far can a hawk's beard fly? Measuring and modelling the dispersal of Crepis praemorsa



    Corresponding author
    1. Department of Biology, University of Oslo, PO Box 1050 Blindern, N-0316 Oslo, Norway
    2. Department of Biology, The Pennsylvania State University, 208 Mueller Laboratory, University Park, PA 16802, USA
      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
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    1. The Norwegian Institute of Nature Research, Dronningens gt. 13, PO Box 736 Sentrum, N-0105 Oslo, Norway
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    1. Department of Biology, University of Oslo, PO Box 1050 Blindern, N-0316 Oslo, Norway
    Search for more papers by this author

    1. Department of Biology, University of Oslo, PO Box 1050 Blindern, N-0316 Oslo, Norway
    Search for more papers by this author

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


  • 1We 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.
  • 2Dispersal 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.
  • 3Four 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.
  • 4Two 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.
  • 5We 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.


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 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 species

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.

dispersal experiments

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.

dispersal models

In this paper we express each dispersal model as a one-dimensional seed shadow (s) (seeds m−2):

image(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.

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
ModelDispersal kernel f(r)References
IPa1r−bBullock & Clarke (2000)
NEa2 exp (−b2r)Clark et al. (1999), Bullock & Clarke (2000)
MIXinline imageBullock & Clarke (2000)
2Dtinline imageClark et al. (1999)
GJinline imageGreene & Johnson (1989)
NSNinline imageNathan et al. (2001)
Table 2.  Definitions of symbols for variables and parameters of the dispersal models in Table 1
SymbolDefinition (unit)
sSeed shadow (seeds m−2)
cSeed count (seeds)
rRadial distance from seed source (m)
 f(r)Dispersal kernel (m−2)
aiScaling parameter in the IP, NE and MIX models
biCurvature parameter in the IP, NE and MIX models
uParameter in the 2Dt model (m2)
pParameter in the 2Dt model
FSeed falling velocity (m second−1)
HSeed release height (m)
dDisplacement height (m)
z0Roughness length (m)
zmMeasurement height for wind speed (m)
UmHorizontal wind speed (m second−1) at measurement height
UHorizontal wind speed (m second−1) between H and d
UgGeometric mean horizontal wind speed (m second−1) between H and d
σUVariance of ln U (m second−1)
WVertical wind speed (m second−1)
g(r)Frequency distribution
AArea of seed trap or ground (m2)

empirical models

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:

image(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 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 = ri + (ri+1 − ri)/2, and A = 2π(inline image).

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:

image(eqn 3)

where ci is the observed seed count in seed trap (or area) i and ĉi 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.

mechanistic models

Two different mechanistic models were considered for Crepis praemorsa (Table 1): (i) Greene & Johnson's 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):

image(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 NSNW. 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:

image(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 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.

model evaluation

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. short-distance vs. long-distance dispersal). Therefore, we also evaluated the models graphically.


dispersal distances

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).

Table 3.  Summary of dispersal distances and wind speeds in the three dispersal experiments for Crepis praemorsa
 Release 1Release 2Trap
  • *

    No seed traps were located closer to the source than 0.7 m.

Dispersal distances r (m)
 Minimum     0.650.030.70*
 Median     1.990.570.70
 Mean     3.510.861.03
 Maximum> 30.003.814.30
Wind speeds U (m second−1)
 Minimum     0.380.010.09
 Median     0.930.400.88
 Mean     0.950.410.90
 Maximum     1.741.492.04

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 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.

empirical models

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 under-predict densities at intermediate distances. The NE model captured the shape of the observed curve over a greater range, but tended to under-predict long-distance 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.

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
ModelParameterRelease 1Release 2Trap
Figure 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).

mechanistic models

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 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).

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
ParameterRelease 1Release 2TrapDistributionSource
  1. NNot significantly different from the normal distribution (Kolmogorov-Smirnov goodness-of-fit test, P > 0.05).

  2. LNNot significantly different from the log-normal distribution (Kolmogorov-Smirnov goodness-of-fit test, P > 0.05).

 F (m second−1)0.25 (0.07)LN0.25 (0.07)LN0.25 (0.07)LNLog-normalLaboratory measurement
 H (m)0.52 (0.05)N0.50 (0.00)0.55 (0.15)NNormalField measurement
 Um (m second−1)2.78 (0.89)N,LN0.75 (0.56)N2.46 (1.08)NNormalField measurement
 U (m second−1)0.97 (0.31)N,LN0.38 (0.29)N1.54 (0.67)NNormalField measurement
 W (m second−1)0.10 (0.35)0.10 (0.35)0.10 (0.35)NormalNathan et al. (2001)
 d (m) (1993)
 z0 (m) (1993)
Figure 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 (dashed-dotted), NSN (dashed), and NSNW (solid).

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.

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 (∞)
ModelRelease 1Release 2Trap
 IP 73.1051.8019.48
 NE 52.5227.7333.52
 MIX 26.7725.8518.28
 2Dt 47.0441.2219.48
 GJ524.9046.56 ∞
 GJW549.4561.11 ∞
 NSN  ∞37.33 ∞
 NSNW 30.3431.55 ∞

model evaluation

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).


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 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 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 high-density 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).


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).