## 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 2π*r* (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.