## Introduction

Dispersal is an extremely important biological process. It affects intra- and interspecific interactions and population dynamics (Dieckmann, Law & Metz 2000; Okubo, Hastings & Powell 2001), rates and patterns of spread (Shaw 1995; Shigesada & Kawasaki 1997; Clark, Lewis & Horvath 2001), genetic variation and structure (Hamrick & Nason 1996; Ibrahim, Nichols & Hewitt 1996), regional species persistence (Brachet *et al*. 1999; van Groenendael, Ehrlén & Svensson 2000), community composition and dynamics (Tilman, Lehman & Yin 1997; van Groenendael, Ehrlén & Svensson 2000) and large-scale biogeographical patterns (Cain, Damman & Muir 1998; Kriticos *et al*. 2003). For population forecasting, conservation and management, for example for biological invasions, it is essential to obtain accurate measures of the distribution of dispersal distances (the dispersal kernel). Unfortunately, dispersal data that allow reliable estimates of dispersal kernel parameters are lacking for most species. This is partly because of a lack of guidelines for dispersal measurement and few attempts at optimizing sampling methods for particular questions (Assunçao & Jacobi 1996; Stoyan & Wagner 2001). In this study, we used Monte Carlo simulations to investigate the efficiency of different sampling designs for dispersal measurement.

Dispersal study design is a decision process. The optimal design will depend on the dispersal-related question, the species, the mode(s) of dispersal it employs and the context. Three main questions need to be answered at the outset of the design process. What is the objective of the dispersal study? What are the options available for measuring dispersal? What are the constraints on the experimental design?

Of these questions, the first is particularly important. Without a clear objective, optimization is not possible. In many dispersal studies the objective is a proper characterization of the entire dispersal kernel, but different aspects of the kernel (such as the mean, the variance or the tail, i.e. long-distance dispersal) may also be of particular interest. In the simulation approach taken in this study we are able to monitor all of these and optimize for any of them.

The options and constraints in an experimental design are likely to be system specific. Options available for measuring dispersal range from tracking individual propagules to estimates based on offspring counts or genetic markers (Turchin 1998; Bullock, Kenward & Hails 2002; Nathan *et al*. 2003). In this study, we focused on seed shadows of realistic dimensions for well-dispersed herbs (Willson 1993), and we confined ourselves to the Eulerian approach for dispersal measurement, i.e. the direct measurement of numbers/densities of propagules at different distances from the source. However, the approach we present is applicable to a wide range of systems and techniques for measurement and estimation.

We used Monte Carlo simulations to assess the efficiency of various trap designs for dispersal measurements under various assumptions about source strengths, sampling efforts and distributions of dispersal distances and directions. While the results of our simulations can be used more or less directly by those designing new field studies for plant point sources, the general guidelines and the simulation approach are applicable to a wide range of organisms, dispersal mechanisms and methods for dispersal measurement; essentially any directly measurable pattern of dispersal that can be characterized by a probability distribution.