## Introduction

Dispersal can be a major determinant of the distribution and abundance of animals, as well as a key mechanism linking behaviour to local and regional population dynamics (Clobert *et al*. 2001). Dispersal plays a fundamental role in regulating local densities, allowing for population spread and redistribution, and ensuring population persistence in highly variable environments (Stacey & Taper 1992). Despite this broad ecological importance, progress in predicting and quantifying dispersal has been hampered by the lack of a general framework for modelling dispersal (Turchin 1998). However, parameter values from fitted models can be used as standardized measures of dispersal for comparing mobility across populations or species and examining the ecological determinants and evolutionary implications of mobility (Rodríguez 2002; Skalski & Gilliam 2003).

Many studies of movement have focused on modelling the dispersal kernel describing the probability distribution of displacement distances from a known point of origin (e.g. Porter & Dooley 1993; Kot, Lewis & van den Driessche 1996). Two continuous probability distributions, the normal and negative exponential, are commonly used as approximations to the decline in frequency of observations as dispersal distance increases away from a source of dispersers (Turchin 1998; Okubo & Levin 2001). Use of the normal distribution as a dispersal model derives from the time-dependent solution of the random walk equation describing diffusive spread.

In contrast to the normal, or diffusion, model, the negative exponential model considers the spatial distribution that arises after a fixed time interval. Various biological mechanisms may generate negative exponential distributions, e.g. when local competition drives dispersal, opposing a natural tendency for philopatry, and animals settle as soon as they find an uncontested patch of suitable habitat (Porter & Dooley 1993). The close connections between diffusion and exponential models sometimes go unrecognized. Exponential dispersal kernels can be derived by modifying diffusion models in various ways, for example, by adding a term for advection toward the origin, or specifying an appropriate compound distribution (Okubo & Levin 2001). Diffusion and exponential models therefore have mechanistic underpinnings and a stronger conceptual grounding than do more phenomenological models (Turchin 1998).

Two key features of dispersal kernels, the kurtosis and the shape of the tails of the distribution, have received particular attention. Dispersal distributions are often strongly leptokurtic because displacements over very short or very long distances are more frequent, and those over intermediate distances more rare, than expected under a normal distribution (Kot *et al*. 1996). Both the diffusion model (mesokurtic; kurtosis = 0) and the negative exponential model (leptokurtic; kurtosis = 3) may be poor fits if the observed kurtosis greatly exceeds 3. Leptokurtic dispersal is often modelled by incorporating intrapopulation heterogeneity in movement rates in two-group models that combine separate dispersal distributions, one for a sedentary component of the population, the other for a mobile component (Skalski & Gilliam 2000; Rodríguez 2002).

A variety of ecological processes depend critically on the shape of the tails of the dispersal kernel, which determines the frequency of long-distance dispersal events. For example, such events have important implications for propagation velocity of invading species, persistence in fluctuating and heterogeneous environments, preservation of metapopulation dynamics in fragmented landscapes, maintenance of genetic diversity, and population responses to climate change (Stacey & Taper 1992; Kot *et al*. 1996; Latore, Gould & Mortimer 1999; Cain, Milligan & Strand 2000; Clobert *et al*. 2001; Clark *et al*. 2003). Consequently, the ability of dispersal models to accurately characterize the frequency of long-distance dispersers is of theoretical interest (Paradis, Baillie & Sutherland 2002). One-group models that assume homogeneous dispersal behaviour have thin kernel tails and often underestimate the probability of long-distance movements (Kot *et al*. 1996; Skalski & Gilliam 2000; Rodríguez 2002), potentially leading to misinterpretation of the role of dispersal at landscape and larger scales. In contrast, two-group models have heavier tails that decline more slowly with distance and can therefore hold a greater proportion of the dispersing population.

For many animal species, natural dispersal is difficult to monitor and model. Common limitations in field studies include the difficulty of delimiting the population and potential range of dispersal, the need for handling the organisms in mark–recapture studies, and the small sample sizes and number of dispersal distributions examined. However, the natal dispersal of lake-dwelling brook charr *Salvelinus fontinalis* (Mitchill) has simplifying characteristics that facilitate the application and testing of dispersal models. Spawning often occurs at only one littoral location with groundwater upwelling, providing a single source of natal dispersers for the entire lake (Ridgway & Blanchfield 1998). The cohort emerges gradually from the spawning site and spreads slowly around the lake margins over a period spanning approximately 2 months (Snucins, Curry & Gunn 1992; Curry, Noakes & Morgan 1995). High rates of predation in open waters result in restriction of young of year (YOY) dispersal to a shallow corridor of inundated shoreline around the lake margins, where large numbers of fish can be readily observed by snorkelling during the emergence and dispersal of a cohort (Biro & Ridgway 1995; Biro, Ridgway & Noakes 1997). Therefore, frequency distributions of dispersal distance can be obtained repeatedly without marking the fish, and movement can be conveniently modelled along a single spatial dimension. To our knowledge, this unique pattern of dispersal of a whole cohort over its entire range has not been studied before in any vertebrate population. Previous studies have documented fine-scaled intrapopulation variation in movement behaviour of dispersing YOY salmonids in still-water environments (McLaughlin, Grant & Kramer 1992; Biro & Ridgway 1995; Biro *et al*. 1997), providing further motivation for comparing one- and two-group dispersal models in this system. Additionally, knowledge about the tails of the dispersal distribution is helpful in assessing whether YOY salmonids can avoid detrimental warm temperatures by reaching cool groundwater refugia distant from the spawning site.

The present study examines the usefulness of simple dispersal models for summarizing and predicting the lake-wide dispersal of an emerging cohort of YOY brook charr, over 12 surveys conducted during a 2-month period. The models are based on two types of dispersal kernel, the normal distribution arising from a simple diffusion process, and a Laplace distribution depicting exponential decay of the frequency of dispersers away from the point of origin. In all, four models were assessed: one-group diffusion and exponential models assuming homogeneous dispersal behaviour, and two-group diffusion and exponential models accounting for intrapopulation heterogeneity. The models assume that dispersal proceeds from an instantaneous point source, along a single spatial dimension, independently of habitat heterogeneity, and at rates determined by constant diffusion or displacement coefficients. Behavioural interactions among individuals and density dependence are not considered. The two-group models allow for intrapopulation heterogeneity in dispersal rates, but assume that the proportions of sedentary and mobile individuals are fixed. A cross-validation approach based on calibrating the models to the distributions from the first two surveys only and then validating them on the remaining 10 distributions was used to evaluate model predictions of the spatial distribution of fish counts and four aggregate measures of dispersal. The robustness of predictions to violation of model assumptions was also assessed.