## Introduction

The linking of movement behaviour with the quantitative description of dispersal has become increasingly important in the development of dispersal ecology. In parallel, dispersal has taken a key position in the development of spatially explicit theoretical concepts and models of populations. This review explores the ways in which the different elements have been brought together and how mechanistic analysis has aided understanding. The emphasis is on the distribution of dispersal distances and how random walk models have and could explain the observed distributions. The evolution of dispersal distances is considered in relation to landscape dynamics and the consequences for population dynamics.

The development of metapopulation ecology has been dependent upon a move away from the simple and unrealistic assumptions about dispersal in classical metapopulation theory and has required the quantitative description of dispersal in relation to landscape features and dynamics (Hanski 1999). The way in which dispersal has been incorporated in metapopulation models has however remained relatively simple and despite or possibly because of that, metapopulation thinking has made major contributions to the development of ecology and its applications, particularly in conservation biology. The substantial body of work on spatially realistic models was initiated by Hanski (1994) with the development of the incidence function model. Dispersal was represented by an emigration rate parameter and a dispersal distance distribution based on the negative exponential. The Virtual Migration (VM) model (Hanski, Alho & Moilanen 2000) has continued to utilize these features and has been used to estimate survival and dispersal parameters from empirical data in butterflies (Petit *et al*. 2001; Wahlberg *et al*. 2002; Schtickzelle & Baguette 2003; Mennechez *et al*. 2004; Wang *et al*. 2004; Cizek & Konvicka 2005; Keyghobadi *et al. *2005; Baguette & Van Dyck 2007) and the red milkweed beetle *Tetraopes tetraophthalmus*, (Matter 2006) and the connectivity measure of the model has been applied to the chinook salmon *Oncorhynchus tshawytscha*, (Isaak *et al*. 2007). Stochastic simulation of metapopulation dynamics and evolution has also used the negative exponential as a dispersal function (Ronce, Perret & Olivieri 2000; Murrell, Travis & Dytham 2002; Hanski & Heino 2003; Hanski & Ovaskainen 2003; Ovaskainen & Hanski 2003; Travis 2003; Moilanen 2004; Ozgul *et al*. 2006) although more recently, diffusion approximations of random walk have been used (Ovaskainen 2004; Hanski, Saastamoinen & Ovaskainen 2006).

The derivation of the negative exponential was based on the assumption that movement is linear (Buechner 1987) but the use has been as a phenomenological model without test of the assumption. The list of species where the negative exponential underestimates frequencies in the tail and sometimes overestimates at short distances is growing (Buechner 1987; Hill, Thomas & Lewis 1996; Kot, Lewis & van den Driessche 1996; Thomas & Hanski 1997; Baguette, Petit & Quéva 2000; Roslin 2000; Baguette 2003; Kuras *et al*. 2003). Chapman, Dytham & Oxford (2007a) concluded on the basis of goodness-of-fit that the negative exponential should not be considered the function of automatic choice in describing distance distributions and modelling metapopulations. A more general criticism has been made of the way in which metapopulation theory has developed by Bowler & Benton (2005) who concluded that dispersal has been treated too simplistically and that emigration, interpatch movement and immigration should be considered as a series of condition-dependent processes. They also concluded that dispersal cannot be collapsed into a single parameter derived from a simple function. Given the centrality of dispersal in metapopulation models, unsurprisingly the dispersal function has been found to affect dynamics (Casagrandi & Gatto 2006; Heinz, Wissel & Frank 2006).

The debate about the way in which dispersal has been considered in metapopulation theory chimes with a more general debate about the limitations of phenomenological models and the value of mechanistic ones. Turchin (1998) and Wein (2001) concluded that an understanding of the fundamentals of dispersal required the study of movement behaviour. The value of mechanistic over phenomenological models is that they may permit the comparison of different concepts of dispersal processes and the rejection of inappropriate models on the basis of data collected in the field (Turchin 1998). Mechanistic home-range analysis was developed by Moorcroft & Lewis (2006) because phenomenological models have limited predictive capability. In developing an understanding of what causes oscillations in populations, Turchin (2003) used a combination of mechanistic and phenomenological approaches.

There is a need for quantitative descriptions of dispersal which fit distance data well and incorporate realistic assumptions about the underlying movement behaviour. Much work has been carried out on behaviourally based analytical models of movement (Skellam 1951a; Okubo 1980; Turchin 1998; Okubo & Levin 2001) which potentially could bridge the gap between behaviour and the quantitative description of dispersal. The starting point of this review is that an understanding of movement behaviour can improve the way in which dispersal is described and increase understanding of process. Mechanistic approaches may not only inform dispersal ecology but also facilitate the linkage of dispersal and population dynamics.

The review starts with a clarification of what is usefully included under the heading of dispersal. Random walk models are then described and their potential value in explaining distance distributions assessed. The causes of leptokurtosis in distance data is explored as a means of establishing how to elaborate diffusion models to provide better descriptions of dispersal. Some current models of distance distributions are then compared and conclusions drawn about which models have potential to provide new insights about dispersal ecology and population processes. The possible outcomes of individual variation in dispersal propensity are considered in the context of the evolution of dispersal and population dynamics. In conclusion, a set of candidate principles for dispersal ecology are proposed and suggestions made for the further development of research.