## Introduction

Dispersal affects both the population dynamics and the population genetics of species. Reciprocally, the dynamics and the genetics of species are likely to act on this behaviour. The forces that may select for higher probabilities of dispersal include (1) temporal variability in the habitat (Van Valen, 1971); (2) avoidance of inbreeding depression (Bengtsson, 1978); (3) kin competition (Hamilton, 1964; Hamilton & May, 1977). As a first approximation, the evolution of dispersal may be described by a balance between these forces and a cost due to either increased mortality during the dispersal phase or during the settling period in a novel habitat. All these factors are often pooled in a single parameter: the cost of dispersal. Further, it is very convenient to assume an island model of migration, where dispersers are redistributed randomly among the different populations. Under this simplifying assumption dispersal is fully characterized by a single parameter, the dispersal probability, which measures the fraction of the progeny leaving its natal site.

However, dispersal does not usually follow an island model and the distribution of dispersal distances is often of interest in itself. It is known that the shape of the tail of the distribution of dispersal distance (long-distance dispersal) determines the rate of spread of colonizing populations as shown by studies of disease epidemics or of post-glacial rates of advance in many plant species (e.g. (Mollison, 1991; Kot *et al*., 1996; Shigesada & Kawasaki, 1997; Clark *et al*., 2001). Although the evolution of the dispersal probability has been analysed assuming dispersal is localized, as in ‘stepping stone’ models, it was still assumed that the distribution of dispersal distance itself was fixed and that the cost was independent of distance (Comins, 1982; Gandon & Rousset, 1999). However, the distribution of dispersal distances itself is subject to selection. Here, our analysis assumes that there is no *a priori* constraint on the dispersal distribution: any distribution is a possible strategy. This contrasts to models where a more restricted family of dispersal distributions is considered (Ezoe, 1998; Bolker & Pacala, 1999; Gandon & Rousset, 1999; Harada, 1999).

The different selective pressures acting on the shape of the dispersal distribution vary with the dispersal distance (Ronce *et al*., 2001). In this paper we provide a formal basis for the evolution of dispersal distribution in the case where only two forces are acting: kin competition and distance-dependent cost of dispersal. In a stable and homogeneous habitat there is always a benefit to disperse far because it decreases the risk of competing with related individuals. On the other hand, the cost of dispersal may be an increasing function of dispersal distance. This cost will have a direct effect on the evolution of dispersal but it will also feed back on the distribution of genetic variation and, therefore on kin competition and the benefit of long-distance dispersal. Kin competition and distance-dependent cost of dispersal are ubiquitous forces acting on this distribution and our aim is to provide a model which may serve as a reference against which one can evaluate the importance of additional factors that may affect the evolution of dispersal distributions.

We first present a general treatment of the evolution of dispersal with any number of demes and for any function of dispersal cost. This analysis is then used in one- (1D) and two-dimensional (2D) models to study the effects of two factors on the evolution of dispersal: (1) deme size; (2) the shape of the cost of dispersal function. We find that long-distance dispersal is selected for even if the survival cost of dispersal is very high, provided the probability of survival does not vanish at long distances. This analysis relates the evolutionarily stable dispersal distribution to a given distribution of distance-dependent cost of dispersal.