## Introduction

Gene flow among and within plant populations concerns evolutionary ecologists, conservationists and ecosystem managers. Spatial patterns of pollen and seed dispersal are important components of gene flow in plants which facilitate connectivity between individuals and populations and create a template on which post-dispersal processes such as selection, competition, predation and exogenous disturbances operate (Linhart & Grant 1996; Nathan & Muller-Landau 2000; Kalisz *et al.* 2001; Vekemans & Hardy 2004). Under current scenarios of rapid human-mediated landscape change, there is increasing interest to better understand and quantify the effects of restricted pollen dispersal and spatial genetic structuring (e.g. Koenig & Ashley 2003). This requires a precise characterization of the seed or pollen shadow, i.e. the distribution of seeds or pollen with distance from the source plant, and its uncertainty.

One important theoretical tool for characterizing seed or pollen shadows is the kernel function, defined as the probability density function of dispersal distances from individual plants (Levin & Kerster 1974). The dispersal kernel has important analytical and modelling applications, such as comparing dispersal and reproductive success parameters across populations with contrasting individual distributions (Burczyk, Adams & Shimizu 1996; Oddou-Muratorio, Klein & Austerlitz 2005; Slavov *et al.* 2009), quantifying metapopulation connectivity (Klein, Lavigne & Gouyon 2006) and forecasting introgression risk from crops or exotic plantations into natural ecosystems in different demographic settings (e.g. Kuparinen & Schurr 2007).

Obtaining accurate estimates of dispersal kernels remains a challenging problem (Jordano 2007; Robledo-Arnuncio & García 2007; Jones & Muller-Landau 2008). Different approaches have been used to address this problem. First, forward predictions which directly model the behaviour of the dispersal agent (e.g. wind, birds, mammals; Schurr *et al.* 2005; Russo, Portnoy & Augspurger 2006; Will & Tackenberg 2008) contribute to a functional understanding of dispersal and allow construction of dispersal kernels. Second, classical non-genetic estimates of seed dispersal kernels rely on inverse modelling (Ribbens, Silander & Pacala 1994; Clark *et al.* 1999; Nathan & Muller-Landau 2000) and reconstruct dispersal kernels and fecundity from the spatial patterns of mother trees, size of the mother trees and the spatial distribution of seeds. This approach does not require identification of the mother plant of each seed and assumes that all potential sources contribute to dispersal into a given point, proportionally to their distance and fecundity. Recent methods allow accounting for the effects of explicit spatial structures of the landscape on the movement of seeds (e.g. Schurr, Steinitz & Nathan 2008). Third, genetic paternity analysis allows for identification of individual mothers (and/or fathers) and improves the basis for estimating dispersal kernels (Oddou-Muratorio, Klein & Austerlitz 2005; Burczyk *et al.* 2006; Goto *et al.* 2006; Robledo-Arnuncio & García 2007; Jones & Muller-Landau 2008). In the case of pollen dispersal, fitting dispersal kernels requires individual genotypes from a sample of seeds of known maternal origin and from all potential pollen donors within the study area.

The simplest approach based on paternity analysis is to use the observed mating distance distribution, obtained from categorical paternity assignment, as estimate for the dispersal kernel. However, this approach (Shimatani *et al.* 2007; Fortuna *et al.* 2008) is not a good estimate of the true dispersal kernel in general (Robledo-Arnuncio & García 2007; Jones & Muller-Landau 2008) because the observed distribution of dispersal distances is an outcome of both the normally unobservable kernel function and the relative spatial distribution of pollen sources (fathers) and recipients (mothers) (Robledo-Arnuncio & Austerlitz 2006). Other potential problems of pollen kernel fitting include the cases where the genotype of a pollen gamete is incompatible with that of all pollen donors within the study area (i.e. potential immigrants, genotyping errors or mutation), ambiguous paternity assignments for some seeds, and, once genotyping errors have been accounted for, how to deal with immigrants from outside the study area (Jones & Muller-Landau 2008). In most cases, the spatial origin of immigrants is unknown, and thus fitted functions are within-population kernels in the sense that they do not incorporate immigrants.

The more general methods based on paternity analysis yield kernel parameter estimates that maximize the likelihood of the observed sample of seed paternal haplotypes, given the Mendelian transition probabilities and given the spatial distribution of pollen donors relative to maternal plants (Adams, Griffin & Moran 1992; Burczyk, Adams & Shimizu 1996; Oddou-Muratorio, Klein & Austerlitz 2005). These methods, usually referred to as neighbourhood models, mating models or full-probability models, do not require a categorical assignment of paternity, and are thus especially useful for low-resolution genetic assays. During the last years, the development of highly polymorphic markers and automated fragment length detection methods substantially improved paternity assignments. Using several independent loci with large number of alleles increases the amount of seed for which either a single pollen donor can be assigned or all potential donors within the study area can be excluded as fathers. Such data sets, thus, comprise the locations of all potential pollen donors within a study area, the locations of selected mother plants and the paternal origin of seed samples harvested from these mothers. However, when there is a large number of candidate fathers, it is still not possible to assign one or exclude all fathers for a given seed, so that the paternity of an important proportion of the sample remains ambiguous (Ashley 2010; Christie 2010).

Besides providing an individually explicit characterization of mating patterns, categorical paternity assignments may also offer potentially useful insights into pollen dispersal kernel estimation. In particular, we propose here the estimation of nonparametric within-population dispersal kernels (which we will denote ‘empirical’ kernels) within the framework of point pattern analysis (Stoyan & Wagner 2001; Illian *et al.* 2008) to complement traditional kernel estimation. An advantage of this approach over curve fitting is that it does not require assumptions about the shape of the dispersal kernel and therefore does not rely on proper model selection. It is well known that kernel functions with very different means and tails may fit, about equally well, the data over the spatial scale of analysis (Smouse, Robledo-Arnuncio & González-Martínez 2007). Moreover, it allows confronting the empirical estimate of the dispersal kernel to (empirical) kernels expected by null models such as the random mating model. Comparison of the empirical kernel estimate to Monte Carlo simulation envelopes of the null model (a standard approach in point pattern analysis) allows for uncertainty assessment of the derived kernel. The width of the simulation envelopes provides information about the shapes of kernels that are, for example, compatible with the random mating null model. In addition, a formal goodness-of-fit (GoF) test can be used to explore if the empirical pollen dispersal kernel differs at selected distance intervals significantly from that expected under the null model.

Thus, point pattern analysis can contribute to dispersal ecology in providing (i) techniques to derive nonparametric kernel estimates free of assumptions about the shape of the dispersal distribution and (ii) assessment of uncertainty in kernel estimates, which has received surprisingly little attention in the literature. The main goal of this article is to provide methods for these purposes. Our objectives were to characterize spatial patterns of pollen flow in a population of *Populus nigra* in Germany, where the species is of conservation concern, and to present new nonparametric point pattern methods to estimate pollen dispersal kernels from categorical paternity assignment and to evaluate the associated uncertainty. We then compared the results of point pattern analysis with those of established methods of parametric kernel fitting. More specifically, we fitted within-population dispersal kernels based on: (i) a general mating (neighbourhood) model, which considers spatial effects and uses all seed data including ambiguous cases, (ii) a simplified mating model which uses only data in which the offspring can be categorically assigned to the parent (Hardy *et al.* 2003; Robledo-Arnuncio & Gil 2005) and (iii) techniques of point pattern analysis, assessing the uncertainty in the kernel estimates using the data set used in (ii).