Study sites
Our study was carried out in 2007 in Brittany (western France), where sea beets colonize coastal areas (Fievet et al., 2007). Exhaustive sampling of all flowering plants was carried out at two study sites within two isolated coves separated by 30 km. The first site, MOR, extends over c. 300 m (N 48°34’168’’; E –2°34’831’’) and the second one, PAL, over c. 600 m (N 48°40’497’’; E –2°52’911’’). At both sites, individuals appeared to be strongly clustered in space (Fig. 1). In MOR, there were four geographical patches (MA, MB, MC and MD), totalling 1094 sampled individuals, plus four isolated plants growing outside these patches (see Fig. 1 and Table 1). In PAL, 592 individuals were clustered within five patches (PA, PB, PC, PD and PE), and 23 plants were geographically isolated (Fig. 1 and Table 1). No B. vulgaris individuals were found for at least 1000 m on either side of these sites.
Table 1. Major characteristics of the different geographical patches of Beta vulgaris ssp. maritima within the two study sites MOR (a) and PAL (b) (a) 

 MA  MB  MC  MD  Overall  

N_{TOTAL}  69  635  281  109  1094 (4)  
N_{PHENOLOGY}  16  45  191  14  266  
N_{POLLEN}  16  45  191  14  266  
N_{INVESTMENT}  0  0  191  0  191  
N_{F}/N_{HCMS}/N_{HNonCMS}  0/3/62 (4)  0/0/633 (2)  17/60/195 (9)  0/0/106 (3)  17/63/996 (18)  
Sex ratio  0  0  0.063  0  0.016  
N_{MOTHER PLANTS}  6  6  32  6  50  
N_{OFFSPRING}  143  133  773  150  1199  
Mean _{PROGENY SIZE} (± SD)  23.83 (± 1.33)  22.17 (± 6.94)  24.16 (± 2.85)  25 (± 0)  23.98 (± 2.29)  
(b) 

 PA  PB  PC  PD  PE  Overall 


N_{TOTAL}  146  18  300  33  95  592 (23) 
N_{PHENOLOGY}  12  0  120  8  78  218 
N_{POLLEN}  12  0  120  8  78  218 
N_{INVESTMENT}  0  0  221  0  92  313 
N_{F}/N_{HCMS}/N_{HNonCMS}  0/0/142 (4)  6/6/2 (4)  61/45/185 (9)  0/0/32 (1)  3/18/71 (3)  70/69/432 (21) 
Sex ratio  0  0.429  0.21  0  0.033  0.123 
N_{MOTHER PLANTS}  12  0  15  8  15  50 
N_{OFFSPRING}  297  0  357  197  368  1219 
Mean _{PROGENY SIZE} (±SD)  24.75 (± 0.87)  –  23.80 (± 4.13)  24.63 (± 1.06)  24.53 (± 1.81)  24.38 (± 2.51) 
At the two study sites, all flowering plants were genotyped for RFLP markers diagnostic for CMS cytoplasms (see De Cauwer et al., 2012) and gynodioecy appeared to be based on a simple cytonuclear determination, with fertile cytoplasms and (except for two plants) only one CMS type (CMS E, which is the most common source of male sterility in this region; see Dufay et al., 2009). Finally, strong spatial genetic structure was detected within both sites (see De Cauwer et al., 2012).
Modelling the dispersal kernel and heterogeneity in male fecundity
The aim of the study was to disentangle the effects of various sources of variation on male fecundity (i.e. ability to produce offspring). A spatially explicit mating model was used to investigate how male reproductive success depended on (1) the physical distance between individuals, (2) the geographical patch, (3) flowering synchrony, (4) the investment in reproduction, (5) pollen production, (6) cytotype (CMS vs nonCMS) and (7) the probability of carrying restorer alleles. The method used here is related to fractional attribution of paternity, where each offspring is partially assigned to each of the possible candidate fathers on the basis of their relative likelihoods of parentage (see OddouMuratorio et al., 2005; Jones et al., 2010). Analyses were carried out independently for each study site (MOR and PAL).
Following Burczyk et al. (2002) and OddouMuratorio et al. (2005), we assumed that each offspring could be the result of either selfpollination, pollen coming from outside the study sites or pollen coming from one of the sampled individuals. A seed o sampled on a mother plant j_{o} with genotype g_{jo} was then expected to have the genotype g_{o} with probability:
 (Eqn 1)
(s, the selfing rate; m, the rate of incoming pollen flow; (1 –m–s), the probability that the pollen donor is inside the study site; π_{jk}, the composition of the pollen pools (described below; Eqn 3).) The allele frequencies (AF) in the incoming pollen pool were measured independently in MOR and PAL from the estimated contribution of individuals located outside the study sites (i.e. through offspring that have no compatible father inside the study sites). The transition probabilities T (..,.) are the Mendelian likelihoods of observing a genotype for a seedling, conditional on the genotype of the parents (Meagher, 1986). The information carried by the genotypes of all seeds sampled in one site was combined into a loglikelihood function, assuming that all fertilization events were independent,
 (Eqn 2)
The proportion of pollen from each father k in the pollen pool of each mother j originating from all known fathers, π_{jk}, was assumed to follow the massaction law,
 (Eqn 3)
Disp _{jk} is taking into account the effect of physical distance between individuals. This effect was modelled using a dispersal kernel, describing the probability density that a pollen grain lands at a given position away from the source. As suggested in previous studies (OddouMuratorio et al., 2005; Fénart et al., 2007), we investigated several shapes for the dispersal kernel, including the exponentialpower function and the logistic function (reviewed in Austerlitz et al., 2004).
The exponentialpower kernel used to model the effect of distance on mating probability was given by:
 (Eqn 4)
(d_{jk}, the distance between mother j and father k; Γ, the Gamma function; a, a scale parameter for distance; b, a shape parameter (Clark, 1998).) The average pollen dispersal distance (δ) is then given by δ = [aΓ(3/b)/Γ(2/b)].
The logistic function was given by:
 (Eqn 5)
Pop _{k} in (0,+∞) takes into account the differences in individual male fecundities resulting from potential microenvironmental variations among the different geographical patches within each study site (see Fig. 1).
Pheno _{jk} takes into account the effect of flowering synchrony and is given by a Gaussianlike curve,
 (Eqn 6)
(Δ_{jk}, the difference in phenology between mother j and father k; Δ_{opt}, the optimal difference in phenology; σ_{Δ}², the variance of the Gaussian curve.) A positive Δ_{opt} means that fathers flowering earlier than a given mother have a higher reproductive success than males flowering synchronously.
Fec _{jk} takes into account the effects of pollen production (PP_{k}) and investment in reproduction (RI_{k}) through an exponential relation,
 (Eqn 7)
(b_{PP} and b_{RI}, selection gradients describing the effect of pollen production and investment in reproduction on male fecundity.) For the individuals that were not scored for pollen production or investment in reproduction, the corresponding exponential term was replaced by a parameter NS_{PP} (different in the different patches) or NS_{RI} (independent of the patch).
Finally, the effect of the cytotype (CMS vs nonCMS) and the effect of the probability of carrying restorer alleles were also taken into account. To assess the effect of the cytotype, we set Sex_{k} = F = 0 for females, Sex_{k} = H_{CMS} for restored CMS hermaphrodites, Sex_{k} = H_{NonCMS} = 1 for nonCMS hermaphrodites and Sex_{k} = NT for nontyped individuals (i.e. individuals for which the sexual phenotype was not scored).
The effect of the probability of carrying restorer alleles was considered through the local CMS restoration frequency around individual k (P_{Freq,k}), using the following equation:
 (Eqn 8)
where b_{Freq,k} is b_{HCMS} if individual k is a restored CMS hermaphrodite, b_{HNonCMS} for a nonCMS hermaphrodite, and 0 for females and nontyped individuals.
By maximizing the loglikelihood logL (Eqn 2), we jointly estimated the following parameters: levels of selfing (s) and of incoming pollen flow (m), the dispersal parameters (a and b), the effect of geographical patches (Pop_{1}, Pop_{2}…Pop_{max}), the effect of flowering phenology (Δ_{opt} and σ_{Δ}), the effect of investment in reproduction (b_{RI} and NS_{RI}), the effect of pollen production (b_{PP}, and NS_{PP1}, …NS_{PPmax}), the effect of the cytotype (H_{CMS} and NT) and the effect of the probability of carrying restorer alleles (b_{HCMS} and b_{HNonCMS}). The loglikelihood function was maximized numerically using the quasiNewton algorithm in Mathematica 7.1 (Wolfram Research, Champaign, Illinois, United States). Several contrasted initial values for the maximization of the parameters were used to confirm that we reached a global maximum (when the initial values actually led to the different maxima, we kept the one that reached the higher loglikelihood). Parameters describing the relative male reproductive success of nonscored individuals (NS_{RI} and NS_{PP1}, …NS_{PPmax}) and of nongenotyped individuals (NT) were incorporated in the models to improve the fit to data, but these results will not be described further in the paper.
The significance of each effect was tested with a Type III likelihoodratio test (LRT). For each test, the loglikelihood of a model without the tested effect was computed. The deviance (i.e. twice the difference between the loglikelihood obtained for the complete model and the loglikelihood obtained for the model without the tested effect) was then compared to a χ^{2} distribution, with the number of degrees of freedom equal to the difference in the number of parameters between the two models.
The confidence intervals for the parameters were obtained through a bootstrap procedure using mother plants as sampling units. For each replication, we successively sampled mother plants with replacement and equal probabilities and kept all their seedlings’ genotypes, until we reached the same total number of seedling genotypes in the bootstrapped data set as in the real data set. To reach exactly the same total number of seedlings, we sampled without replacement the correct number of seedlings among those of the last mother drawn. We then estimated the parameters on the bootstrapped data set by maximizing the loglikelihood (Eqn 2). For each study site, we derived 95% symmetric confidence intervals for all parameters from 250 bootstrapped data sets.