With model 1, we estimated from equation (8) the reduction of effective male population density (d^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGKbazgaqcaaaa@2E0D@_em/d_obs) when only accounting for variation in phenotypic traits as 0.32 when the assumed genotyping error rate was set at 0%, 0.24 for a rate of 0.1%, and 0.27 for a rate of 2.5%.

The mean reduction of effective male population density due to phenotypic traits and temporal assortative mating was estimated at (d^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGKbazgaqcaaaa@2E0D@_em/d_obs) = 0.30, 0.23 and 0.27 (with an error rate of 0%, 0.1% and 2.5% respectively). Among pollen-recipients, regardless of genotyping error, the highest reductions were estimated from the pollen clouds of pollen-recipients belonging to the two latest phenological groups (4 and 5), and particularly that of group 5 (almost two-fold decrease, results not shown).

Finally, the mean reduction of effective male population density due to phenotypic traits, temporal assortative mating and non-random dispersal was estimated to (d^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGKbazgaqcaaaa@2E0D@_em/d_obs) = 0.09, 0.07 and 0.06 with an error rate of 0%, 0.1% and 2.5% respectively.

The overall selfing rate estimated with model 1or 2 was 10% and was slightly affected by genotyping error (Table 1). It varied among phenological groups (Figure 3) and among families, as estimated by MLTR: the mean outcrossing rate (t_m) was significantly different from 1 within half of the families, with values ranging from 31% to 88% (i.e. selfing rates ranging from 12% to 69%). The selfing rate was close to 20% in group 2 (t_m = 82.7%, standard deviation = 7.7%). In group 3, the selfing rate was estimated at 7% (t_m = 93%, SD = 0.066): the family-level t_m values were significantly different from 1 within 3/8 families, ranging between 62.5% and 94%. The selfing rate was estimated at zero in the two latest flowering groups (4 and 5).

According to Araki and Blouin 33, assignment error can have a strong effect on the estimation of the relative reproductive successes of different groups, particularly by increasing type II errors (assignment of an untrue parent) when the proportion of unsampled parents is high. Avoiding these errors is generally difficult 35. Here, we accounted for genotyping errors more realistically than previous methods 36, where a mistyped allele can be of any size compared to the original allele. Instead, we assumed that a genotyping error could be only of ± 1 repeat unit, which is more realistic for microsatellite loci.

As expected, some estimated parameter values varied with different assumed rates of error. However, the general trends remained quite similar: the estimates of dispersal parameters were relatively stable, and the ranking of relative phenology parameters, as well as fertility parameters of various phenotypic trait classes, were barely affected, except for sexual type. The main difference lay in the significance of effects, which was certainly due to the increased information provided by accounting for genotyping errors.

Similarly, the value of the estimated rate of external pollen (m) when genotyping errors were assumed may be more reliable than without errors. The very high value (~80%) obtained when these errors were ignored illustrates a particular caveat of paternity analyses using microsatellite markers: higher allele numbers generally produce higher estimated immigration rates. Indeed, using paternity exclusion methods, the apparent gene flow was estimated here between 45% and 55% with four loci, depending on the level of polymorphism of the selected loci, and increased up to 70% with six loci (Gérard et al., unpublished results). Increasing the level of polymorphism of the chosen loci improves the stringency of paternity analysis, by reducing the rate of multiple assignments and type II errors. At the same time, however, more polymorphic loci increase the risk of overestimating external pollen flow by increasing the error rate.

We found a high level of pollen immigration, even when typing errors were accounted for (external pollen flow ~60% for the highest assumed genotyping error rate). This has also been found in other anemophilous forest tree species such as Pseudotsuga menziesii 37, or Quercus robur and Q. petraea 38. Moreover, the best-fitting dispersal kernels were rather fat-tailed, which appears increasingly as a common feature in plants, including perennial herbs (e.g. 3940), crops (e.g. 41) and forest trees (e.g. 4243). Long-distance dispersal may tend to connect distant populations, homogenizing differentiated gene pools. Here it is likely to have acted on the evolution of the hybrid zone of the two Fraxinus species, by connecting remote populations, increasing local genetic diversity 7 and possibly counteracting the effect of local adaptation if assortative mating is not strong enough. This long-distance dispersal has probably contributed to the creation and maintenance of a large-scale hybrid zone, as detected all along the Loire valley 25.

The fertility of hybrids (i.e. the intermediate phenological groups), and in particular individuals from group 2, is higher than the fertility of either parental species (Groups 1 and 5). Differences in some phenotypic traits may participate in this increased fertility.

We found in our previous study 28 that intermediate flowering hybrids (groups 2–4) produced more flowers and had a large DBH (Table 4). Here we show that larger tree diameters and higher flowering intensities increased male fertility (Table 3), as expected for wind-pollinated species 46, and also detected in other tree species (e.g. 24). This may contribute to increase the fertility of group 2.

Surprisingly, we did not detect any effect of the sexual type on fertility, probably due to the similar values of siring success of different classes of hermaphrodites (i.e. individuals with a majority of staminate flowers did not sire more seeds than perfect hermaphrodites). Thus male mating success did not depend on the relative proportions of staminate vs. perfect flowers. Nevertheless, a higher male fertility of males relative to hermaphrodites was detected in controlled crosses 47 and natural populations 30 of F. excelsior. This was retrieved here by the very high relative male fecundity of pure males estimated with a typing error rate but it should be nuanced by the very low frequency of males in the population (2%).

Optimal sex allocation may also contribute to increase the male fertility of hybrids. Indeed, trees producing many seeds also had the highest relative male fecundity. Our results are consistent with classical predictions of sex allocation theory, i.e. a constant optimal sex allocation for simultaneous hermaphroditism, where individuals simultaneously produce male and female gametes 4849. Nevertheless, many plant species seem to exhibit a gradual shift in sex allocation, and thereby in functional gender, with increasing size 465051. A positive correlation between male and female reproductive success has been detected in very few cases only 525354. Here we present a case where individuals with a high male mating success also show a high female success. Further long-term work is required to confirm that this observation remains true through time and for other hybrid populations.

The higher relative fertility of group 2 may be partly influenced by its lower mean level of gall attacks, compared with those of the other groups 28. As gall mites (Eriophyes fraxiniflora) infect male ash flowers 55, it is expected that high attack levels would influence the relative male fecundity. Indeed, low levels of gall attacks seemed to have little effect but high rates strongly decreased relative fecundity.

Flowering phenology also generates asymmetrical pollen flow: trees are quite successful in fertilizing the individuals with a flowering period that overlaps their own but starts later. In contrast, they show quite limited success on the individuals that flower earlier than they do, even if the flowering times overlap. This may result from the protogyny of the Fraxinus species: early flowering trees may participate in reproduction mainly as pollen donors. Mite attacks may also contribute to the observed asymmetry of gene flow by pollen since late flowering trees are more infected than early flowering trees 28. They may also favour hybridization, by reducing the male fertility of late flowering trees. This asymmetry may provide a demographic advantage to early flowering hybrids because of the greater distances of pollen dispersal compared to seed dispersal in forest trees. On the other hand, although F. angustifolia individuals (Group1) had poor fertility during the year of study (which is likely due to higher susceptibility to late winter frosts, also affecting their fruit production), they may benefit from favourable years and contribute significantly to young seedling generations (Gérard et al. in preparation). Moreover, F. angustifolia shows reduced dormancy compared to F. excelsior 56, possibly conferring another demographic advantage. Thus, the invasive potential of F. angustifolia through hybridization in this region may be highly modified in the case of global warming.

Further work, however, is still needed to assess the relative fitness of seeds produced by different types of individuals, either selfed or outcrossed. Moreover, as pollen dispersal seems to occur over large distances, a larger sampling effort is perhaps needed to get more accurate estimates of the dynamics of the metapopulation in this hybrid zone. As the evolution of partially cross-fertile plant communities is greatly influenced by the strength of assortative mating and demographic characteristics 57, theoretical work is also needed to better understand the interaction of short- and long-distance dispersal and assortative mating, as well as environmental fluctuations (e.g. climate) in plant hybrid zones.

Common ash (F. excelsior L.) and narrow-leaved ash (F. angustifolia Vahl) are closely related species 58 with contrasted distributions across Europe: F. angustifolia has a Southern Mediterranean distribution, whereas F. excelsior occurs at more Northerly latitudes. They are in sympatry in several regions in France, such as the Loire and Saône valleys 2559. Both species are post-pioneer forest trees with a colonizing behavior and a discontinuous spatial distribution. They require abundant water, especially F. angustifolia which is often found in low elevation riparian forests, particularly in central France 5960. Both are protogynous, although anther dehiscence occurs while the stigma is still receptive, and pollen and fruits are wind-dispersed. Fraxinus excelsior has a complex trioecious breeding system: sexual types vary across a continuum from pure male individuals to pure females, with all kinds of hermaphrodites in between 475561. Much less is known regarding the mating system of F. angustifolia, which rather seems to be androdioecious 28. The species differ in their phenology: initiation of flowering occurs between mid-March and early April for F. excelsior, and between mid-December and late January for F. angustifolia, with larger among-year variation 2627.

The study site is an unmanaged, naturally regenerated population covering almost 7 ha, located at Saint-Dyé-sur-Loire (latitude 47° 39' 10" N, longitude 1° 28' 40" E, elevation 78 m.a.s.). It is along the Loire river in central France in a zone of sympatry between the two species (see 28 for all details on the stand along with a map). The population was composed of 269 flowering adult trees, and we observed a unimodal distribution of flowering dates, which ranged between the extreme phenologies of the two species. All trees were assigned to one of five phenological groups according to their flowering date (i.e. bud flush, just before pollen offset, 1 group = 2 weeks). These five phenological classes were validated by a Canonical Discriminant Analysis on morphological traits and microsatellite allelic frequencies, and by linkage disequilibrium estimation within and among groups (28 and Table 4). We also measured the diameter at breast height (DBH), and grouped them into five discrete classes (< 40 cm to > 160 cm, with 40 cm intervals). We finally measured for each individual two fitness components, the flowering and fruiting intensity (each as a visually-assessed 5-level class variable), as well as the intensity of floral gall attacks (also as a visually-assessed 5-level class variable: class 0 includes trees without any gall whereas in class 4, more than 80% of flowers were infected) (see Table 4). For each individual, we also recorded its sexual type, according to one of the four following categories: MM, MH, HM, HH, which correspond either to pure males (MM) or to three types of hermaphrodites, with either a majority of male flowers (MH) or a majority of hermaphroditic flowers (HM) or only hermaphroditic flowers (HH). We followed the classification of Morand-Prieur 30, who also observed three additional categories of trees (three types of females) that we did not observe here.

We harvested 432 seeds on 27 fruiting trees (16 seeds per tree) in the autumn of 2003. The distances between the sampled mother trees ranged from 14 to 1775 m, with a mean of 538 m. We tried to sample individuals of each phenological group, but the earliest flowering trees (first group) did not produce sufficient numbers of fruits. Seeds were rehydrated and sterilized as described in Raquin et al. 62, and embryonic tissues were dehydrated in a 1:1 ethanol-acetone solution. Total DNA was extracted from dried tissue using a DNeasy^® 96 Plant Kit (Qiagen).

All flowering individuals in the population and all seeds were genotyped at eight microsatellite loci: M-230, FEMSATL 4, 8, 10, 11, 12, 16 and 19 6364, under conditions described by Morand et al. 65 and Gérard et al. 28. The theoretic exclusion probability over the eight loci was 0.9995 for the 269 flowering adult trees.

Paternity analysis allows one to gauge the heterogeneity in male fecundity by estimating the effect of ecological and/or phenotypic variables on this fecundity: for example the effect of the floral phenotype 66, of the inflorescence morphology 67 or of the floral phenology 37. Here we adapted a mating model developed by Oddou-Muratorio et al. 24, which stems from the neighbourhood model 3132. This method allows one to estimate jointly the dispersal curve, level of pollen immigration, selfing rate and the heterogeneity in male fertility. It avoids type I and type II errors occurring in categorical paternity assignment (for a review see 68).

The reduction in effective male population density can have strong consequences: it may for example increase drift in natural populations and influence the rate of accumulation of beneficial and deleterious mutations 2474. In fact, unequal male fecundities and asynchronous flowering may drastically reduce the effective male population density 74. Using male fecundities estimated as described above, we thus estimated the reduction in effective male population density caused by heterogeneity in male fecundity under the complete model following Oddou-Muratorio et al. 24. This density is defined by d_em = N_em /A, where N_em is the effective male population size (i.e. the inverse of the probability that two pollen grains come from the same male) and A is the area covered by the study population. The reduction in effective male population density can be expressed as:

d e m d o b s = 1 N ( ∑ k ∈ Ψ ( ∏ i = 1 n f c ( k ) i ) ) 2 ∑ k ∈ Ψ ( ∏ i = 1 n f c ( k ) i ) 2       ( 8 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdsgaKnaaBaaaleaacqWGLbqzcqWGTbqBaeqaaaGcbaGaemizaq2aaSbaaSqaaiabd+gaVjabdkgaIjabdohaZbqabaaaaOGaeyypa0ZaaSaaaeaacqaIXaqmaeaacqWGobGtaaWaaSaaaeaadaqadaqaamaaqafabaWaaeWaaeaadaqeWbqaaiabdAgaMnaaDaaaleaacqWGJbWycqGGOaakcqWGRbWAcqGGPaqkaeaacqWGPbqAaaaabaGaemyAaKMaeyypa0JaeGymaedabaGaemOBa4ganiabg+GivdaakiaawIcacaGLPaaaaSqaaiabdUgaRjabgIGiolabfI6azbqab0GaeyyeIuoaaOGaayjkaiaawMcaamaaCaaaleqabaGaeGOmaidaaaGcbaWaaabuaeaadaqadaqaamaarahabaGaemOzay2aa0baaSqaaiabdogaJjabcIcaOiabdUgaRjabcMcaPaqaaiabdMgaPbaaaeaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGUbGBa0Gaey4dIunaaOGaayjkaiaawMcaamaaCaaaleqabaGaeGOmaidaaaqaaiabdUgaRjabgIGiolabfI6azbqab0GaeyyeIuoaaaGccaWLjaGaaCzcamaabmaabaGaeGioaGdacaGLOaGaayzkaaaaaa@6E05@

where d_obs is the observed male population density, fc(k)i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGMbGzdaqhaaWcbaGaem4yamMaeiikaGIaem4AaSMaeiykaKcabaGaemyAaKgaaaaa@33E9@ is the fecundity of male k belonging to the c^th class of the phenotypic trait i, and n is the number of traits under consideration. We also estimated the reduction in effective male population density due to dispersal features and phenology, including in the above sums all p_jk's and all g_d(kj)'s.

We performed a mixed-mating model analysis using the software MLTR 3.0 70, to assess the relation between selfing rate and flowering time, by estimating the mean outcrossing rates t_m in each phenological group (from the second to the fifth, because we had no seeds for the first group) from the multilocus genotypes of the seeds harvested on the mother trees, using an EM algorithm. Standard errors were computed by performing 1000 bootstrap replicates using families as resampling units. We also estimated the family-level t_m values (i.e. the mean outcrossing rates among seeds harvested from each mother-tree) 70.

If α was the error rate and if the paternal allele of the offspring was unambiguously known, we assigned as father with a probability α, a male homozygous for an allele that differ by 1 repeat unit from the offspring, and with a probability 1–2α, a male homozygous for the same allele.

This means that in equation 8, the standard transition probabilities T(g_o|gjo MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGNbWzdaWgaaWcbaGaemOAaO2aaSbaaWqaaiabd+gaVbqabaaaleqaaaaa@312B@, g_l) defined by Meagher (1986) were replaced by T(go|gjo,gl)=∏lociTloc(go,loc|gjo,loc,gl,loc) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@67C4@, where for each locus loc, and with the notations from 70:

T l o c ( { o 1 , o 2 } | { m 1 , m 2 } , { f 1 , f 2 } ) = D o 1 m 1 m 2 D ˜ o 2 f 1 f 2 + D o 2 m 1 m 2 D ˜ o 1 f 1 f 2 ( 1 + δ o 1 o 2 ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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r7aKnaaBaaaleaacqWGVbWBdaWgaaadbaGaeGymaedabeaaliabd+gaVnaaBaaameaacqaIYaGmaeqaaaWcbeaaaOGaayjkaiaawMcaaaaacqGGSaalaaa@818A@

with δ_ij = 1 if i = j and 0 if i ≠ j, Dom1m2=(δom1+δom22) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGebardaqhaaWcbaGaem4Ba8gabaGaemyBa02aaSbaaWqaaiabigdaXaqabaWccqWGTbqBdaWgaaadbaGaeGOmaidabeaaaaGccqGH9aqpdaqadaqaamaalaaabaacciGae8hTdq2aaSbaaSqaaiabd+gaVjabd2gaTnaaBaaameaacqaIXaqmaeqaaaWcbeaakiabgUcaRiab=r7aKnaaBaaaleaacqWGVbWBcqWGTbqBdaWgaaadbaGaeGOmaidabeaaaSqabaaakeaacqaIYaGmaaaacaGLOaGaayzkaaaaaa@4479@ is the Mendelian contribution from the mother, without any mistyping, and the contribution from the father accounting for possible mistyping:

D ˜ o f 1 f 2 = ( ( α δ o f 1 + 1 + α δ o f 1 − 1 + ( 1 − 2 α ) δ o f 1 ) + ( α δ o f 2 + 1 + α δ o f 2 − 1 + ( 1 − 2 α ) δ o f 2 ) 2 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGebargaacamaaDaaaleaacqWGVbWBaeaacqWGMbGzdaWgaaadbaGaeGymaedabeaaliabdAgaMnaaBaaameaacqaIYaGmaeqaaaaakiabg2da9maabmaabaWaaSaaaeaacqGGOaakiiGacqWFXoqycqWF0oazdaWgaaWcbaGaem4Ba8MaemOzay2aaSbaaWqaaiabigdaXaqabaWccqGHRaWkcqaIXaqmaeqaaOGaey4kaSIae8xSdeMae8hTdq2aaSbaaSqaaiabd+gaVjabdAgaMnaaBaaameaacqaIXaqmaeqaaSGaeyOeI0IaeGymaedabeaakiabgUcaRiabcIcaOiabigdaXiabgkHiTiabikdaYiab=f7aHjabcMcaPiab=r7aKnaaBaaaleaacqWGVbWBcqWGMbGzdaWgaaadbaGaeGymaedabeaaaSqabaGccqGGPaqkcqGHRaWkcqGGOaakcqWFXoqycqWF0oazdaWgaaWcbaGaem4Ba8MaemOzay2aaSbaaWqaaiabikdaYaqabaWccqGHRaWkcqaIXaqmaeqaaOGaey4kaSIae8xSdeMae8hTdq2aaSbaaSqaaiabd+gaVjabdAgaMnaaBaaameaacqaIYaGmaeqaaSGaeyOeI0IaeGymaedabeaakiabgUcaRiabcIcaOiabigdaXiabgkHiTiabikdaYiab=f7aHjabcMcaPiab=r7aKnaaBaaaleaacqWGVbWBcqWGMbGzdaWgaaadbaGaeGOmaidabeaaaSqabaGccqGGPaqkaeaacqaIYaGmaaaacaGLOaGaayzkaaaaaa@7C24@