Most flowering plants mate and disperse offspring locally because of their sessile habit. As a consequence, the density and spatial distribution of individuals within populations play an important role in determining variation in reproductive success (Levin & Kerster 1969; Antonovics & Levin 1980; Kunin 1993; Groom 1998; Davis et al. 2004). Plant species that are sexually polymorphic are likely to be particularly sensitive to the contingencies of spatial distribution. This is because polymorphic populations are reproductively subdivided into separate sexes (e.g. dioecy), or into distinct mating groups that differ in floral morphology (e.g. heterostyly). As a result, a sexual morph's reproductive success may depend not only on patch density, but also on the local frequency of the sexual morphs with which it can mate (Wyatt & Hellwig 1979; Barrett & Thomson 1982; Meagher 1991; Heilbuth et al. 2001; Wilson & Harder 2003). Previous studies have documented frequency-dependent selection in the female fertility of polymorphic species (Eckert et al. 1996; McCauley & Brock 1998), as well as density dependence in monomorphic species (Silander 1978; Groom 1998). However, to our knowledge, no study has examined the joint effects of density- and frequency-dependent processes on plant fitness, even though pollinators are known to respond to both the density of flowers and the frequency of floral morphs (Smithson & MacNair 1997).

Here we employ neighbourhood models (Mack & Harper 1977; Weiner 1982; Watkinson et al. 1983; Pacala & Silander 1985) to investigate the spatial determinants of mating success in Narcissus assoanus, a sexually polymorphic flowering plant with stigma-height dimorphism. Populations of this species are composed of two floral morphs that differ in style length (Baker et al. 2000a,b). In long-styled plants (hereafter L-morph), the stigma is positioned at the same height or above the two stamen levels, whereas in short-styled plants (hereafter S-morph), the stigma is positioned well below the two stamen levels. In Narcissus species with stylar polymorphisms, including N. assoanus, plants are usually self-incompatible, but fully cross-compatible with other plants in a population regardless of morph identity (Barrett et al. 1996). This condition differs from typical heterostylous species, which usually exhibit both self- and intra-morph incompatibility (Barrett & Cruzan 1993). As a result, in Narcissus floral morphology has a much stronger influence on governing patterns of outcrossed mating and fertility than in heterostylous plants with self- and intra-morph incompatibility (Barrett & Harder 2005).

Population surveys of N. assoanus have revealed wide variation in the relative frequencies of the L- and S-morphs (Baker et al. 2000a), which is thought to reflect asymmetries in the mating patterns of the two morphs (Barrett et al. 1996; Baker et al. 2000b; Cesaro et al. 2004). Specifically, the S-morph is likely to be less proficient at intra-morph mating than the L-morph because the large spatial separation of stigmas and anthers (herkogamy) in the S-morph reduces the precision of intra-morph pollen transfer. Manipulative field experiments in N. assoanus provide evidence to support this hypothesis as seed set is reduced in patches containing only the S-morph (Thompson et al. 2003). Morph-specific differences in mating and fertility imply that the spatial distribution and frequencies of floral morphs within local neighbourhoods are likely to play an important role in determining variation in mating success. Previous studies on Narcissus have largely dealt with population-level morph ratios (e.g. Baker et al. 2000a,b; Arroyo et al. 2002; Barrett et al. 2004) and have ignored the reproductive consequences of within-population variation in morph structure.

Here we test two hypotheses that deal with the relation between local floral morph composition and female fertility, and are consequences of morph-specific differences in floral morphology. First, if well developed herkogamy in the S-morph limits intra-morph pollen transfer, we would predict that the fertility of this morph should be insensitive to the number of S-morph neighbours, despite being fully compatible with them. In contrast, the fertility of the S-morph should increase with the number of L-morph neighbours. Second, we predict that because plants of the L-morph can engage in both inter-morph and intra-morph mating, the fertility of this morph should be insensitive to morph identity and fertility will increase similarly with both L- or S-morph neighbours. These hypotheses therefore propose that most mating in the S-morph results from inter-morph pollen transfer (disassortative), whereas mating in the L-morph involves both inter- and intra-morph (assortative) pollen transfer. In essence, mating in the L-morph should be random with respect to the morph composition of local neighbourhoods, whereas mating in the S-morph is likely to be largely disassortative. These hypotheses, while specific to our study system, have broader implications because, if supported, they would demonstrate a level of context-dependence not generally recognized in studies of plant mating.

With restricted seed dispersal and the proposed patterns of mating and fertility, we also predict differences in the local clustering of morphs within populations of N. assoanus. Because of the inheritance of style length in Narcissus (L-morph ss; S-morph S-; see Baker et al. 2000b) and mostly L-biased morph ratios of populations in our study area (Baker et al. 2000a), plants of the two morphs are likely to segregate different average ratios of the L- and S-morphs. Specifically, high levels of disassortative mating in the S-morph should result in equal frequencies of L- and S-styled plants, whereas due to random mating, the progeny of the L-morph should reflect local morph ratios. We tested this hypothesis by investigating the patterns of style morph clustering within natural populations of N. assoanus.

Zero-inflated count data. Count data are often analysed using standard Poisson regression methods, based on the assumption that the data exhibits a unimodal distribution. However, visual inspection of our data revealed that seed set was not unimodal because approximately 20% of plants failed to set seed. Therefore, we used a ZIP distribution (Lambert 1992; Heilbron 1994) to account for the moderately high incidence of reproductive failure. The ZIP distribution accommodates zero-inflation by specifying the probability of zero and non-zero observations separately

2.1a

2.1b

Regressing seed set against local density. In standard Poisson regression, only the mean of the Poisson distribution, λ, varies as function of covariates. In a ZIP regression, however, both λ and p may vary as a function of covariates. Thus, for the L-morph, we estimated λ and p as linear functions of local density

2.2a

2.2b

2.3a

2.3b

It is reasonable to assume that seed set will vary with the local density of plants because pollen supply has previously been shown to limit seed set (Baker et al. 2000c; Thompson et al. 2003). However, it is not known how far pollen is dispersed by insects, so it was difficult to determine a priori how large R should be for calculation of N_l and N_s. Nevertheless, the scale at which neighbours influence seed set can be inferred from the data itself (Pacala & Silander 1985). For example, it is possible to calculate N_l and N_s using a range of different values for R (e.g. 1m to the maximum distances in populations), and then determine which value of R provides the best fit to the observed seed set. Thus, R becomes a parameter, hereafter called neighbourhood radius that we estimated along with the other parameters listed above.

To estimate the parameters, we used a simulating annealing algorithm to search for parameter values that maximized the log-likelihood of the data, thereby obtaining a maximum likelihood estimate for each of the parameters included in equations (2.2a,b) and (2.3a,b) (α, β, ϕ, σ, θ, and R). We also calculated confidence intervals for each parameter as follows. First, we generated 10000 random values for each of the model parameters, and combined them to obtain 10000 unique sets of the model parameters. Then, we calculated the log-likelihood of each parameter set, and its deviance from the maximum log-likelihood (2(L−L_max)), and excluded those sets for which the deviance exceeded the critical value of the χ² distribution (α=0.05, d.f.=1). Finally, we selected the minimum and maximum parameter values from the remaining sets and used these as the 95% confidence limits.

Hypothesis tests. The regression models outlined above (equations (2.2a,b) and (2.3a,b)) embody two assumptions that correspond to the first two hypotheses outlined in §1. First, pollen transfer in the S-morph is assumed to be disassortative, because λ_s and p_s are allowed to covary with N_l but not N_s (equations (2.3a,b)). Second, pollen transfer in the L-morph is assumed to involve both assortative and disassortative components, because λ_l and p_l are allowed to covary with N_s as well as N_l (equations (2.2a,b)).

To test these two assumptions, and the related hypotheses, we formulated two alternative regression models and compared them to our null models (equations (2.2a,b) and (2.3a,b)) using likelihood ratio tests. The first alternative model allows λ_s and p_s to covary with N_s as well as N_l, as would be expected if the S-morph were capable of significant assortative pollen transfer

2.4a

2.4b

The second alternative model allows λ_l and p_l to covary with N_l but not N_s, as would be expected if L-morphs were not capable of assortative pollen transfer

2.5a

2.5b

The maximum log-likelihood of these alternative models was computed as described previously, then subtracted from the maximum log-likelihood of the full model (equations (2.2a,b) and (2.3a,b)). The alternative model was deemed to fit the data better than the null model if the difference in the log-likelihood exceeded twice the critical value of the χ² distribution (α=0.05, d.f.=q, where q is the difference in the number of model parameters).

We also predicted that pollen transfer in the L-morph was random with respect to morph (i.e. seed set increases to an equal degree with the local density of plants of the L- and S-morphs). An assumption of equal slopes was not built into the null model (equations (2.2a,b)) because preliminary analyses showed that, among plants that set seed, the number of seeds (λ_l) increased faster with the local density of the L-morph than it did with the local density of the S-morph (i.e. θ>). However, it was also noted that the probability of observing surplus zeros (p_l) decreased with the local density of the S-morph (i.e. β<0), thereby compensating for the fact λ_l increased faster with respect to N_l than N_s. In other words, the function used to specify total seed set () increases approximately equally with both N_l and N_s, even though λ_l and p_l do not.

Unfortunately, it was not possible to conduct a formal test of the equal slopes hypothesis because is a compound function of λ_l and p_l, and is therefore slightly nonlinear with respect to both N_l and N_s (i.e. there is no single slope with respect to N_l or N_s). However, we plotted the fitted function in three dimensions ( versus N_l and N_s) to visually evaluate whether the slopes are approximately equal (see §3). We also calculated the average slope of with respect to N_l and N_s at equally spaced locations (i=1,…,25) on the N_l–N_s plane

2.6a

2.6b

The null models (equations (2.2a,b) and (2.3a,b)) embody an additional assumption that we did not discuss above, i.e. that seed set of both morphs increases to an equal degree with the local density of the L-morph (because α, β, σ and θ are common to both equations). To evaluate this assumption, we formulated alternative models that allow λ_s and p_s to differ from λ_l and p_l even in the absence of S-morph neighbours

2.7a

2.7b

These models were compared to the null models using likelihood ratio tests as described previously.

Clustering of morphs. The clustering of morphs is likely to influence seed set if both pollen and seed dispersal are local. To test for clustering, we calculated the average difference between the population and neighbourhood morph ratios as follows

2.8a

2.8b

Table 1 Sample sizes and means for the measurements of plants from eight populations of Narcissus assoanus studied in southwestern France. (Number of individuals per floral morph (N L, N S), number of total individuals, number of flowers, the morph ratios (number (more ...)

Figure 1 The mean seed set of (a, b) the L-morph and (c, d) S-morph in eight populations of N. assoanus in relation to the number of plants of the L- and S-morph in a neighbourhood radius R of 10.04m. We calculated mean values by dividing the number of (more ...)

Figure 2 The fitted relation between morph-specific seed set and the number of neighbours in eight populations of Narcissus assoanus. (a) The fitted relation between the seed set of the L-morph and the number of neighbours (equations (2.3a,b)). Data points indicate (more ...)

The observed seed set of the S-morph increased with the number of neighbouring L-morphs (figure 1c), but was insensitive to the number of neighbouring S-morphs (figure 1d). This implies that most pollen transfer to the S-morph was disassortative (figure 1c). This was corroborated by the likelihood ratio test: the alternative model allowing both assortative and disassortative pollen transfer (equations (2.4a,b)) did not fit the data significantly better (p>0.05, d.f.=2) than the null model (equations (2.3a,b); strictly disassortative pollen transfer). Thus, the null model was the most parsimonious model for explaining fertility relations in the S-morph (figure 2b).

The final likelihood ratio test also confirmed that seed set of the two morphs increased to an equal degree with the number of neighbouring L-morphs. In particular, the alternative model (equations (2.8a,b), in which λ_s and p_s are allowed to differ from λ_l and p_l) did not provide a significantly better fit (p>0.05, d.f.=4) than the null model (equations (2.3a,b)).

Figure 3 The mean difference between the population morph ratio and the morph ratio of plants surrounding individuals of the L- and S-morph in eight populations of Narcissus assoanus calculated for each of 30 different neighbourhood radii (1–30m; (more ...)

Morph ratios in sexually polymorphic plants are a visible signature of the mating patterns that have occurred in populations during preceding generations. The distributions of morphs within a population also provide the spatial template for subsequent mating patterns since pollen dispersal is usually local, particularly in animal-pollinated species. In N. assoanus, we have demonstrated that the floral morph composition of local neighbourhoods has an important influence on variation in female fertility. By inferring expected patterns of pollen transfer based on floral morphology, we predicted the relations between local morph ratios and patterns of seed set. Below we discuss the likely mechanisms responsible for our results and also highlight the strong context-dependence of mating opportunities in structured plant populations that arise from density- and frequency-dependent processes. We argue that such context-dependence may be a general feature of the ecology of plant mating.

Symmetrical disassortative mating and 1:1 morph ratios occur rarely in Narcissus because floral morphology rather than physiological mechanisms governs mating patterns (Barrett & Harder 2005). Imperfect sex-organ reciprocity and the associated differences in herkogamy between the floral morphs have been proposed as a major cause of asymmetrical mating patterns and biased morph ratios (Barrett et al. 1996; Baker et al. 2000a,b). This explanation assumes that, unlike typical heterostylous species, the floral morphs in Narcissus are not functionally equivalent as mating partners. This hypothesis was supported by our results. There were significant differences between the floral morphs in their reproductive behaviour, depending on the morph identity of neighbours (figure 2). Pollen transfer and mating in N. assoanus is, therefore, context-dependent because of the spatial heterogeneity of morph frequencies at a local scale. This spatial heterogeneity is further reinforced because asymmetrical mating results in different patterns of segregation in the floral morphs.

The fertility of L-styled plants increased significantly with the total number of plants in local neighbourhoods. This pattern occurred regardless of morph identity implying that the L- and S-morph were functionally equivalent as paternal mating partners (figures 1a,b and 2a). In the L-morph, density-dependent rather than frequency-dependent processes play a more significant role in governing fertility variation. In contrast, the fertility of the S-morph was influenced by the morph identity of plants in local neighbourhoods. Increased fertility was positively associated with the number of neighbouring L-styled plants, whereas the number of S-styled plants had no influence on variation in seed set (figures 1c,d and 2b). Hence, both density- and frequency-dependent control of fertility variation was evident in the S-morph. This difference in mating behaviour of the floral morphs may be associated with their contrasting evolutionary histories and numerical occurrence in contemporary populations (reviewed in Barrett & Harder 2005).

What specific features of sex-organ deployment promote asymmetrical mating in N. assoanus? Both floral morphs possess upper-level anthers positioned for pollen donation to stigmas of the L-morph, but lower-level anthers differ consistently in position (see fig. 1 in Baker et al. 2000a). In the S-morph, lower-level anthers are positioned just below upper-level anthers and both stamen levels probably function primarily in pollen donation to stigmas of the L-morph. In contrast, lower-level anthers of the L-morph are positioned further down the floral tube and probably provide most of the pollen that fertilizes ovules of S-styled plants. Because of these differences in lower stamen position, the S-morph is probably mostly engaged in inter-morph mating, whereas stamen differentiation in the L-morph promotes both intra- (via upper-level anthers) and inter-morph (via lower-level anthers) mating. The morph-specific differences in the relations between local morph ratios and seed set that we observed in our study support these inferences on the influences of stamen position on pollen transfer. Moreover, phenotypic models of pollen transfer (Barrett et al. 1996) and genetic mating models (Baker et al. 2000b) that take into account these specific details of morphology in Narcissus demonstrate that asymmetrical mating can result in the patterns of morph-ratio variation that is observed in N. assoanus.

We also observed differences in the clustering of N. assoanus morphs that is consistent with inferred morph-specific patterns of mating and restricted seed dispersal (Hamrick & Nason 1996; Stehlik & Holderegger 2000; Kalisz et al. 2004). On the scale of less than 5m, average morph ratios in the vicinity of S-morph individuals were less L-biased than the mean population morph ratio, whereas average morph ratios in the vicinity of L-morph individuals were more L-biased than the average population morph ratio (figure 3). This result suggests that the difference in spatial clustering may have contributed to the observed average lower fertility of the S-morph (table 1). However, deviations from the average population morph ratio were small, with maximum differences of less than 5% for both morphs (figure 3). Furthermore, at a scale of 10m (the scale we have estimated to be most relevant to seed set), neighbourhood morph ratios were not discernibly different than population morph ratios. This result implies that pollen dispersal distances in N. assoanus exceed the size of the clusters created by local seed dispersal.

We have used neighbourhood models to examine the relation between female fertility and the local morph composition of neighbourhoods. These models enabled us to test several hypotheses concerning the patterns of pollen transfer and mating in populations of N. assoanus. Although similar methods have been employed to infer spatial patterns of plant competition (Mack & Harper 1977; Weiner 1982; Watkinson et al. 1983; Pacala & Silander 1985), this is their first application to the study of mating patterns in plant populations. Measurements of pollen dispersal and patterns of assortative and disassortative mating are technically demanding since they require pollen markers and hypervariable genetic loci, respectively, that are not available for most plant species. The modelling framework used here in combination with manipulative field experiments (e.g. Thompson et al. 2003) can provide an alternative approach for determining the complex ecological and genetic mechanisms that govern mating success in plants.

We thank Rolf Holderegger and Suzanne Barrett for field assistance and John Thompson for logistical advice. We are grateful to Kathryn Hodgins and Angela Baker for comments on the manuscript. This research was funded by NSERC Discovery grants to J.P.C. and S.C.H.B.; I.S. was supported by a post-doctoral fellowship from the Canada Research Chairs Program to S.C.H.B