We next investigated whether certain flower-colour genotypes sire more offspring than would be expected given local genotype frequencies, and whether this varies across the hybrid zone. We did this by comparing ’observed’ mating events, based on genetic and dispersal data, to ’expected’ mating events simulated based on dispersal only for mothers in each of five bins of approximately 300m (n=3, 6, 21, 27, 3 mothers per bin; figure \ref{120180}). We calculated observed and expect mating probabilities for each of 1000 sets of dispersal parameters and proportions of missing fathers from the trimmed MCMC results.

To calculate observed mating probabilities we used the likelihoods from FAPS that an individual candidate father had sired at least one offspring with a maternal plant based on genotype data, his distance from the mother, and dispersal parameters, integrating over uncertainty in sibling relationships \cite{Ellis2018}. For expected mating probabilities under random mating we calculated the probability of mating between each pair of mothers and candidate fathers based on the distance between them and dispersal parameters. We then summed these values over all candidates of each Rosea and Sulfurea genotypes, and over mothers within each spatial bin. We normalised likelihoods for each maternal genotype to sum to one to give relative probabilities than mothers in each bin received pollen from males of each flower colour genotype.

One explanation for apparent leptokurtosis in dispersal is incorrect assignment of paternity to candidates far from maternal plants, either because the true father was not sampled or due to insufficient statistical power of the marker set. This would inflate the tail of the dispersal kernel even if true dispersal were not leptokurtic (i.e. cause the shape parameter of the GND to be less than one, even when it is really greater or equal to one).

To test this, we first examined how often we should expect incorrect paternity assignment using this marker set, and the extent to which incorrect assignment and missing fathers would inflate apparent dispersal. We used FAPS to simulate offspring based on observed maternal and candidate male genotypes \cite{Ellis2018}. For each of the 60 mothers we sampled mates in proportion to their distance, assuming that the pollen dispersal distances are exponentially distributed with a mean of 10, 50, 100 and 200m. Since clustering offspring into sibships is the most computationally demanding part of the analysis, we instead consider paternity of individuals rather than sibships; this is valid because he are primarily interested in mating events, not of sibling relationships themselves. We simulated one offspring for each of N fathers for each mother, where N is the most likely number of full sibship families in observed families rounded to the nearest integer (figure \ref{167264}), to give a total of 433 offspring. We then calculated (1) the posterior probability of paternity for each true sire in the absence of information about dispersal, (2) mean distance between mothers and known true sires and (3) weighted mean distance between mothers and possible all sires, weighted by probability of paternity for all candidates with a probability >0.001 of having sired at least one offspring. To test dependency on sampling effort we then removed 10%, 20%, 30% or 40% of true sires at random from the dataset and recalculated weighted-mean distances between mothers and sires. We repeated this procedure on 100 replicate simulated datasets.

We then used a second set of simulations to investigate the extent to which a joint analysis of paternity and dispersal could mitigate the inflation of leptokurtosis. We generated offspring as described above assuming exponential dispersal of 50m only, and performed a simplified joint-analysis of paternity and dispersal by grid interpolation. We multiplied the matrix of paternity probabilities from genetic data by a matrix of dispersal probabilities from \( \Pr(d_{mj}|a,b,\lambda)\) using combinations of scale (\(5 \leq a \leq 150\)) and shape (\(0.1 \leq b \leq 2\)) parameter values, with fixed \(\lambda\) at the posterior mean of 0.82. We calculated the likelihood of each combination of parameters by summing joint probabilities of paternity over each candidate father, and multiplying those likelihoods over all 433 offspring. We then recorded the combination of dispersal parameters with the highest (log) likelihood. We also repeated this procedure for datasets excluding true sires as described above. We repeated these analyses on 100 replicate simulated datasets.

FAPS grouped the 937 offspring into between 434 and 437 full-sibling families of between one and 16 offspring, which showed very little dependence on dispersal parameters (figure \ref{167264}). We identified a total of 1879 possible mating events between candidate fathers and the 60 mothers for those sibships, of which identified 316 had posterior probabilities greater than 0.99 and 1050 had posterior probabilities less than 0.01. The data were compatible with broad range of plausible values for the proportion of missing fathers (mean=0.21, 95% credible intervals = 0.06, 0.42; figure \ref{266841}), but the posterior distribution was very similar to the prior distribution (figure \ref{804996}). For all plausible full sibling families in each maternal family the posterior probability that offspring alleles were drawn from population allele frequencies was lower than the posterior probability of paternity for at least one sampled candidate.

Pollen dispersal distances are characterised by many dispersal events between nearby plants, and a long tail of more distant mating events (figure \ref{731441}). 50% of the high-probability fathers were within 40m of the mother; distances to remaining fathers decayed slowly up to a maximum of 2398m. Consistent with this, the full pollen dispersal kernel inferred jointly with sibships and paternity consistently showed shape parameters less than one (posterior mean=0.40, 95% credible intervals: 0.329-0.483) with posterior-mean average dispersal distance of 78m (95% credible intervals: 52-140m) . Mixture parameter was strongly weighted towards signal coming from generalised normal dispersal rather than random draws from the population (mean=0.82, 95% credible intervals = 0.77, 0.93; figure \ref{266841}). These results indicate strong leptokurtosis in the pollen dispersal kernel.