Flower colour and male
fitness
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.
Power
analyses
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.
Results
Dispersal is
leptokurtic
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.