Double decomposition: decomposing the variance in subcomponents of male extra-pair reproductive success

1. Extra-pair reproductive success (EPRS) is a key component of male fitness in socially monogamous systems and could cause selection on female extra-pair reproduction if extra-pair offspring (EPO) inherit high value for EPRS from their successful extra-pair fathers. However, EPRS is itself a composite trait that can be fully decomposed into subcomponents of variation, each of which can be further decomposed into genetic and environmental variances. However, such decompositions have not been implemented in wild populations, impeding evolutionary inference. 2. We first show that EPRS can be decomposed into the product of three life-history subcomponents: the number of broods available to a focal male to sire EPO, the male's probability of siring an EPO in an available brood and the number of offspring in available broods. This decomposition of EPRS facilitates estimation from field data because all subcomponents can be quantified from paternity data without need to quantify extra-pair matings. Our decomposition also highlights that the number of available broods, and hence population structure and demography, might contribute substantially to variance in male EPRS and fitness. 3. We then used 20 years of complete genetic paternity and pedigree data from wild song sparrows (Melospiza melodia) to partition variance in each of the three subcomponents of EPRS, and thereby estimate their additive genetic variance and heritability conditioned on effects of male coefficient of inbreeding, age and social status. 4. All three subcomponents of EPRS showed some degree of within-male repeatability, reflecting combined permanent environmental and genetic effects. Number of available broods and offspring per brood showed low additive genetic variances. The estimated additive genetic variance in extra-pair siring probability was larger, although the 95% credible interval still converged towards zero. Siring probability also showed inbreeding depression and increased with male age, while the numbers of available broods and offspring per brood did not. 5. Our results indicate that the probability that a male will sire an EPO in an available brood is the primary source of genetic variation in male EPRS, implying that the evolution of female extra-pair reproduction could be facilitated by genetic covariance with this subcomponent of EPRS.