Methods
System
We systematically monitored a population of house sparrows Passer domesticus on Lundy Island in the Bristol Channel, UK (51.11N, 4.40W), since 2000 (see Nakagawa et al. 2007; Ockendon, Griffith, and Burke 2009; Schroeder et al. 2012; Dunning et al. 2023). The sparrows on Lundy breed in nesting boxes, arranged into neighbourhoods broadly defined by building infrastructure or linear features. Females are socially monogamous, but genetically promiscuous (Schroeder et al. 2016), and, on Lundy, most have 2-3 broods of 4-5 eggs per breeding season (Westneat et al. 2014).
We collected tissue samples from nestlings at the natal site and from recaptured birds post-fledging and used the DNA extracted from those to allocate paternity with the help of >22 microsatellite loci (Dawson et al. 2012). We then constructed a near-complete genetic pedigree (see Schroeder et al. 2015), spanning 19 years, 2000 - 2019. All sparrows are fitted with a unique sequence of three coloured leg rings and a British Trust for Ornithology (BTO) coded metal ring (for details see Simons et al. 2015), which allowed us to later identify social pairs at the nest box. Dispersal to and from Lundy Island is limited. This, and our systematic and thorough monitoring, allowed us to determine the exact age of birds in years, and to know when they died, either from the rings of birds found dead or, defined as when ringed birds were not observed for more than two years (see Simmons et al 2015).
To measure female divorce, we first excluded females that only had a single brood, and thus, no opportunity to divorce their social mate. We also excluded 17 females that divorced mates following the death of their social partner, where the death of a social male occurred during the female breeding year. We removed offspring whose parents (either social or genetic) were missing or uncertain. We defined a divorce event where a female paired socially with a new male to that of her previous social partner, between broods but within years. This resulted in 353 female breeding years. These female years represented 920 broods by 190 females, 205 social fathers and 309 genetic sires between 2004 and 2019.
We defined a chick as extra-pair where they survived to the point of sampling on day two and the confirmed social- differed from the genetic father (the sire) in our pedigree. We counted the number of extra-pair offspring and the number of both social and extra-pair fathers within females within years (female years).
Models and permutations
To empirically test if females that divorce more often were also more likely to produce extra-pair offspring, as implied by intrasexual antagonistic pleiotropy theory, we ran two GLMM models with Bayesian Markov Monte-Carlo simulations, using MCMCglmm in R (Hadfield 2010; R Core team 2023):
The association of divorce with extra-pair paternity - To examine the link between female year divorce and extra-pair paternity, and because MCMCglmm cannot easily deal with proportion data, we ran a multinomial model with the number of extra-pair and social offspring per female year as response variables (see Hadfield 2010). We fitted female divorce, measured as the number of social partners within a female year, the number of broods she initiated, to control for increased opportunity for extra-pair offspring, and her age in years since hatching, to compensate for the effect of age on reproductive value (Hsu et al. 2017), as fixed effects. We also included Female ID and breeding year as random effects on the intercept to account for variation within those groups.
The effect of divorce on extra-pair male engagement - To examine the link between female divorce and engagement of extra-pair partners, we used a bivariate model structure, with the number of extra-pair partners within a given female year as the response variable. We again fitted female divorce, the number of broods initiated and her age in years since hatching as fixed effects. Female ID and breeding year were again modelled as random intercepts to account for variation within those groups. We first ran models using a Poisson distribution and logit link function, but those models failed to converge. Instead, we used a Gaussian distribution and link function and output estimates between the Poisson and Gaussian models were equivalent.
For all models, we used the default priors of the MCMCglmm package, and ran over 343,000 iterations, with a burn-in of 34,000 and a thinning interval of 200. We checked the posterior trace plots to ensure that autocorrelation was below 0.1 and that the effective sample sizes ranged between 1,000 and 2,000. The fixed effects were considered statistically significant when the 95% credible interval (CI) of its posterior distribution did not span zero.
To test that our results were biologically meaningful, and not the outcome of random chance, we ran a series of permutations. We removed the link between female ID and reproductive traits by building random matrices between males and females to re-run our models. First, we simulated 1000 breeding events, by shuffling the number of offspring and extra-pair offspring between females while maintaining age structure. We then repeated these steps to simulate the number of extra-pair partners for each female. For each permutation, we ran an identical GLMM model to those described above. We dropped the bottom 2.5% of the lower credible intervals, and the top 2.5% of the upper credible intervals, to leave 95% of the 1000 credible intervals. We then extracted the minimum lower and maximum upper credible interval and the mean estimate. We interpreted significance – that is, our results were unlikely to have occurred by chance – where the observed posterior mean fell outside the span of the permuted credible intervals.
Results
From 533 female breeding years, we identified 4963 offspring of known social and genetic parentage including 932 extra-pair offspring, 1.7 per female year (0 – 11, sd: 1.77), from 1403 broods (2.6 per female year, sd: 0.68). Females who engaged in extra-pair behaviour had a mean of 2.5 extra-pair offspring per female year (1 – 11, sd: 1.6). Within female breeding years, 120 females divorced their social partners on at least one occasion (110 once, and 10 twice, 1.24 per social partners female year: sd 0.47), where 413 remained faithful to a single social partner over multiple broods within a female year.
Neither the number of social fathers nor the number of broods was significantly linked with the proportion of extra-pair offspring that hatched within a female year. However, the number of social fathers and the number of broods per female year were positively linked to the number of extra-pair partners she chose within a breeding year. The log odds of having an extra-pair partner increased by 0.43, or 1.54 extra-pair partners, per social partner, and by 0.24, or 1.27 extra-pair partners per brood respectively (Figure 1). Female age was not linked to either the proportion of extra-pair offspring or the number of extra-pair sires.
Neither the number of broods nor the number of social fathers was associated with the number of extra-pair partners in our randomizations. The observed estimates all fell outside of the simulated confidence intervals (Figure 1). Observed posterior means fell outside of simulated 95% confidence intervals for both broods (0.27, -1.29 – 0.13) and social fathers(0.81, -0.00 – 1.86), implying that our results are not the result of chance. Our results support the hypothesis that females who divorce social partners regularly also engage more extra-pair males than those who maintain social monogamy.