(C) Age- and density-dependent reproduction and survival
For the age- and density-dependent models, we utilized annual data on reproduction and survival for all individuals present within the studied time periods (Table S1). In total, there were 5247 records from 2729 individuals (1325 females and 1361 males). We studied how annual measures of reproduction and survival resulted in the observed means and (co)variance of individual life-histories by building age- and density-dependent reproduction and survival models. First, we estimated the age- and sex-specific annual number of recruits using univariate mixed-effect models on the following form:
\(\eta_{\text{ghijk}}=c_{g}+\beta_{1g}S_{i}+\beta_{2g}a_{\text{hi}}\ +\beta_{3g}a_{\text{hi}}S_{i}+\ \beta_{4g}{\overset{\overline{}}{n}}_{k}+\beta_{5g}(n_{\text{jk}}-{\overset{\overline{}}{n}}_{k})\ +I_{\text{gi}}+Y_{\text{gj}}+P_{\text{gk}}+e_{\text{ghijk}}\), (eq. S6)
We model survival and annual number of recruits at age h of individual i breeding in year j on population k,.where η1hijk = logit(survival) and η2hijk = log(number of recruits). Both variables were modelled with the same fixed and random effect structure, however the residual error for survival was assumed to be that of a binomial distribution and thus its variance was fixed to 1, whereas the residual variance for the number of recruits was assumed to be that of an over-dispersed Poisson distribution, where we estimated the over-dispersion component. \(\beta_{1g}\) represents the average sex differences in yearly survival and reproduction, and \(\beta_{2g}\)represents age-specific survival and reproduction. Where gdenotes whether the effects are for reproduction or survival. We treated age as a two-level categorical variable (first year breeders versus older individuals) and also fitted an interaction with sex, as we were expecting sex-specific (\(\beta_{3g}\)) patterns of age-dependent reproduction and survival (Stubberud et al. 2017). These models had also as fixed effect \((\beta_{4g}\)) for the mean population size (\({\overset{\overline{}}{n}}_{k}\)) of population (\(k\)) and the effect (\(\beta_{5g}\)) of yearly deviations from the mean population size in number of individuals\((n_{\text{jk}}-{\overset{\overline{}}{n}}_{k})\). This within-subject centering approach allowed us to model density regulation accounting for differences in the mean population size between populations, and allowed us to test for any spatial versus temporal effects of population size in recruitment and survival (van de Pol & Wright 2009). Here, year-specific values \((Y_{\text{gj}})\), population-specific values (\(P_{\text{gk}}\)), individual-specific values \(I_{\text{gi}}\), and within-individual residual deviations\(e_{\text{ghijk}}\), were all assumed to come from separate normal distributions for each life-history trait with variances \(V_{Y_{g}}\),\(V_{P_{g}}\) , \(V_{I_{g}}\), and \(V_{e_{g}}\).
Second, we fitted a multivariate mixed-effects model, where we estimated the covariance between yearly survival and recruit production at the individual and residual levels:
\(\par \begin{bmatrix}I_{s}\\ I_{f}\\ \end{bmatrix}\sim mvn(0,\ \mathbf{V}_{\mathbf{I}})\);\(\mathbf{V}_{\mathbf{I}}\mathbf{=}\par \begin{bmatrix}\text{\ \ }\mathbf{V}_{\mathbf{I}_{\mathbf{1}}}&\\ \mathbf{C}_{\mathbf{I}_{\mathbf{12}}}&\mathbf{V}_{\mathbf{I}_{\mathbf{2}}}\\ \end{bmatrix}\) , (eq. S3a)
\(\begin{bmatrix}e_{s}\\ e_{f}\\ \end{bmatrix}\sim mvn(0,\ \mathbf{V}_{\mathbf{e}})\);\(\mathbf{V}_{\mathbf{e}}\mathbf{=}\begin{bmatrix}\ \mathbf{V}_{\mathbf{e}_{\mathbf{1}}}&\\ \mathbf{C}_{\mathbf{e}_{\mathbf{12}}}&\mathbf{V}_{\mathbf{e}_{\mathbf{2}}}\\ \end{bmatrix}\) , (eq. S3b)
where \(I_{s}\) and \(I_{f}\) represent an individuals average survival and annual number of recruit production in the latent scales, while\(e_{s}\) and \(e_{f}\) represent the deviations of each breeding season for each individuals mean values, also in the latent scale. \(V_{I}\)represents the among individual variance covariance matrix, with elements \(V_{I_{1}}\), \(V_{I_{2}}\) and \(C_{I_{12}}\), representing among-individual variance in survival, annual reproduction and their covariance, respectively. Whereas \(\mathbf{V}_{\mathbf{e}}\) represents the within-individual variance covariance matrix, with elements\(V_{e_{1}}\), \(V_{e_{2}}\) and \(C_{e_{12}}\), representing within-individual variance in survival, annual reproduction and their covariance, respectively. Note that \(V_{e_{1}}\) was fixed to one by convention.