(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.