(A) Statistical models and simulation equations
We used simulations to study how the observed pattern of covariation among life-histories, was caused by the among- and within individual variation in the annual reproduction and survival. More specifically we wanted to assess how the negative covariance between life span, and reproductive rate and generation time and reproductive rate could be caused by a positive covariance between annual reproduction and survival. To this end, we used the estimated parameters in the models of annual survival and reproduction (Table 3) to simulate individual life-histories, and from that, estimate the patterns of covariation between individual life-histories that arise from the annual estimates of annual survival and recruit production.
We used a modified version of equation 2 to simulate individual life-histories through time.
\(\eta_{\text{ghij}}=c_{g}+\beta_{2g}a_{\text{hi}}\ +\ \beta_{4g}{\overset{\overline{}}{n}}_{k}+\beta_{5g}n_{\text{jk}}\ +I_{\text{gi}}+Y_{\text{gj}}+e_{\text{ghij}}\)(eq. S8)
We simulated annual survival and annual number of recruits at ageh of individual i breeding in year j . Where η1hijk = logit(survival) and η2hijk = log(number of recruits). The number of recruits and whether an individual survived to the next year was simulated using values for the different components of this equation that were estimated for our population. Where \(\beta_{1g}\) represents the average sex differences in yearly survival and reproduction, and \(\beta_{2g}\) represents age-specific survival and reproduction. Where g denotes whether the effects are for reproduction or survival. As in our analyses, the effect of age was treated as a two-level categorical variable, where first year breeders had lower reproduction compared to older individuals as a function of the estimated parameters in table 2. For simplicity we modelled the male and female population separately based on the sex specific values of survival and reproduction. We ignored population differences and simulated a single meta-population, were population wide differences where simulated as individual differences. Here, year-specific values \(\left(Y_{\text{gj}}\right)\) for survival and reproduction were simulated from a normal distribution with variance equal to the one we estimated in our models.
Importantly individual specific values and within individual realization in different years, where simulated from multivariate distributions with variance-covariance matrix \(V_{I}\) and \(\mathbf{V}_{\mathbf{e}}\), for the among and within individual effects.
\(\par \begin{bmatrix}\mathbf{I}_{\mathbf{s}}\\ \mathbf{I}_{\mathbf{f}}\\ \end{bmatrix}\sim mvn(0,\ \mathbf{V}_{\mathbf{I}})\);\(\mathbf{V}_{\mathbf{I}}\mathbf{=}\par \begin{bmatrix}\text{\ \ }V_{I_{1}}&\\ C_{I_{12}}&V_{I_{2}}\\ \end{bmatrix}\) , (eq. S9a)
\(\begin{bmatrix}\mathbf{e}_{\mathbf{s}}\\ \mathbf{e}_{\mathbf{f}}\\ \end{bmatrix}\sim mvn(0,\ \mathbf{V}_{\mathbf{e}})\);\(\mathbf{V}_{\mathbf{e}}\mathbf{=}\begin{bmatrix}\ V_{e_{1}}&\\ C_{e_{12}}&V_{e_{2}}\\ \end{bmatrix}\) , (eq. S9b)
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. Where\(V_{I}\) 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. Based on the patterns of survival and reproduction we simulated the next year breeding individuals and so on for 20 years. We simulated 1000 metapopulations and estimated the covariance in life-history trade-offs between scenarios of the observed positive covariance, a scenario of negative covariance (opposite sign of what was observed) and no covariance between survival and reproduction