(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