Calculating post-reproductive representation
To calculate PrR two data series were obtained, the lxand mx series. The first series, lx, is
the probability of survival to a given age x, obtained from the survival
model output (Fig. 2). The second, mx, is calculated as
the proportion of offspring born to females of age x out of all females
that were alive at that age. To get mx we used the age
of females, where known or estimated year of birth was available, and
the age at which females gave birth. If a female was not observed in a
given year it was recorded as “no birth”.
Mathematically, PrR is based on lX and
ex, where ex is the life expectancy at
age x. A multiplication of these terms gives TX (the
total individual years lived after age x). PrR is then calculated from
Tx at age B and M, which are the ages, where 5% and
95% of female fecundity has been realised (inferred from
mx). Thus, the formula for calculating PrR is:
\begin{equation}
PrR=\ \frac{T_{M}}{T_{B}}=\frac{l_{M}}{l_{B}}*\frac{e_{M}}{e_{B}}\nonumber \\
\end{equation}The input was a lifetable consisting of lx (from the
basta model output) and mx (calculated from the
observational data). To test the statistical significance of the PrR
value it was tested against the null hypothesis that survivorship and
fecundity declines with the same rate, which would lead to PrR = 0. We
simulated 9999 populations of 1000 individuals, where this null
hypothesis was true and compared each of these null populations to each
permutation of the observed population. The simulated null populations
were generated based on the demographic parameters of the given killer
whale population. The p-value was obtained by evaluating how many of
these simulated populations had a PrR greater than or equal to the PrR
values obtained form the observed populations, with the number of
samples included in both the numerator and
denominator1,63.