Calculate the probability of observing a parent-offspring pair (calc_POP) and
half-sibling pair (calc_HSP) for closed-kin mark recapture (CKMR) for an age-structured
model.
Arguments
- t
Vector, capture year of parent
i- a
Vector, age at capture of parent
i- y
Vector, birth year of offspring
j- N
Abundance of mature spawners. Matrix by
[y, a]- fec
Fecundity schedule of mature spawners. Matrix by
[y, a]- yi
Vector, birth year of sibling
i. Must be older than siblingj.- yj
Vector, birth year of sibling
j.- Z
Instantaneous total mortality rate. Matrix by
[y, a]
Parent-offspring pairs
The parent-offspring probability is calculated from Bravington et al. 2016, eq 3.4:
$$p_{\textrm{POP}} = 2 \times \dfrac{f(y_j,y_j - (t_i - a_i))}{\sum_a f(y_j,a) N(y_j,a)}$$
where \(y_j - (t_i - a_i)\) is the parental age in year \(y_j\). Scalar 2 accounts for the fact
that the parent could be either a mother or a father.
calc_POP is vectorized with respect to t, a, and y.
Half-sibling pairs
The half-sibling probability is calculated from Bravington et al. 2016, eq 3.10, and expanded by Hillary et al. 2018, Supplement S2.8.1 for age-specific survival and fecundity of the parent:
$$p_{\textrm{HSP}} = 4 \times \sum_a\left( \dfrac{N(y_i, a)f(y_i, a)}{\sum_{a'} N(y_i, a')f(y_i,a')}\times \exp(-\sum_{t = 0}^{y_j - y_i - 1} Z(y_i + t,a + t))\times \dfrac{f(y_j,a+y_j-y_i)}{\sum_{a'} N(y_j,a')f(y_j,a')} \right) $$
The first ratio is the probability that a fish at age \(a\) in year \(y_i\) is the parent of \(i\).
The exponential term is that fish's survival from year \(y_i\) to \(y_j\).
The second ratio is the probability that the parent of \(i\), age \(a+y_j-y_i\) in year \(y_j\), is the parent of \(j\).
The parent is not observed in the HSP, so we sum the probabilities over all potential ages in year \(y_i\).
calc_HSP is vectorized with respect to yi and yj.
References
Bravington, M.V. et al. 2016. Close-Kin Mark-Recapture. Stat. Sci. 31: 259-274. doi:10.1214/16-STS552
Hillary, R.M. et al. 2018. Genetic relatedness reveals total population size of white sharks in eastern Australia and New Zealand. Sci. Rep. 8: 2661. doi:10.1038/s41598-018-20593-w