Appendix A. Four different gametic patheways.
Eq. 2 is a simplified version of Hill’s results for separate sexes. More generally, with separate sexes it can be important to consider the four different pathways by which gametes can be transmitted across generations: male parent – male offspring (denoted by mm), male parent – female offspring (mf), female parent – male offspring (fm) and female parent – female offspring (ff). In a stable population, the means for the mm and ff pathways are _{} = 1 and the other two means are dictated by the sex ratio of newborns (_{}). Finally, let the covariance of the numbers of male and female offspring from each male parent be Cov(mm,mf) and the corresponding value for female parents be Cov(fm,ff). Then, effective size can be calculated as (Hill 1972, Eq. 17; Hill 1979, Eq. 9):
(A.1) |
Rao et al. (1973) showed that the covariance in the numbers of males and females in a family is a simple function of the sex ratio of newborns and the difference between the mean and variance of family size:
(A.2) |
From Eq. A.2 it is easy to see that if reproductive variance is Poisson the covariance is 0, but it is positive if _{}
The associated variance terms (V_{mm}, V_{mf}, V_{fm}, V_{ff}) for each pathway in Eq. A.1 depend on sex-specific means and variances in reproductive success, the newborn sex ratio, and the Poisson scaling factor. Specifically, it can be shown that
These equations show that the variances for each pathway are a product of _{} for that pathway and a scaling term that has two components: the Poisson scaling factor (a) that characterizes the relationship between mean and variance of lifetime reproductive success among same-aged, same-sex individuals, and a term that represents additional variance arising from differences in _{} for individuals with different age at death.
Table 4 shows values of these variables based on the life table data and sex-specific means and variances of k. Using these results, we can calculate terms A and B in Eq. A.1 as A = 8.07 and B = 9.59 and hence N_{e} as:
Thus, in spite of the skewed sex ratio and different vital rates in the two sexes, we get the same result obtained by the simpler Eq. 2. This equivalence, however, depends on the assumption of Poisson variance in reproductive success among same-aged, same-sex individuals (i.e., α = 1). Under the more likely scenario of overdispersed variance in reproductive success (α > 1), the covariance terms become more important and effective size calculated using Eq. A.1 will be smaller than that from Eq. 2. This is illustrated in Table A1, which is comparable to Table 4 but assumes α = 4. Under those conditions, N_{e} = 510 from Eq. A.1, about 20% lower than the result from Eq. 2 (635).
TABLE A1. As in Table 4, but assuming overdispersed variance in reproductive success (V_{x} = 4_{x}).
Age at death | _{}_{x} | V_{x} | D_{x} | _{}_{x}D_{x} | SSD_{Ix} | Δ_{x} | SSD_{Gx} | SSD_{x} |
Males | ||||||||
1 | 0.399 | 1.596 | 280 | 111.7 | 446.9 | -1.03 | 296.8 | 743.7 |
2 | 1.197 | 4.789 | 210 | 251.4 | 1005.6 | -0.23 | 11.2 | 1016.8 |
3 | 2.394 | 9.577 | 126 | 301.7 | 1206.7 | 0.97 | 117.5 | 1324.2 |
4 | 3.990 | 15.962 | 84 | 335.2 | 1340.8 | 2.56 | 551.3 | 1892.1 |
Totals | 1.429 | 7.110 | 700 | 1000.0 | 4976.8 | |||
Females | ||||||||
1 | 0.000 | 0.000 | 120 | 0.0 | 0.0 | -3.33 | 1333.3 | 1333.3 |
2 | 1.526 | 6.105 | 72 | 109.9 | 439.6 | -1.81 | 235.1 | 674.7 |
3 | 4.579 | 18.315 | 43 | 197.8 | 791.2 | 1.25 | 67.0 | 858.2 |
4 | 10.684 | 42.735 | 65 | 692.3 | 2769.2 | 7.35 | 3501.1 | 6270.3 |
Totals | 3.333 | 20.455 | 300 | 1000.0 | 9136.5 |
Pathway^{c} | ||||
mm | mf | fm | ff | |
V_{k} | 4.684 | 1.840 | 17.723 | 5.541 |
1.000 | 0.429 | 2.333 | 1.000 |
LITERATURE CITED
Rao, B.R., S. Mazumdah, J.H. Waller, and C.C. Li. 1973. Correlation between the numbers of two types of children in a family. Biometrics 29:271–279.