Effective Population Size Ne with unequal sex ratio

One means of estimating the size of a population is a "mark & recapture" census, in which one or more individuals are marked and released back into nature. Individuals are then recaptured from the wild: the estimated population size is the inverse of the proportion of marked individuals recaptured. For example, if a large number of individuals are captured, marked, and released, recapture of five marked individuals among 5,000 recaptures over time implies a population size 1/ N = (5 / 5,000) and N = 1,000. The sample principal can be applied by re-sampling alleles from a population.

A population comprises Nm and Nf dioecious males and females, respectively. In the tide pool model, the probability that two alleles chosen from any two different individuals in the present generation are both derived from any male in the previous generation is 1/4, or from any female is also 1/4. The probability that the two alleles are from the same male is 1/Nm, or from the same female is 1/Nf. Then the probability that two alleles came from the same individual (male or female) is the inverse of the sum of these probabilities, so

(1 / Ne) = (1 / 4)(1 / Nm) + (1 / 4)(1 / Nf) = (1 / 4Nm) + (1 / 4Nf)

To isolate
Ne from this equation, first provide a common denominator for the right side by multiplying the two terms by Nf / Nf and Nm / Nm, respectively, so

(1 / Ne) = (1 / 4Nm) + (1 / 4Nf) = (Nf + Nm) / 4NmNf

for which the reciprocal is

Ne = 4NmNf / (Nf + Nm)

Check: if the numbers of males and females are equal, Nm = Nf = N / 2, and from the formula Ne = (4)(N/2)(N/2) / (N/2 + N/2) = (4)(N2/4) / N = N

Where the sex ratio of breeding individuals is unequal, this departs rapidly from expectation. In the extreme case with one breeding male, where Nm = 1, Ne = 4NmNf / (Nf + Nm= (4)(1)(Nf)/ (Nf + 1) so if Nf >> 1 then Ne = 4Nf / (Nf ) = 4. The same is true if Nf = 1.

HOMEWORK: Use the Excel spreadsheet to calculate
Ne for various scenarios of Nm & Nf as instructed.

© 2022 by Steven M Carr