Because the source populations provide no further contributions after the founding generation, unlike in the general case, the mean admixture fraction does not change with time.
However, when a hybrid population is founded in a single admixture event, we have found that the mean admixture fraction is constant across generations and is therefore uninformative about the time since the founding of the hybrid population.
An important observable quantity that can be estimated in modern admixed populations and used for understanding historical aspects of the admixture process is the mean admixture fraction from a source population.
Therefore, for g ≥ 0, w g = P 1 r 1 g + P 2 r 2 g + P 3, and the closed-form expression for the X-chromosomal female mean admixture fraction in generation g ≥ 1, E [ H 1, g, f X ], is z g = P 1 r 1 g + 1 + P 2 r 2 g + 1 + P 3. We report this result in the main text as Equation 17, using it to obtain E [ H 1, g, m X ] in Equation 18.
For recent admixture in the single-admixture model, or in a model with continuing admixture after the founding of the admixed population, the mean admixture fractions from S 1 for the X chromosome depend on the sex-specific contributions in a more complex way, incorporating the sex-specific contributions from S 2 (Equations 12, 13, and 17 20).
Considering admixture processes that are constant in time, we demonstrate that surprisingly, although the mean autosomal admixture fraction from a specific source population does not reveal a sex bias in the admixture history, the variance of autosomal admixture is informative about sex bias.
Figure 5 plots the mean X-chromosomal admixture fraction for females and males (Equations 17 and 18) with the mean autosomal admixture fraction when both source populations have a female bias in contributions, s 1 f > s 1 m and s 2 f > s 2 m.
As the mean X-chromosomal admixture fraction is generally near its limit for g in this range, both in the single-admixture model and in the constant-admixture model, the specific g ∈ [ 14, 20 ] only minimally changes the results.
Here, we obtain the closed form for the mean X-chromosomal admixture fraction in a random female and a random male from the admixed population in the special case of constant admixture over time.
The limit of the mean X-chromosomal admixture fraction fits the 2 1 ratio by which the X chromosome is inherited, following Equation 1, with the variables in Equation 1 viewed as the initial admixture values s 1, 0 f and s 1, 0 m.
A simple framework noting that in a population with equally many females and males, two-thirds of X chromosomes appear in females, suggests that the mean X-chromosomal admixture fraction is a linear combination of female and male admixture parameters, with coefficients 2/3 and 1/3, respectively.
