Background: Two seminal studies in the 1980s, by Preston and Coale (1982) and Arthur and Vaupel (1984), generalized the Lotka equations developing the variable-r methods. Objective: Time changes in sex ratios (males:females) are studied from the perspective of the variable-r method to estimate the contributions of fertility, mortality, and net-migration. Methods: The time change in sex ratios can be calculated as a comparison of the growth rates of the sexes. The difference is then decomposed into population composition, and the age-specific components of fertility, mortality, and net migration. Thirteen countries with long historical demographic series are used to illustrate the time trends in sex ratios. Results: Most countries are moving towards a greater number of males per females. The greatest changes were observed in Norway and Sweden, with males catching up with females at older ages due to survival. Meanwhile, the sex differential in net-migration flows explains the decline in sex ratios in Spain and Australia and the increase in the United States. Conclusions: Our results shed light on the debate on female–male population imbalances and how fertility, mortality, and net migration contribute to these disparities. Contribution: Vaupel and Canudas-Romo (2003) wrote a paper in honor of Vaupel’s mentor Nathan Keyfitz on his 90th anniversary. The paper focused on a common topic of interest of all three researchers, namely mathematical demography. In honor of the legacy of the work of Jim Vaupel, we revive that idea with a collection of mathematical derivations that are simple, yet powerful in their demographic interpretation.

中文翻译：

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性别比例中的变量 r：纪念 Jim Vaupel 的公式（作者：Vladimir Canudas-Romo、Wen Su、Mike Hollingshaus）

背景：Preston 和 Coale (1982) 以及 Arthur 和 Vaupel (1984) 在 20 世纪 80 年代的两项开创性研究推广了 Lotka 方程，开发了变量 r 方法。目的：从变量r法的角度研究性别比（男性：女性）的时间变化，以估计生育率、死亡率和净移民的贡献。方法：性别比的时间变化可以通过性别增长率的比较来计算。然后将差异分解为人口构成以及生育率、死亡率和净移民的年龄组成部分。使用具有长期历史人口序列的 13 个国家来说明性别比的时间趋势。结果：大多数国家正在朝着男女比例增加的方向发展。在挪威和瑞典观察到最大的变化，由于生存的原因，男性在年龄较大时赶上了女性。与此同时，净移民流中的性别差异解释了西班牙和澳大利亚性别比例下降以及美国性别比例上升的原因。结论：我们的结果揭示了关于女性与男性人口失衡以及生育率、死亡率和净移民如何造成这些差异的争论。贡献：Vaupel 和 Canudas-Romo (2003) 在 Vaupel 的导师 Nathan Keyfitz 90 周年之际写了一篇论文。这篇论文重点关注了三位研究人员共同感兴趣的话题，即数学人口统计学。为了纪念吉姆·沃佩尔 (Jim Vaupel) 的工作遗产，我们通过一系列简单但在人口统计解释上强大的数学推导来复兴这一想法。