Yellin J, Samuelson P A
Proc Natl Acad Sci U S A. 1974 Jul;71(7):2813-7. doi: 10.1073/pnas.71.7.2813.
We analyze a simple, deterministic model of the dynamics of population changes in a bisexual, reproductive system based on marriage. Our model is one of a general class, special cases of which have been previously discussed within the framework of population biology by D. G. Kendall, L. A. Goodman, J. H. Pollard, and others. Here, we extend and complete previous analyses of systems characterized by first-degree homogeneous, unbounded marriage functions, allowing for arbitrary birth and death rates. The dynamics of the model is determined by three coupled first-order, nonlinear differential equations, similar to those used in the description of chemical reactions and of radioactive decay chains. Solutions of the differential equation system are classified according to the associated patterns of birth and death rates of the two sexes, and growth and stability properties are discussed. This preliminary report gives conditions sufficient to insure the existence of a unique, exponential mode of population growth or decay, with a finite ratio of the sexes. We also exhibit other conditions which, in contrast to the standard, linear demographic analysis of Lotka, guarantee that the sex ratio asymptotically becomes infinite. The model manifests a delicate balance between the vital parameters that alerts one to the possibility of selfaggravating distortions of the sex ratio, once a monogamous society's fertility falls below the replacement value.
我们分析了一个基于婚姻的两性生殖系统中种群变化动态的简单确定性模型。我们的模型属于一类通用模型，其特殊情况先前已由D. G. 肯德尔、L. A. 古德曼、J. H. 波拉德等人在种群生物学框架内进行过讨论。在此，我们扩展并完善了先前对以一阶齐次、无界婚姻函数为特征的系统的分析，同时考虑了任意的出生率和死亡率。该模型的动态由三个耦合的一阶非线性微分方程确定，这与用于描述化学反应和放射性衰变链的方程类似。微分方程组的解根据两性相关的出生率和死亡率模式进行分类，并讨论了增长和稳定性特性。这份初步报告给出了足以确保存在唯一的指数型种群增长或衰减模式且两性比例有限的条件。我们还展示了其他条件，与洛特卡的标准线性人口分析不同，这些条件保证性别比渐近地变为无穷大。该模型显示出关键参数之间的微妙平衡，这提醒人们一旦一夫一妻制社会的生育率降至更替值以下，性别比就有可能出现自我加剧的扭曲。
