Objective: Few methods are available for analyzing populations with changing rates. Here hyperstable models are presented and substantially extended to facilitate such analyses. Methods: Hyperstable models, where a known birth trajectory yields a consistent set of age-specific birth rates, are set out in both discrete and continuous form. Mathematical analysis is used to find new relationships between model functions for a range of birth trajectories. Results: Hyperstable population projection matrices can create bridges that project any given initial population to any given ending population. New, explicit relationships are found between period and cohort births for exponential, polynomial, and sinusoidal birth trajectories. In quadratic and cubic models, the number of cohort births equals the number of period births a generation later, with a modest adjustment. In sinusoidal models, cohort births equal the number of period births a generation later, modified by a factor related to cycle length. Contribution: Because of their adaptability, structure, and internal relationships, hyperstable birth models afford a valuable platform for analyzing populations with changing fertility. The new relationships found provide insight into dynamic models and period–cohort connections and offer useful applications to analysts.

中文翻译：

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分析超稳定人口模型（罗伯特·舍恩）

目标：很少有方法可用于分析具有变化率的群体。这里提出了超稳定模型并对其进行了大幅扩展以促进此类分析。方法：超稳定模型，其中已知的出生轨迹产生一组一致的特定年龄出生率，以离散和连续形式提出。数学分析用于寻找一系列出生轨迹的模型函数之间的新关系。结果：超稳定人口投影矩阵可以创建将任何给定初始人口投影到任何给定最终人口的桥梁。在指数、多项式和正弦曲线出生轨迹的周期出生和队列出生之间发现了新的、明确的关系。在二次和三次模型中，队列出生数等于一代后的周期出生数，并进行适度调整。在正弦曲线模型中，队列出生数等于一代人之后的经期出生数，并根据与周期长度相关的因素进行修改。贡献：由于其适应性、结构和内部关系，超稳定生育模型为分析生育率变化的人群提供了一个有价值的平台。发现的新关系提供了对动态模型和时期群组联系的洞察，并为分析师提供了有用的应用程序。