BACKGROUND
Kinship models, from the pioneering work of Goodman, Keyfitz, and Pullum to the recent matrix-oriented approach of Caswell, have assumed time-invariant demographic rates, and computed the kinship structures implied by those rates. In reality, however, demographic rates vary with time and it is of interest to compute the consequences of such variation for kinship structures.

OBJECTIVE
Our goal is to develop a matrix model for the dynamics of kinship networks subject to arbitrary temporal variation in survival, fertility, and population structure.

METHODS
We develop a linked set of equations for the dynamics of the age structure of each type of kin of a Focal individual. The matrices that describe survival and fertility are given as functions of time. The initial conditions required for the time-invariant model are replaced with a set of boundary conditions for initial time and initial age.

RESULTS
The time-varying model maintains the kinship network structure of the time-invariant model. In addition to the results provided by the time-invariant model, it provides kinship structures by period, cohort, and age. It applies equally to historical sequences of past demographic rates and to projections of future rates. As an illustration, we present an analysis of the kinship structure of Sweden from 1891 to 2120.



中文翻译：

---

亲属关系的正式人口统计学 III：随时间变化的人口统计率的亲属关系动态（作者：Hal Caswell, Xi Song）

背景
亲属关系模型，从 Goodman、Keyfitz 和 Pullum 的开创性工作到最近 Caswell 的面向矩阵的方法，假设人口统计率随时间变化，并计算这些比率所隐含的亲属关系结构。然而，在现实中，人口比率随时间而变化，计算这种变化对亲属结构的影响是很有趣的。

目标
我们的目标是为亲属网络的动态发展一个矩阵模型，该模型受到生存、生育率和人口结构的任意时间变化的影响。

方法
我们为焦点个体的每种类型的亲属的年龄结构的动态发展了一组相关联的方程。描述存活率和生育率的矩阵作为时间的函数给出。时不变模型所需的初始条件由一组初始时间和初始年龄的边界条件代替。

结果
时变模型保持了时不变模型的亲属关系网络结构。除了时不变模型提供的结果外，它还提供了按时期、队列和年龄划分的亲属结构。它同样适用于过去人口比率的历史序列和未来人口比率的预测。作为例证，我们对瑞典从 1891 年到 2120 年的亲属结构进行了分析。