About 50 years ago, Keyfitz (1971) asked how much further a growing human population would increase if its fertility rate were immediately to be reduced to replacement level and remain there forever. The reason for demographic momentum is an age–structure inertia due to relatively many potential parents because of past high fertility. Although nobody expects such a miraculous reduction in reproductive behavior, a gradual decline in fertility in rapidly growing populations seems inevitable. As any delay in fertility decline to a stationary level leads to an increase in the momentum, it makes sense to think about the timing and the quantum of the reduction in reproduction. More specifically, we consider an intertemporal trade-off between costly pro- and anti-natalistic measures and the demographic momentum at the end of the planning period. This paper uses the McKendrick–von Foerster partial differential equation of age–structured population dynamics to study a sketched problem in a distributed parameter control framework. Among the results obtained by applying an appropriate extension of Pontryagin’s Maximum Principle are the following: (i) monotony of adaptation efforts to net reproduction rate and convex decrease/concave increase (if initial net reproduction rate exceeds 1/is below 1); and (ii) oscillating efforts and reproduction rate if, additionally, the size of the total population does not deviate from a fixed level.

大约 50 年前，Keyfitz（1971）询问，如果生育率立即降低到更替水平并永远保持在这个水平，不断增长的人口还会增加多少。人口动力的原因是年龄结构惯性，由于过去的高生育率，潜在父母相对较多。尽管没有人预料到生殖行为会出现如此奇迹般的减少，但快速增长的人口中生育率的逐渐下降似乎是不可避免的。由于生育率下降到稳定水平的任何延迟都会导致势头增加，因此考虑生育率下降的时间和数量是有意义的。更具体地说，我们考虑了成本高昂的促生育措施和反生育措施与规划期结束时的人口势头之间的跨期权衡。本文使用年龄结构群体动力学的 McKendrick-von Foerster 偏微分方程来研究分布式参数控制框架中的一个概述问题。通过应用庞特里亚金最大原理的适当扩展所获得的结果如下：（i）对净繁殖率和凸减少/凹增加的适应努力的单调性（如果初始净繁殖率超过1/低于1）；(ii) 如果总人口规模不偏离固定水平，则努力和繁殖率会出现波动。