Birth trajectory under changing fertility conditions

An initially stable population of women is assumed to shift its reproductive behavior to a different level either abruptly or in a prescribed gradual fashion. A closed-form solution which is exact up to about 30 years after the reproductive adjustment is found for the resulting birth trajectory. Exact expressions are also found for the long-time asymptotic behavior of both the birth trajectory and the total population size when the shift in reproductive behavior is to bare replacement level. Accurate approximations to these asymptotic results are then derived and used to illustrate why a growing population continues to grow even after shifting to bare replacement reproductive behavior.