The momentum of a population whose birth rates gradually change to replacement levels

A formula is given for computing the ultimate size of a population whose birth rates change over a finite time to replacement levels. This formula is derived in the context of the usual discrete-time version of the one-sex model of population growth and represents an extension of work by Keyfitz and others. When birth and death rates tend over an infinite time to replacement levels, necessary and sufficient conditions are investigated for the ultimate population size to be finite and nonzero.