The long-term effects of time-dependent maternity behavior

Using the integral population model of Sharpe and Lotka, it is demonstrated that if the time variation of the maternity function is assumed to only affect the parent population, then standard methods of obtaining the long-term behavior may still be used. Further, if the net maternity function has explicit time dependence, in contrast to age dependence, only for time less than the minimum age of childbearing, the standard techniques still may be used. It is shown that the recent extensions of Cerone and Keane to include exponential time dependence may be applied, together with the above observations, to define piecewise time-dependent net maternity functions and to determine the resulting long-term asymptotic behavior of the population. The population management problem of determining the time path of change needed to approach a given population also is considered in some detail.