Populations with Sinusoidal Birth Trajectories

Starting with a sinusoidal birth function, we derive an explicit expression for the associated wave of net maternity (R(t)). The phase shift and relative amplification ofR(t) depend heavily on the cycle length of oscillation (T) relative to the mean age of net maternity (μ) and on the amplitude of oscillation of the birth function. For short cycle lengths (up to about 12 years), the birth andR(t) functions are virtually identical. When the cycle length is long (over 120 years),R(t) is essentially equal to the ratio of the number of births at timetto that at timet−μ. TheR(t) function oscillates with the same cycle length as the birth function, but it is not sinusoidal and its asymmetry increases as the amplitude of birth function increases. This paper breaks new ground by providing a complete description of the dynamics of the model, including an analysis of the sensitivity ofR(t) to the amplitude of the birth function, which was not accessible to previous approaches.