The Metastable Birth Trajectory

Abstract The metastable model generalizes the stable population model by allowing net maternity to change exponentially over age and time. As a result, the metastable model generates an exponentially quadratic birth trajectory, which is characterized by a constant proportion of births by age of mother. The metastable model is well suited to analyzing steady fertility declines and transitions between two regimes of fixed vital rates. (ProQuest: ... denotes formulae omitted.) 1. Introduction Demography is built on its models. The basic life table, dating from 1662, followed a single birth cohort to the death of its last member. The life table was later conceptualized as a stationary (zero growth) population, a view expanded by the stable population which captures the implications of any given set of age-specific birth and death rates. Recently, the stable population has been extended to the metastable population, which allows the vital rates to change over time. In the stable population, constant vital rates yield (linear) exponential growth in the number of births, while in the metastable model net maternity that increases exponentially over age and time produces quadratic exponential growth in births. This note describes the metastable model, specifies its underlying vital rates and their pattern of change, and provides the model's birth trajectory over time. 2. The stable model Lotka's stable population was the foundational model of 20th century mathematical demography. One elegant equation, generally known as the model's renewal or characteristic equation, captures its essential features. That equation can be written (1) 1 = ∫ e^sup -rx^p(x) f(x) dx where the integration spans all ages x, p(x) is the probability of surviving from birth to exact age x under the specified regime of age-specific mortality, f(x) is the force of fertility at exact age x under the specified regime of age-specific fertility, and r, a constant, is Lotka's intrinsic growth rate (Lotka 1998; Preston, Heuveline, and Guillot 2001; Schoen 1988). The stable model is typically applied to a female only population. The stable population Net Reproduction Rate (R0) is then given by (2) R^sub 0^ = ∫ p(x) f(x) dx where the integral ranges over all reproductive ages and p(x)f(x) is the net maternity function. Equation (1) can be interpreted in a manner that emphasizes its renewal or regenerative nature. Let us scale the stable population so that there is one birth in our reference year. Then that unit birth arises from the sum, over all reproductive ages, of the eirx births that occurred x years ago who survived to age x under the specified mortality regime and then gave birth in the reference year according to the specified fertility regime. The renewal equation thus shows how an exponential birth trajectory is perpetuated by constant agespecific fertility and mortality. The realization that constant vital rates yield a population that grows exponentially but maintains a constant age composition is probably the most profound insight in mathematical demography. 3. The metastable model The stable population shows the implications of a given set of age-specific fertility and mortality rates, but is essentially a static model. The metastable model is dynamic because it accommodates monotonic exponential change in vital rates over age and time. Let the metastable net maternity function be written as (3) φ(x, t) = p(x, t) f(x, t) where φ(x; t) is the net maternity of women age x at time t, p(x; t) is the probability that women born at time t - x will survive to attain age x at time t, and f(x; t) is the force of fertility for women age x at time t. Over time, metastable net fertility changes according to (4) φ(x, t) = φ0(x) e^sup rx^ e^sup htx^ where φ0(x) describes a time invariant net maternity function whose R0 = 1, r is the intrinsic growth rate implied by φ(x; 0), and h is a new metastable parameter that specifies how net maternity changes over both age and time. …