Populations with quadratic exponential growth

Stable population models, based on fertility and mortality rates that do not change over time, are too unrealistic and inflexible to capture the dynamics of many observed populations. Dynamic models, which allow vital rates to change over time, are needed to systematically analyze such populations. Here we examine dynamic—hyperstable—models with increasing or decreasing vital rates, providing the first detailed analysis of a closed form model of monotonic demographic change. Using two different approaches, we demonstrate how exponentiated quadratic birth trajectories are related to exponentially increasing vital rates. Focusing on the plausible assumption of a fixed proportional distribution of births by age of mother, we show how convergence to hyperstability parallels convergence to classical stability. Our analysis focuses on net maternity rates, allowing considerable flexibility in patterns of change in either fertility or mortality. Under the assumption of constant mortality over time, we specify the hyperstable population's age structure and its relationship to its associated stable population at every point in time. The hyperstable and associated stable populations are in dynamic equilibrium, as the Kullback distance, which measures the degree the two age distributions differ, remains constant over time. The exponentiated quadratic provides a straightforward model of equilibrated change. Its flexibility and relatively simple structure give it significant potential as an analytical tool.