The Stable Population Model

Abstract

In this chapter, we consider the most basic linear one-sex age-structured population model, known as the stable population model. In the eighteenth century, Euler developed a difference equation model to show that an age-structured population with constant fertility and mortality will grow geometrically. Moreover, Euler derived relations among various demographic indices under this geometrical growth and suggested that these relations could be used to estimate incomplete data. This brilliant discovery would have been the starting point of modern demography had it not been lost until the stable population theory was formulated in the twentieth century. Although the stable population model is very simple, it can be successfully applied to real populations that exhibit exponential growth. The stable population model has not only become a central tenet of modern demography, but has also stimulated a wide variety of mathematical studies in population biology, epidemiology, and social sciences. In this chapter, we formulate the stable population model as an initial-boundary value problem of the McKendrick partial differential equation. We investigate the basic properties of the model based on Lotka’s integral equation, which also gives an alternative formulation of the basic model. Our main purpose is to prove the Fundamental Theorem of Demography (the Sharpe–Lotka–Feller theorem/the strong ergodicity theorem). By introducing the dual system, we then derive some stochastic interpretations of the Fundamental Theorem. Next, we present some applications in real human demography. Finally, we discuss the age-profile dynamics of quasi-stable populations.