Spatial dynamics of populations with changing birth, death, and migration rates: a generalization of multiregional stable population theory

Kim's recent reformulation in discrete time of Preston and Coale's continuous time generalized age-distribution model for any population is extended to the multiregional case by the introduction of interregional migration and the simultaneous consideration of multiple interacting regional populations.The paper begins with the use of a vector notation to describe the age - region distribution of any multiregional population and develops generalized multiregional Lotka equations that converge, over time, to the corresponding equations for a stable multiregional population. But the vector notation does not lead to a method for estimating relevant demographic parameters. To overcome this difficulty, a more powerful notation is introduced: one that substitutes matrices for vectors. This extension is carried out by defining a second subscript that designates the place of residence at some previous fixed moment in time. Such an extension, however, creates alternative generalizations that depend directly on the characteristics of the available data.If the data consist of <i>two sets of lifetime migration data</i> (that is, two sets of population counts cross-classified by place of birth), two possible definitions of the lambda-matrix -- the multiregional counterpart to the growth ratio in the single-region case -- result in two alternative multiregional extensions of the single-region generalized system. One leads to a simpler formulation of the multiregional Lotka equations, whereas the other facilitates the derivation of formulas used for estimating various demographic parameters.If the data include <i>a single set of period migration data</i> (that is, consist of two population counts at time <i>t</i>-1 and <i>t</i>, the second being cross-classified by place of residence at the first), two definitions of the lambda-matrix are again possible. Only one, however, permits a generalization that recovers the classical results of multiregional stable population theory.