A Discrete‐time Dynamic Analysis of Interregional Population Systems
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abstract
This paper uses a system of simultaneous linear difference equations to explain how various dynamic properties of a closed interregional population system are determined by (1) the system's fixed natural growth and migration structure and (2) the initial population distribution. A matrix containing natural growth rates and interregional migration rates is defined as a structural matrix. Based upon structural matrices, interregional population systems are classified into (1) ZPG, (2) Malthusian, (3) mixed, and (4) vanishing systems. The analysis is accomplished by using the Z‐transform method, which can be translated into an efficient computer algorithm. For any closed interregional population system with a fixed structural matrix, it is found that the existence and stability of an equilibrium distribution, the long‐run proportional distribution, the long‐run growth rate, and the location of the “dominant region” are all independent of the initial population size and distribution but are completely determined by the largest eigenvalue and the associated eigenvector of the system's structural matrix. However, the actual population sizes at equilibrium, the saturation and doubling times, and the fluctuations of the system's time paths depend both on the structural matrix and on the initial population size and/or distribution. Analogous to the mean and standard deviation in descriptive statistics, the eigenvalues and eigenvectors extracted from the structural matrix are useful summarizing indices in linear dynamic systems.
