# Analytical Representation of Eigensystem in Multiregional Leslie Matrix Model: Application to Sensitivity Analysis of Population Declining in Japan

The multiregional Leslie matrix is a transition matrix model in which matrices, including interregional migration, represent the fertility and survival entries of the Leslie matrix. By integrating the age structure and interregional migration, this model allows us to evaluate the impact of interregional migration on the population growth rate. We have found a theorem that expresses the eigensystem of the multiregional Leslie matrix using eigenvalues and matrix elements. Using this theorem, we can estimate the contribution of each region to the descendants and the percentage of ancestors in each region in a steady state. For example, in Japan, which has been in a phase of population decline since 2010, there is not only a declining birthrate and an aging population but also a problem of uneven population movement due to the concentration of the population in Tokyo. A sensitivity analysis based on our analysis results and the national census can clarify the mathematical structure of population decline. In this study, we apply our theory to Japanese governmental statistics from 2010 to 2020 to numerically evaluate the impact of age-specific and regional migration and fertility rates on population decline.