Regime-Dependent Diffusion–Gravity Dynamics in a Declining Population System: A Network-Based Analysis of Internal Migration in Japan

Population decline has become a defining feature of many regional systems, with Japan’s total population entering a sustained decline from 2009 onward, yet limited attention has been paid to how diffusion-based migration models perform under prolonged demographic contraction and systemic shocks. This study examines internal migration dynamics in Japan over a 22-year period (2004–2025) by modeling inter-prefectural population flows across the country’s 47 prefectures as a spatial network governed by diffusion processes with long-range interaction effects. Annual prefecture-to-prefecture migration matrices are constructed and analyzed within a graph-based framework using a discrete Laplacian formulation augmented with gravity-type interaction terms. To account for structural changes over time, the analysis adopts a regime-based approach that distinguishes periods of demographic stability, sustained population decline, exogenous shock, associated with the COVID-19 period (2020–2021), and post-shock reconfiguration (2022–2025). Regimes are defined based on demographic and systemic transitions rather than equal temporal lengths, while model estimation is conducted on annual transitions to maintain consistency across periods. This framework allows for a systematic comparison of migration dynamics across contrasting demographic conditions. The results indicate that diffusion-based dynamics provide a reasonable representation of migration adjustment during periods of relative demographic stability, but exhibit reduced explanatory capacity during prolonged population decline and systemic shocks. In contrast, hybrid diffusion–gravity formulations are more robust across regimes by incorporating both local spatial interactions and long-distance hierarchical linkages. Post-shock periods display mixed migration behavior, suggesting partial re-stabilization without a full return to pre-decline diffusion patterns. By framing internal migration as a regime-dependent dynamical system, this study contributes to applied mathematics by clarifying the conditions under which diffusion operators remain suitable in declining population systems and when extended formulations are required. Although Japan serves as the empirical case, the proposed framework also applies to other regions experiencing demographic decline and spatial reorganization.