Memory and Spatial Coupling in Regional Population Dynamics: A Fractional-Order Lotka–Volterra Geosimulation on the NUTS-3 Graph

Regional population trajectories are shaped by two properties that standard projection tools treat only implicitly: the memory of past demographic regimes and the interaction between neighbouring regions. This study proposes a geosimulation framework in which the annual population of NUTS-3 regions evolves according to a fractional-order Verhulst–Lotka–Volterra system defined on the regional graph. Each region has its own intrinsic growth rate, carrying capacity and Caputo derivative order in the relaxation–oscillation range (0, 2), so that the memory or inertia of its trajectory becomes a region-specific, mappable parameter, while a gravity-type coupling term lets neighbouring populations compete for, or contribute to, regional growth. The model is calibrated on Eurostat population series for the 42 NUTS-3 regions of Romania over 2000–2025 as a coupled system, for 1,070 NUTS-3 regions of the European Union and EFTA over 2003–2023 without coupling, and as coupled systems for the 28 regions of Bulgaria, the 10 regions of Lithuania and the 20 regions of Hungary, so that Romania is compared with the other declining countries of Europe, including Latvia through its uncoupled estimates. Integer-order and uncoupled variants are nested within the model, so the contribution of memory and of spatial interaction is tested with information criteria, a parameter-recovery experiment and rolling-origin forecast evaluation against trend and drift benchmarks. The estimated memory orders are analysed with global and local Moran statistics and the fitted Romanian system is projected to 2035 under baseline, memoryless and uncoupled scenarios. Freeing the memory order reduces the residual error of the Romanian system by a factor of 7.9, and the fractional model is preferred for 79% of 1,070 European regions; the estimated orders are predominantly inertial (median 1.78 in Europe), and the order is strongly clustered in Europe (Moran’s I = 0.38) but not in Romania (I = -0.16). Coupling improves the fit in 36 of the 42 Romanian regions. Bulgaria, Lithuania, Hungary and Latvia need memory as much as Romania, with residual errors reduced by factors of 6.3, 4.9, 4.4 and 1.7, whereas coupling is spatially identified only in Romania, weakly identified in Bulgaria and absent in Lithuania and Hungary; the memory order is clustered in Bulgaria (I = 0.44) and negatively autocorrelated in Lithuania and Hungary (I = −0.46 and −0.52). In all five countries growth is confined to the capital region and a few large cities, whose share of the national population rises throughout the window. Memory-based forecasts are competitive with extrapolations but not superior to a five-year drift. The 2035 baseline projects a national decline of 3.9%, with growth in the capital region and decline in the south. The framework is reproducible from public data and transfers directly to other spatially interacting processes on administrative graphs.