Variational approach in modeling the rural historical settlement: a model of a two-level hierarchy of centers

Abstract

The subject of this study is the modeling of the spatial distribution of the medieval population. To take into account the conditions of heterogeneity of the territory in modeling, the author develops a variational approach proposed by S.M. Huseyn-Zadeh. The model is based on the idea of settlement, placement of the population on the territory as a process in which each of its participants is guided in their activities by achieving goals dictated by natural considerations, for example, maximum accessibility to their cultivated area, maximum proximity to the central settlement (center). Using the methods of calculus of variations and the concept of Pareto optimality within this approach makes it possible to identify and numerically describe the relationships between the optimal parameters of the system. In the previous paper, the author proposed a model of a one-level hierarchy of the placement of centers (central settlements). Additional analysis of historical material leads to the need to refine the model, namely, adding a hierarchy of centers to it. In this regard, this paper proposes a model of spatial placement with a two-level hierarchy of centers. The approbation of the proposed model is carried out on the material of the scribal books of the Shelonskaya Pyatina of the Novgorod land of the end of the XV century. The high degree of conformity of the obtained theoretical and empirical data allows us to consider mathematical modeling as an adequate and convenient auxiliary tool in studying the nature and dynamics of historical rural settlement systems. The possibilities of the proposed modeling also in terms of filling data gaps are demonstrated by the example of the task of reconstructing the approximate population of the Shelonskaya Pyatina of the considered time period.