Mathematical Models of the Manifestation of the Sustainable Development of Territories

Sustainable development (SD) is a process of preserving and safely harnessing the capabilities of the geographical environment, understood as a set of conditions and resources of natural, industrial, and social origin. The state and sustainable change of the geographical environment are measured in terms of national and regional wealth indicators, which include the assessment of available natural resources, produced capital, and human capital. The value of national wealth (NW) is divided into discrete, persistent, and continuously positively changing parts, characterizing the geographical environment. Using the mathematical apparatus, the theoretical problem of how SD is possible when implementing contradictory trends in nature conservation, rates of economic growth, and social development is solved. The current accumulation of NW is described by a weighted sum of independent time functions, which are partial solutions of a linear differential equation with additive control. The NW growth functions take into account the accumulated rents for investments in improving the natural environment and in production and creativity. Taking into consideration the share of participation of each component in the NW increment, this article proposes a classification of countries and regions in areal coordinates. The sovereign and self-sufficient state of the NW of Russia and its regions is characterized by a harmonious (with respect to the set of assets) middle position in the classification, which determines its connecting role in the organization of the global economy to ensure SD. The mathematical model describes NW accumulation from different income sources and allows comparing the real and calculated trends of long-term changes in the geographical environment of SD using the example of the Russian Federation and the United States. New opportunities are opening up for applying mathematical tools to explain the contradictory qualities of SD in different countries, taking into account the uniqueness of the geographical environment in its expanded understanding.