MATHEMATICAL MODEL DESCRIBING THE TRANSFORMATION OF THE PROTO-KARTVELIAN POPULATION

This work discusses the second period of transformation of the Proto-Kartvelian population, when the population divided into three parts: Proto-Svan; speaking the Colchian-Georgian language and the third part was scattered on the European continent. The second period is described by two different mathematical models: a part of the Proto-Kartvelian speaking population went to Europe and slowly began the process of their assimilation on the European continent. The unknown function that determines the number of Proto-Kartvelian-speaking people in Europe at the time is described by the Bernoulli equation with variable coefficients that also take the assimilation process into account. The analytical solution of the Cauchy problem is found in quadratures. The population that remained primarily in former Asia and the Caucasus region was gradually divided into two groups: those who spoke the Proto-Svan and those who spoke Colchian-Georgian languages. To describe their interference and development, a mathematical model is used, which is described by a nonlinear dynamic system with nonlinear terms of self-limitation and takes into account the unnatural reduction of the Colchian-Georgian population as a result of hostilities with neighboring peoples. For a dynamic system without nonlinear terms of self-constraint, in the case of certain relationships between variable coefficients, the first integral was found, by means of which the Bernoulli equation with variable coefficients was obtained for one of the unknown functions. In the case of constant coefficients of the dynamic system, for certain dependences between the coefficients, the dynamic system follows the system of Lotka-Volterra equations with corresponding periodic solutions. For the general mathematical model (nonlinear terms of self-limitation and unnatural reduction of the Colchian-Georgian population due to hostilities with neighboring peoples) in two cases of certain interdependencies between constant coefficients, it is shown that the divergence of an unknown vector-function in the physically meaningful first quarter of the phase plane changes the sign when passing through some half-direct one. Taking into account the principle of Bendixson, theorems have been proved, on the variability of the divergence of the vector field and the existence of closed trajectories in some singly connected domain of the point located on this half-direct (starting point of the trajectory).