Mathematical Model Describing the Process of Interaction Between Georgian, Laz, Mingrelian, and Svan Population

Abstract

This paper examines the fourth period of Kartvelian population transformation, during which Laz and Mingrelian languages emerged from the Zan linguistic base, leading to the formation of four Kartvelian languages: Georgian, Laz, Mingrelian, and Svan. This process involves internal linguistic-cultural interactions and external influences (assimilation, dissimilation).
A mathematical model, formulated as a four-dimensional nonlinear dynamic system, describes the interactions and demographic changes of Kartvelian populations, incorporating demographic parameters, linguistic-cultural interactions, and external factors (wars, migrations, epidemics).
The model enables analysis of historical development. Methods for finding first integrals in four- and three-dimensional systems are discussed. Using two first integrals, the system is reduced to a two-dimensional system, capturing Georgian-Laz linguistic dynamics, with Mingrelian and Svan population sizes derived algebraically.
Theorems establish the coexistence dynamics of Georgian, Laz, Mingrelian, and Svan populations. The Poincaré-Bendixson theorem confirms closed integral trajectories, ensuring no Kartvelian language group becomes extinct.