Mathematical modeling of the competition of two ideologies with internal conflicts

When studying social processes, it is of great interest to predict the behavior of society or its individual components. At present, methods of mathematical modeling and corresponding mathematical models are being actively developed for this purpose. The purpose of the work is to build a mathematical model of the competitive struggle of two ideologies taking into account the spontaneous and forced transitions of individuals between ideologies, to analyze the resulting model to determine the scenarios for the development of ideologies, иand also to find the conditions under which this or that scenario is realized. Methods. In this paper, parametric studies of the development of ideologies over time are carried out for various values of the model parameters. To determine the conditions for the existence of different scenarios for the development of ideologies the stability of the model is studied. Results. A model of the competitive struggle of two ideologies is proposed  taking into account spontaneous and forced transitions of individuals between ideologies. In the considered model all ideologies eventually come to stable stationary states. It is shown that the development of ideologies can occur only according to three scenarios: (A) both ideologies survive and coexist; (B) both ideologies die out, and (C) one of the ideologies survives while the other dies out. The conditions for the existence of each of the scenarios for the development of ideologies are determined. Conclusion. Despite the fact that the real system under consideration is discrete with a large number of elements (adherents of ideologies) a transition to a continuous model is possible. The equations obtained in the considered model are the modified LotkaVolterra equations. The analysis of the model made it possible to derive criteria for the existence of various scenarios for the behavior of ideologies, to determine the boundaries by the parameters of the model that separate the scenarios for the development of ideologies. Unlike similar works this work takes into account spontaneous and forced transitions between ideologies including those due to internal conflicts. The constructed model can be used to analyze electoral processes, predict the emergence and development of terrorist groups, various religious communities, etc.

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