Construction and analysis of a mathematical model of the dynamics of competence formation in the learning process

Abstract

The article proposes a mathematical model describing the dynamics of competence formation in the educational process. The model is developed based on an analogy with compartmental and epidemiological approaches and considers five levels of competence development: entry, low, medium, sufficient, and high. Both progressive and regressive transitions between these levels are taken into account, including contact and non-contact interaction mechanisms. The resulting model is formulated as a system of nonlinear ordinary differential equations that describe the temporal evolution of student groups across different competence levels. For the proposed system, the corresponding Cauchy problem is analyzed. In particular, the boundedness of solutions is established, and the existence and uniqueness of solutions are proved using standard results from the theory of differential equations, based on the Lipschitz continuity of the right-hand side. The developed model provides a formal framework for describing the competence formation process and can serve as a foundation for further investigations, including stability analysis, sensitivity analysis of model parameters, and optimal control of the educational process.