Methodological foundations for teaching the discipline «Mathematical modelling of positive systems» in the master's programme in applied mathematics

Abstract

Objective. The paper addresses the development and theoretical substantiation of an integrated methodological framework for teaching the discipline «Mathematical Modelling of Positive Systems» to master's-level students enrolled in the Applied Mathematics programme. The focus is on design a pedagogically grounded didactic structure that transcends the mere transmission of the formal apparatus of positive systems theory and instead cultivates in students the capacity to independently identify algebraic structure in concrete phenomena and transfer it across diverse disciplinary contexts, that is the main research competency at the master's level of higher education. Methods. The investigation rests on a two-level methodological synthesis. The first level comprises a systematic analysis of the mathematical structure of positive linear systems theory, encompassing Metzler matrices, nonnegative matrices, the Perron-Frobenius theorem, and M-matrices, alongside their principal applied instantiations in compartmental epidemic models, Leontief's inter-industry balance model, and population dynamics. The second level concerns pedagogical design drawing on active learning principles, problem-based learning, and the competency-based framework stipulated by the national higher education standard for speciality 113 «Applied Mathematics». Results. A three-phase course architecture is proposed: «Mathematical Foundations», «Model Construction», «Applied Analysis». A three-tier competency profile is substantiated: mathematical, modelling, and research competencies, each furnished with explicit assessment criteria. Pilot implementation within a master's programme demonstrated an increase in the mean score for the criterion «conceptual depth of analysis»; 73% of students independently formulated a non-trivial interdisciplinary analogy between compartmental epidemic and economic equilibrium models. Conclusions. An effective methodology for teaching positive systems theory is organised not around the reproduction of theorems but around cultivating students' ability to perceive the structural unity of the class of positive dynamical systems across the diversity of their disciplinary interpretations. The spiral architecture of the course, whereby each key mathematical object recurs in multiple applied contexts, ensures a robust conceptual knowledge network and constitutes a necessary condition for the formation of research competencies at the second cycle of higher education.