Towards a mathematical theory of athletic training. Part 2. Mathematical modeling of adaptation processes under the influence of training loads: the approach of Viktor Nikolaevich Seluyanov

Relevance. The development of a comprehensive mathematical theory of athletic training continues to be a significant challenge in the field of sports science. Viktor Nikolaevich Seluyanov’s approach, based on compartment modeling, is a unique attempt to understand adaptation processes at all levels of the body, from cellular to systemic. Despite the potential for prediction, his models have not been widely adopted and require further analysis. The objectives are to systematize, reconstruct, and critically analyze the hierarchical system of mathematical models for short-term and long-term adaptation to training loads, developed by V.N. Seluyanov, and to assess their internal consistency, predictive power, and applicability. Methods. The study is based on the analysis and reconstruction of mathematical models derived from Seluyanov’s work. Systems of ordinary differential equations that describe shortterm and long-term adaptations, such as muscle bioenergetics and gas transport, as well as endocrine, immune, and digestive systems, have been reconstructed. Numerical modeling was performed using MATLAB and solvers for stiff systems. Results. The hierarchy of Seluyanov’s models has been organized. The short-term adaptation model accurately reproduces key physiological responses to exercise, such as the dynamics of creatine phosphate and ATP levels, lactate accumulation, and changes in cardiovascular and respiratory parameters. The long-term adaptation model shows the nonlinear dynamics of muscle fiber and mitochondrial growth, as well as how hormonal and immune status influence structural changes. It is shown that training at 70% of maximum capacity primarily stimulates muscle strength rather than aerobic fitness. Conclusions. The theoretical approach of V.N. Seluyanov represents an internally consistent mathematical theory of adaptation in sports, with significant predictive and educational potential. Reconstructed models enable a transition from empirical training planning to forecasting through simulation modeling. However, there are several limitations, such as the large number of empirical coefficients that require individual calibration, and a number of simplified assumptions. Despite these limitations, this approach continues to be relevant for the development of personalized training programs and decision support systems in elite sports.