Scientific relevance and purpose. This research looks at the urgent task of modeling the structure of the human respiratory system and processes occurring in it, in order to predict the changes in physiological parameters occurring under different mechanical actions. Results. This paper suggests mathematical model based on the description of equations of the mass flow and mass flow rate in the pulmonary channels in cases, when airways are branched in accordance with the prin-ciple of regular dichotomy with regard to the equations of work dynamics of the respiratory muscles and the ability to model different stresses in the breathing circuit, caused by trainers. The research examined the stresses generated by muscles in the radial and axial direction of the equivalent hollow cylinder, which represented the chest with regard to the elastic stress component in the cylinder wall and variable muscle stress in the circumfe-rential direction. The paper contains the results of mathematical modeling of breathing without stress, the graphs of volume and mass flow in lungs generations and pressure-flow diagram. Conclusions. The developed mathematical models enable more precise multi-parameter modeling of the dynamics of functioning of complex biotech system "respiratory muscles trainer - human", which enables the implementation of the prediction of shifts of physiological and mechanical properties from the values of the normal process and to adjust the control actions on this basis