Abstract: A nonlinear system of three differential equations is studied that describes photosynthesis in autotrophic systems. An area is identified that is invariant with respect to motion along the trajectory of the system with increasing time. In this area, the existence of a unique stationary solution is established and questions of its stability are investigated. At present, due to the exponential growth of the population, industrial progress and, as a consequence, an increase in the general pollution of the biosphere, the study of the resistance of plant organisms to anthropogenic pollution is acquiring the most important practical and theoretical significance. At the same time, the problem of a qualitative study of the processes of photosynthesis has become extremely urgent. In problems related to photosynthesis, it is of great interest to determine the laws of functioning of he system, as well as the choice of methods of mathematical and computer modeling. This is the process of converting the absorbed light energy into chemical energy of organic compounds, the only process in the biosphere leading to an increase in the energy of the biosphere due to an external source - the Sun, which ensures the existence of both plants and all heterotrophic organisms. The most important external factors affecting the processes of photosynthesis and photorespiration are temperature, photosynthetic active radiation, water regime, the regime of plant mineral nutrition, as well as the content of carbon dioxide and oxygen in the surrounding space. In recent decades, there has been an increase in the concentration of carbon dioxide in the atmosphere and a change in the thermal regime on a planetary scale. In this regard, the problem of predicting changes in the intensity of photosynthesis of plants caused by changes in the concentration of atmospheric carbon dioxide and temperature is urgent.
For citation: Mukhamadiev, E. M., Nurov, I. D. and Sharifzoda, Z. I. Qualitative Analysis and Stability of the Dynamics of Photosynthesis in Autotrophic Systems, Vladikavkaz Math. J., 2021, vol. 23, no. 3,
pp. 113-125 (in Russian). DOI 10.46698/u3748-9762-0471-a
1. Riznichenko, G. Yu. Lekcii po matematicheskim modelyam v biologii
[Lectures on Mathematical Models in Biology],
Moscow, Ijevsk, Regular and Chaotic Dnamics, 2011, 558 p. (in Russian).
2. Rubin, A. B. Biophysic. Vol. 2. Biophysics of Cellular Processes, Moscow,
Publishing House of Moscow State University, 2000, 468 p.
3. Riznichenko, G. Yu., Belyaeva, N. E., Kovalenko, I. B. and Rubin, A. B. Mathematical
and Computer Modeling of Primary of Photosynthetic Processes, Biophysics,
2009, vol. 54, no. 1, pp. 10-22.
4. Nurov, I. D. and Sharifzoda, Z. I. Qualitative System of Kinetic Equations Describing
the Interaction of One-Electron and Two-Electron Carrier, XXVI Mezhdunar.
konf. "Matematika. Komp'juter. Obrazovanie" (Pushhino, 28 janvarja - 2 fevralja 2019)
[Twenty-sixth International Conference "Mathematics. Computer. Education"
(Pushino, January 28 - February 2, 2019)], 2019, pp. 17 (in Russian).
5. Mukhamadiev, E. M., Sharifzoda, Z. I. and Nurov, I. D. Qualitative Investigation of the Nonlinear Problem of Photosynthesis, Doklady Akademii Nauk Respubliki Tadzhikistan [Reports of the Academy of Sciences of the Republic of Tajikistan], 2019, vol. 62, no. 9-10, pp. 511-518 (in Russian).
6. Kurosh, A. G. Kurs vysshej algebry [Course of Higher Algebra],
Moscow, Nauka, 1968, 431 p. (in Russian).
7. Bautin, N. N. and Leontovich, E. A. Metody i priemy kachestvennogo issledovaniya
dinamicheskih sistem na ploskosti [Methods and Tricks for the Qualitative Study of Dynamical Systems on the Plane],
Moscow, Nauka, 1976, 496 p. (in Russian).
8. Hartman, P. Ordinary Differential Equations, New York, John Wiley and Sons, 1964.
9. Arabov, M. K. Analysis of the Stability of a Singular Point of a Second Order Quasilinear Equation,
Izv. AN Resp. Tadzhikistan. Otd. fiz.-mat., khim., geol. i tekhn. nauk [News of the
Academy of Sciences of the Republic of Tajikistan. Department of Physical
Mathematical, Chemical, Geological and Technical Scienses],
2015, vol. 158, no. 1, pp. 42-49 (in Russian).
10. Ilolov, M. and Kuchakshoev, Kh. S. On Abstract Equations with Unbounded
Nonlinearities and their Applications, Doklady Mathematics,
2009, vol. 80, no. 2, pp. 694-696. DOI: 10.1134/S1064562409050160.
11. Mukhamadiev, E. M., Nurov, I. D. and Khalilova, M. Sh. Limiting Cycles
of Piece-Linear Second Order Differential Equations, Ufa Mathematical Journal,
2014, vol. 6, no. 1, pp. 80-89. DOI: 10.13108/2014-6-1-80.
12. Mukhamadiev, E. M., Gulov, A. M. and Nurov, I. D. Analysis of Limit Cycle
Appearing for one Class of Non-Linear Second Order Differential Equation,
Proceedings of Voronezh State University. Series: Physics. Mathematics, 2016, no. 1, pp. 118-125 (in Russian).
13. Yumagulov, M. G.Vvedenie v teoriyu dinamicheskikh sistem: Uchebnoe posobie [Introduction to the Theory of Dynamical Systems: Tutorial], Saint Petersburg, Lan' Publishing, 2015, 272 p. (in Russian).
14. Gulov, A. M. and Sharifzoda, Z. I. Certification: That they are the Authors
of the Computer Program of the "A Software Package for Constructing Phase
Portraits of Dynamic System" from 21.06.2019.