Phase portraits of two nonlinear models of circular gene networks. (Russian. English summary) Zbl 1540.37109
Summary: For two dynamical systems of dimensions 4 and 5 which simulate circular gene networks with non-linear degradation of their components we find conditions for existence of periodic trajectories and construct invariant domains which contain all these trajectories. Interiors of both domains are homeomorphic to torus, and the boundary of each of them contains a unique equilibrium point of the corresponding dynamical system.
MSC:
| 37N25 | Dynamical systems in biology |
| 70K05 | Phase plane analysis, limit cycles for nonlinear problems in mechanics |
| 92D10 | Genetics and epigenetics |
| 92D15 | Problems related to evolution |
Keywords:
circular gene network model; phase portrait; nonlinear dynamical system; equilibrium point; invariant domain; periodic trajectoryReferences:
| [1] | Glass L., Pasternack J. S. Stable oscillations in mathematical models of biological control systems // J. Math. Biol. 1978. V. 6. P. 207-223. · Zbl 0391.92001 |
| [2] | Likhoshvai V. A., Kogai V. V., Fadeev S. I., Khlebodarova T. M. On the chaos in gene networks // J. Bioinform. Comput. Biol. 2013. V. 11. Article ID 1340009. |
| [3] | Gedeon T., Pernarowski M., Wilander A. Cyclic feedback systems with quorum sensing cou-pling // Bull. Math. Biol. 2016. V. 78, N 6. P. 1291-1317. · Zbl 1348.92021 |
| [4] | Глызин С. Д., Колесов А. Ю., Розов Н. Х. Существование и устойчивость релаксацион-ного цикла в математической модели репрессилятора // Мат. заметки. 2017. Т. 101, № 1. С. 58-67. |
| [5] | Hastings S., Tyson J., Webster D. Existence of periodic solutions for negative feedback cellular control system // J. Differ. Equ. 1977. V. 25. P. 39-64. · Zbl 0361.34038 |
| [6] | Ayupova N. B., Golubyatnikov V. P. On two classes of non-linear dynamical systems: the four-dimensional case // Sib. Math. J. 2015. V. 56, N 2. P. 231-236. · Zbl 1357.37043 |
| [7] | Golubyatnikov V. P., Gradov V. S. Non-uniqueness of cycles in piecewise-linear models of circular gene networks // Sib. Adv. Math. 2021. V. 31, N 1. P. 1-12. |
| [8] | Golubyatnikov V. P., Minushkina L. S. On uniqueness and stability of a cycle in one gene network // Sib. Electron. Math. Rep. 2021. V. 18, N 1. P. 464-473. · Zbl 1470.37112 |
| [9] | Ayupova N. B., Golubyatnikov V. P. On structure of phase portrait of one 5-dimensional circular gene network model // J. Appl. Ind. Math. 2021. V. 15, N 4. P. 376-383. · Zbl 1512.37100 |
| [10] | Golubyatnikov V. P., Golubyatnikov I. V., Likhoshvai V. A. On the existence and stability of cycles in five-dimensional models of gene networks // Numer. Anal. Appl. 2010. V. 3, N 4. P. 329-335. |
| [11] | Гантмахер Ф. Р. Теория матриц. М.: Физматгиз, 1967. |
| [12] | Khlebodarova T. M., Kogai V. V., Fadeev S. I., Likhoshvai V. A. Chaos and hyperchaos in simple gene network with negative feedback and time delays // J. Bioinform. Comput. Biol. 2017. V. 15, N 2. Article ID 1650042. |
| [13] | Арнольд В. И., Варченко А. Н., Гусейн-Заде С. М. Особенности дифференцируемых отображений. М.: МЦНМО, 2009. |
| [14] | Гайдов Ю. А. Об устойчивости периодических траекторий в некоторых моделях генных сетей // Сиб. журн. индустр. математики. 2008. Т. 11, № 1. С. 57-62. · Zbl 1223.92017 |
| [15] | Казанцев М. В. О некоторых свойствах графов доменов динамических систем // Сиб. журн. индустр. математики. 2015. Т. 18, № 4. С. 42-49. |
| [16] | Голубятников В. П., Кириллова Н. Е. Фазовые портреты двух моделей генных сетей // Мат. заметки СВФУ. 2021. Т. 28, № 1. С. 3-11. |
| [17] | Ivanov V. V. Attracting limit cycle of an odd-dimensional circular gene network model // J. Appl. Ind. Math. 2022. V. 16, N 3. P. 409-415. · Zbl 1529.37019 |
| [18] | Кириллова Н. Е. Об инвариантных поверхностях в моделях генных сетей // Сиб. журн. индустр. математики. 2020. Т. 23, № 4. С. 69-76. |
| [19] | Golubyatnikov V. P., Likhoshvai V. A., Ratushny A. V. Existence of closed trajectories in 3-D gene networks // J. Three Dimensional Images 3D Forum. Japan. 2004. V. 18, N 4. P. 96-101. |
| [20] | Петровский И. Г. Лекции по теории обыкновенных дифференциальных уравнений. М.: изд-во Моск. ун-та, 1984. |
| [21] | Хартман Ф. Обыкновенные дифференциальные уравнения. М.: Мир, 1970. |
| [22] | Чумаков Г. А., Чумакова Н. А. Гомоклинические циклы в одной модели генной сети. Мат. заметки СВФУ. 2014. Т. 21, № 14. C. 97-106. · Zbl 1374.92039 |
| [23] | Поступила в редакцию 22 марта 2023 г. После доработки 22 марта 2023 г. Принята к публикации 29 мая 2023 г. |
| [24] | Аюпова Наталья Борисовна Институт математики им. С. Л. Соболева СО РАН, пр. Академика Коптюга, 4, Новосибирск 630090 ayupova@math.nsc.ru Голубятников Владимир Петрович Институт математики им. С. Л. Соболева СО РАН, пр. Академика Коптюга, 4, Новосибирск 630090; |
| [25] | Новосибирский военный институт им. И. К. Яковлева войск национальной гвардии РФ Ключ-Камышенское плато, 6/2, Новосибирск 630114 golubyatn@yandex.ru REFERENCES |
| [26] | Glass L. and Pasternack J. S., “Stable oscillations in mathematical models of biological control systems,” J. Math. Biol., 6, 207-223 (1978). · Zbl 0391.92001 |
| [27] | Likhoshvai V. A., Kogai V. V., Fadeev S. I., and Khlebodarova T. M., “On the chaos in gene networks,” J. Bioinform. Comput. Biol., 11, article No. 1340009 (2013). |
| [28] | Gedeon T., Pernarowski M., and Wilander A., “Cyclic feedback systems with quorum sensing coupling,” Bull. Math. Biol., 78, No. 6, 1291-1317 (2016). · Zbl 1348.92021 |
| [29] | Glyzin S. D., Kolesov A. Yu., and Rozov N. Kh., “Existence and stability of the relaxation cycle in a mathematical repressilator model,” Math. Notes, 101, No. 1, 71-86 (2017). · Zbl 1365.92027 |
| [30] | Hastings S., Tyson J., and Webster D., “Existence of periodic solutions for negative feedback cellular control system,” J. Differ. Equ., 25, 39-64 (1977). · Zbl 0361.34038 |
| [31] | Ayupova N. B. and Golubyatnikov V. P., “On two classes of non-linear dynamical systems: the four-dimensional case,” Sib. Math. J., 56, No. 2, 231-236 (2015). · Zbl 1357.37043 |
| [32] | Golubyatnikov V. P. and Gradov V. S., “Non-uniqueness of cycles in piecewise-linear models of circular gene networks,” Sib. Adv. Math., 31, No. 1, 1-12 (2021). |
| [33] | Golubyatnikov V. P. and Minushkina L. S., “On uniqueness and stability of a cycle in one gene network,” Sib. Electron. Math. Rep., 1, No. 1, 464-473 (2021). · Zbl 1470.37112 |
| [34] | Ayupova N. B. and Golubyatnikov V. P., “On structure of phase portrait of one 5-dimensional circular gene network model,” J. Appl. Ind. Math., 15, No. 3, 376-383 (2021). · Zbl 1512.37100 |
| [35] | Golubyatnikov V. P., Golubyatnikov I. V., and Likhoshvai V. A., “On the existence and stability of cycles in five-dimensional models of gene networks,” Numer. Anal. Appl., 3, No. 4, 329-335 (2010). |
| [36] | Gantmacher F. R., The Theory of Matrices, vol. 1, Chelsea Publ. Co. (1984). |
| [37] | Khlebodarova T. M., Kogai V. V., Fadeev S. I., and Likhoshvai V. A., “Chaos and hyperchaos in simple gene network with negative feedback and time delays,” J. Bioinform. Comput. Biol., 15, No. 2, article No. 1650042 (2017). |
| [38] | Arnold V. I., Gusein-Zade S. M., and Varchenko A. N., Singularities of Differentiable Maps, Birkhäuser, Boston (1985). · Zbl 0554.58001 |
| [39] | Gaidov Yu. V., “On the stability of periodic trajectories in some gene network models,” J. Appl. Ind. Math., 4, No. 1, 43-47 (2010). |
| [40] | Kazantsev M. V., “On some properties of the domain graphs of dynamical systems [in Rus-sian],” Sib. Zh. Ind. Mat., 56, No. 4, 42-48 (2015). · Zbl 1349.92061 |
| [41] | Golubyatnikov V. P. and Kirillova N. E., “Phase portraits of two gene networks models [in Russian],” Mat. Zamet. SVFU, 28, No. 1, 3-11 (2021). · Zbl 07820934 |
| [42] | Ivanov V. V., “Attracting limit cycle of an odd-dimensional circular gene network model,” J. Appl. Ind. Math., 16, No. 3, 409-415 (2022). · Zbl 1529.37019 |
| [43] | Kirillova N. E., “On invariant surfaces in gene network models,” J. Appl. Ind. Math., 14, No. 4, 666-671 (2020). · Zbl 1509.37035 |
| [44] | Golubyatnikov V. P., Likhoshvai V. A., and Ratushny A. V., “Existence of closed trajectories in 3-D gene networks,” J. Three Dimensional Images 3D Forum, Japan, 18, No. 4, 96-101 (2004). |
| [45] | Petrovskii I. G., Lectures on the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1970). |
| [46] | Hartman Ph., Ordinary Differential Equations, 2nd ed., SIAM (1982). · Zbl 0476.34002 |
| [47] | Chumakov G. A. and Chumakova N. A., “Homoclinic cycles in a gene network model [in Russian],” Mat. Zamet. SVFU, 21, No. 4, 97-106 (2014). · Zbl 1374.92039 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
