Research output: Contribution to journal › Article › peer-review
On Oscillations in a Gene Network with Diffusion. / Golubyatnikov, Vladimir; Ayupova, Natalia; Kirillova, Natalia.
In: Mathematics, Vol. 11, No. 8, 1951, 04.2023.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - On Oscillations in a Gene Network with Diffusion
AU - Golubyatnikov, Vladimir
AU - Ayupova, Natalia
AU - Kirillova, Natalia
N1 - The study was supported by a grant from the Russian Science Foundation (project no. 23-21-00019).
PY - 2023/4
Y1 - 2023/4
N2 - We consider one system of partial derivative equations of the parabolic type as a model of a simple 3D gene network in the presence of diffusion of its three components. Using discretization of the phase portrait of this system, comparison theorems, and other methods of the qualitative theory of differential equations, we show uniqueness of the equilibrium solution to this system and find conditions of instability of this equilibrium. Then, we obtain sufficient conditions of existence of at least one oscillating functioning regime of this gene network. An estimate of lower and upper bounds for periods of these oscillations is given as well. In quite a similar way, these results on the existence of cycles in 3D gene networks can be extended to higher-dimensional systems of parabolic or other evolution equations in order to construct mathematical models of more complicated molecular–genetic systems.
AB - We consider one system of partial derivative equations of the parabolic type as a model of a simple 3D gene network in the presence of diffusion of its three components. Using discretization of the phase portrait of this system, comparison theorems, and other methods of the qualitative theory of differential equations, we show uniqueness of the equilibrium solution to this system and find conditions of instability of this equilibrium. Then, we obtain sufficient conditions of existence of at least one oscillating functioning regime of this gene network. An estimate of lower and upper bounds for periods of these oscillations is given as well. In quite a similar way, these results on the existence of cycles in 3D gene networks can be extended to higher-dimensional systems of parabolic or other evolution equations in order to construct mathematical models of more complicated molecular–genetic systems.
KW - Poincaré map
KW - cycles
KW - fixed point theorem
KW - gene network models
KW - invariant domains
KW - invariant surfaces
KW - phase portraits
KW - reaction–diffusion equations
KW - stability
KW - systems of non-linear differential equations
UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85153760574&origin=inward&txGid=3f2f0d92d401fe086d2685c3dfca3196
UR - https://www.mendeley.com/catalogue/55d0fa0d-57f8-3b0f-9207-b1a015595083/
U2 - 10.3390/math11081951
DO - 10.3390/math11081951
M3 - Article
VL - 11
JO - Mathematics
JF - Mathematics
SN - 2227-7390
IS - 8
M1 - 1951
ER -
ID: 59248442