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 -