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{\bf MATHEMATICAL AND NUMERICAL MODELING \\
OF GENE NETWORK FUNCTIONING}
\footnote{ The work was supported by RFBR grant 09-01-00070,
by grant 2.1.1/3707 of the AVZP-program
"High-School potential development",
and by interdisciplinary grant 119 of SB RAS.}
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%%% \author{
A.A. Akinshin
{\it Altai state technical university}
e-mail: mayortg@gmail.com
V.P. Golubyatnikov V.P.
{\it Sobolev institute of mathematics SB RAS}
e-mail: glbtn@math.nsc.ru
Golubyatnikov I.V.
e-mail: ivan.golubyatnikov@gmail.com
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Questions of existence of periodic trajectories
in natural gene networks and in their mathematical models
play an important role in the theory of the gene networks.
Actually, similar questions appear
in various domains of pure and applied mathematics,
and even in the case of 2-dimensional dynamical systems
very famous problems, such as the Center-Focus problem, are still open.
Some sufficient conditions of existence of cycles and
corresponding stability questions for odd-dimensional
nonlinear dynamical systems of chemical kinetics
were studied in our previous publications
where these systems were considered as models
of gene networks functioning.
The behavior of trajectories of these systems
in even-dimensional dynamical systems of this type,
or in presence of positive feedbacks
in corresponding gene networks, is much more complicated.
Usually, such systems have several stationary points and cycles.
Some of these points and cycles are stable, and
boundaries between the basins of these attractors
contain unstable stationary points and/or cycles.
Description of the phase portraits of these systems,
visualization of
these boundaries and detection of these unstable
cycles are hard problems both in pure and in numerical mathematics.
At first, we study here simple gene networks models,
where the regulation is realized by the negative feedbacks only.
In this rather simple case, we have detected in our numerical
experiments non-uniqueness of limit cycles.
Then we consider some models of gene networks regulated by
combinations of negative and positive feedbacks.
More complicated gene networks models
can be interpreted
as combinations of these "elementary" models.
We find conditions of existence of stable cycles in
some models of gene networks regulated by negative feedbacks
and by simple combinations of negative and positive feedbacks.
Special algorithms and programs for numerical simulations of
these results are elaborated as well.
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