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.
| Abstracts file: | AkGoLyapAbstr.tex |
| Full text file: | AkGoLyap100.pdf |
| Presentation file: | GolubyatnikovAAL.pdf |