The system of two the first order ordinary differential equations arising in the gene regulatory networks theory is studied. The structure of attractors for this system is described for three important behavioral cases: activation, inhibition, mixed activation-inhibition. The geometrical approach combined with the vector field analysis allows treating the problem in full generality. A number of propositions are stated and the proof is geometrical, avoiding complex analytic. Although not all the possible cases are considered, the instructions are given what to do in any particular situation.

We consider mathematical model of genetic regulatory networks (GRN). This model consists of a nonlinear system of ordinary differential equations. The vector of solutions X(t) is interpreted as a current state of a network for a given value of time t: Evolution of a network and future states depend heavily on attractors of system of ODE. We discuss this issue for low dimensional networks and show how the results can be applied for the study of large size networks. Examples and visualizations are provided

We study a genetic model (including gene regulatory networks) consisting of a system of several ordinary differential equations. This system contains a number of parameters and depends on the regulatory matrix that describes the interactions in this multicomponent network. The question of the attracting sets of this system, which depending on the parameters and elements of the regulatory matrix, isconsidered. The consideration is mainly geometric, which makes it possible to identify and classify possible network interactions. The system of differential equations contains a sigmoidal function, which allows taking into account the peculiarities of the network response to external influences. As a sigmoidal function, a logistic function is chosen, which is convenient for computer analysis. The question of constructing attractors in a system of arbitrary dimension is considered by constructing a block regulatory matrix, the blocks of which correspond to systems of lower dimension and have been studied earlier. The method is demonstrated with an example of a three-dimensional system, which is used to construct a system of dimensions twice as large. The presentation is provided with illustrations obtained as a result of computer calculations, and allowing, without going into details, to understand the formulation of the issue and ways to solve the problems that arise in this case.

In networks, which arise in multiple applications, the inhibitory connection between elements occur. These networks appear in genetic regulation, neuronal interactions, telecommunication designs, electronic devices. Mathematical modelling of such networks is an efficient tool for their studying. We consider the specific mathematical model, which uses systems of ordinary differential equations of a special form. The solution vector X(t) describes the current state of a network. Future states are dependent on the structure of the phase space and emerging attractive sets. Attractors, their properties and locations depend on the parameters in a system. Some of these parameters are adjustable. The important problem of managing and control over the system, is considered also.

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