Abstract:

Networks generally are understood as collections of objects (called nodes) with the prescribed structure. They are everywhere. The most important issues about networks are: 1) how they functionate; 2) how are they organized; 3) are they static or dynamic; 4) which practical conclusions goals do they pursue, and so on. In our paper we restrict ourselves to mathematical modeling of some kind of networks. These networks appear in biology [8],[9], theory of telecommunication networks [7], theory of neuronal interactions [5],[6]. Their modeling is necessary due to the fact that experimental research is hard and expensive. Mathematical modeling brings order and clarity. Since any mathematical model is an idealization of the process being modeled, only models that agree well with the available data should be used. We draw our attention to networks, consisting of any number of nodes, linked with known regulations. Each node is denoted xi, i=1,2, …The values of xican change in time, therefore we consider functions xi(t). These functions are positive valued. Their behavior depends on the total impact of the rest of the network elements.The interaction between elements of a network is predefined and encoded in a regulatory matrix W, which is the square n by n matrixwith real entries. Networks of this kind appear in the theory of genetic regulation. By this is meant the adaptation of cells of living organisms to changes in the environment. These changes include also diseases. Some researchers have considered the mathematical models of genetic regulatory networks (GRN), formulated in terms of ordinary differential equations (ODE). Solutions of equations correspond to extraction of proteins by elements of a network. The combined effect of this helps an organism to adapt well to fluctuating environment and withstand hazardous influences, including diseases. Systems of ODE have the form