Non-linear Systems

All biological systems – biological macromolecules, cells, tissues, and biocenosis – are active distributed systems. The transformation of substances and energy in these systems occurs in individual elementary volumes related to each other by the substance transportation, diffusive or directed under the action of external forces or with the help of special adaptation mechanisms inherent to living organisms. Every elementary volume is a system open with respect to mass and substance that is far from thermodynamic equilibrium, moreover, the energy-carrying substances or other energy sources are distributed in space and connected between themselves by the fluxes of substance and energy. In such systems, the so-called autowave processes are possible: the propagation of pulses and excitation waves, the formation of stationary spatially inhomogeneous distributions of substances, and other self-organization phenomena. 

The waves of electric potentials propagate in the fibres of cordial muscle. Pathological states here in the form of arrhythmia and fibrillation are determined by the appearance of autonomous sources of waves, the reverberators. Other types of autowave processes manifest themselves in the morphogenesis processes in the tissue differentiation. Genetic systems of the protein biosynthesis are local reaction elements of such systems, and the transport processes are performed by the systems of active transmembrane transport. In some communities (collective amoebas), the cellular interaction is performed by secreting the substances-attractants (cyclic AMP). Mutual movement of the cells to a source of signals and their aggregation are of a wave character. Autowave processes are also in the basis of the motions in the walls of blood vessel channels, peristalsis of other sections of gastrointestinal tract, mechanical displacements of the cells on a plane surface, and other processes.

Nonlinear interaction of the components in a system combined with transport processes leads to complex spatial and temporal behavior regimes of the system’s components. The first model of such kind of interaction was examined by Turing in his work «Chemical Basis of Morphogenesis». Alan M. Turing (1912–1954), English mathematician and logician, became famed for his studies in computer logic and the theory of automation. In 1952, he published the first part of an investigation dedicated to mathematical theory of the structure formation in an initially homogeneous system where chemical reactions occur simultaneously, including autocatalytic processes accompanied by the energy consumption, and passive processes of transport–diffusion. The Turing work became classic, and its ideas are in the basis of modern theory of nonlinear systems, theory of self-organization, and synergetics. 

In linear systems, the diffusion is a process that leads to the equalization of concentrations over the whole reaction volume. However, in the case of nonlinear interaction between the variables x and y, the instability of homogeneous stationary state can arise, and complex spatio-time regimes form like the autowaves or dissipative structures. They are represented by stationary in time and inhomogeneous in space concentration distributions, maintained on the account of the dissipation of system’s energy. The appearance of structures in such systems is stipulated by the difference in the diffusion coefficients of reagents, namely, by the presence of a short-range «activator» with a small diffusion coefficient and of a long-range «inhibitor» with a large diffusion coefficient.