V. G. Yakhno

Fronts and pulses: Elementary autowave structures

We will consider models which are most frequently encountered in experiments with autowaves in the form of fronts or excitation pulses (travelling pulses (TP) and travelling fronts (TF)).

Ways of investigation of autowave systems

We do not use the word “methods” in the title of this section since we wish to emphasize that a qualitative theory of AW processes is still to be constructed. This can be illustrated by comparing the undeveloped approaches used for studying AW systems with the state-of-the-art investigation of the nonlinear systems of the second order whose behaviour in the parameter space is known in ample detail. So far, only some particular, though rather important, cases have been considered. However, qualitative and computer methods for studying AW processes simulated by parabolic quasilinear equations seem to be promising. The point is that it is sufficient to take into account a finite (usually small) number of degrees of freedom. Therefore the main problem which arises in analyzing AW processes, is to recognize these several degrees of freedom, to find stationary solutions, and to investigate their stability and transitions between them.

Physical premises for the construction of basic models

The multitude of objects in which AWP have been observed is far too great to dwell on the construction and analysis of mathematical models for each of them. Their common features, with respect to AW phenomena, are revealed in the structure of simplified basic models which are analysed in the subsequent chapters of this book. Evidently, a model must be a sort of a “sketch” of the system concerned, so that it can be studied with a fair degree of completeness and the results obtained would be relevant to a wide class of objects. The basic models considered below meet this requirement. On the other hand, one should bear in mind the limits of applicability of the results obtained in this way to concrete objects. To this end, one should have, of course, a sufficiently complete mathematical description of the object in view, to be able to compare it with the corresponding basic model. Meanwhile, some essential limitations can be considered from a general point of view. These aspects are the subject of the present Chapter where we deal with mathematical models for systems with finite transport velocities, direct interactions between remote elements, AWPs in media with anisotropy in the transport phenomena and with mutual diffusion of the system components.

Noise and autowave processes

The fundamental kinetic equations (1.1) for AW processes are obtained by statistical averaging. Indeed, the “concentrations” or “velocities” of the kinetic variables are certain averaged values. In principle an arbitrary deterministic model corresponds to a stochastic model whose kinetic interactions are described by the probabilities of elementary events that occur in small time intervals in small elementary volumes. Noise arising in discrete interactions will become small after averaging if the number of interacting objects (molecules, living organisms) is large. We refer to such types of noise as “natural noise” in the following. However they can also be important in systems with a small number of interacting objects, some examples are ecological communities, systems of genes and organelles in a living cell. Even small amounts of noise can have a substantial effect in systems with concentrations near bifurcation points. Natural noise also has a pronounced effect on multistationary systems. It is interesting to note that in relaxation-type auto-oscillatory chemical reactions there may be time intervals during which the concentration of reactants decreases by many orders of magnitude. The part played by natural fluctuations increases sharply (Zhabotinskii 1974; Romanovskii et al. 1975).

Spatially inhomogeneous stationary states: dissipative structures

Spontaneous breaking of symmetry and the formation of stationary structures in spatially homogeneous systems with local positive feedback, Eqs. (1.1)–(1.3), were first indicated by A. Turing (1952) in his pioneering work on one of the crucial problems in biology of development — morphogenesis. The title of the work we cite was “Chemical foundations of morphogenesis”. After a decade, in the Sixties, such structures have been investigated further by members of I. Prigogine’s Brussels School (Glansdorff and Prigogine, 1971). In their studies the phenomenon was provided with a name, “dissipative structures” (DS), which is relevant to thermodynamical aspects of the problem: the structures exist owing to an influx of energy, substance etc. from the environment and the dissipation within the system. During the past decade, various aspects of the theory of DS and its applications have been investigated in many laboratories (Vasilev, Romanovskii and Yakhno 1979, Kerner and Osipov 1978, 1980, 1983, Belintsev 1983, Mcolis and Prigogine 1977, Romanovskii, Stepa-nova and Chernavskii 1984; Murray 1977).

The synchronization of auto-oscillations in space as a self-organization factor

Mutual synchronization (self-synchronization) is a remarkable, powerful factor of self-organization in distributed auto-oscillatory systems or in networks of discrete auto-oscillatory systems of collective activity, which are most diverse in nature. By synchronization we mean a spontaneous adjustment to a unique synchronous frequency and the establishment of certain phase relations between oscillations, which are stable with respect to perturbations, in some parts of a distributed system or in partially discrete auto-oscillators.

Autowave mechanisms of transport in living tubes

In this chapter we consider a class of AW mechanisms in biology which effect fluid transfer inside a tube and are completely or partially caused by changes in the shape of the tube. Such a fluid transfer is called a peristaltic current and its objective is the transport and/or mixing of the fluid. In the first case the change in the shape of the tube usually has the character of a travelling wave, while in the second more complex wave processes take place. Technical examples of peristaltic systems are for instance various roll pumps used in devices for artificial blood circulation or for precise dosage. Biological examples are the organs of the gastrointestinal tract (oesophagus, stomach, small intestine, colon), the ureter, myometer, some bloodvessels and other organs with smooth muscle tissue as well as small transport veins in plants or even unicellular organisms.

Autowave processes and their role in natural sciences

In the progress of natural sciences there are stages when different domains, seemingly quite separate, are united by common ideas and methods. Such a stimulating cooperation of sciences has always been fruitful for the domains involved. A remarkable feature of modern developments is an extensive penetration of mathematics, as well as various methods of experimental and theoretical physics, into chemistry and biology. The process of contact between different sciences has always led to the consolidation of common concepts and regularities.

Autonomous wave sources

Autonomous wave sources (AWS) hold a prominent place among the characteristic nonlinear structures in a homogeneous autowave medium. They are regions of finite dimensions where nonstationary motions lead to the generation of TPs. It is well known that there are so-called pacemakers in an inhomogeneous space which are localized regions of a self-sustained oscillatory medium, where the oscillation frequency is higher than anywhere in the surrounding space. The formation of a TP by such sources occurs under any initial conditions.