Spatial Dissipative Structures in Excitable Media (Plankton, Soil Bacteria. . . .)

Spatial Dissipative Structures in Excitable Media (Plankton, Soil Bacteria. . . .)

DOI: 10.4018/978-1-5225-9651-6.ch004

Abstract

A general, the simplest model of a spatial dissipative structure arising in an excitable medium is constructed, containing at least two components interacting with each other with their own mobility. One of these components (active) uses the other component as food. It is shown that such a model leads to a stationary stable spatial distribution of the components in the form of Liesegang bands. As specific examples of the formation of spatial dissipative structures, structures arising in plankton consisting of phytoplankton and zooplankton and in the soil containing the bacterial population and the nutrient substrate are considered. Bifurcation diagrams are constructed in the parameter space, characteristic for each of the considered excitable media, which determine the conditions for the formation of dissipative structures in these media. The existence in the plankton of a strange attractor of a previously unknown shape in four-dimensional phase space has been discovered.
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General Formulation Of The Problem Of Spatial Dissipative Structure In An Excitable Medium

The problem of energy dissipation, dissipative structures is one of the unconditionally significant fundamental problems of natural science. Of paramount importance are the phenomenological special dissipative structures arising from the initially uniform distribution of matter. An example of such structures are the famous Liesegang bands (Liesegang, 1896, 1911a, b, 1923, 1924). Attempts to explain the Liesegang bands have been undertaken for more than 100 years (Ostwald, 1899; Nell, 1905; Hatshchek, 1911, 1914; Brandford, 1916; Jablczynski, 1923; Dogadkin, 1928). The establishment of physical, chemical, as well as, as it became obvious (Kravchenko et al., 1998a, b, c) biological mechanisms for the occurrence of Liezegang fields and the development of a mathematical apparatus that reflects the essence of these mechanisms is extremely important. A convincing explanation for the Liezegang bands suggested by G.V. Zhizhin (Zhizhin, 2004a, b, 2005). The explanation is based on solutions of systems of parabolic differential equations taking into account the provisions of chemical kinetics, diffusion, and reversibility of chemical reactions.

It was possible to establish that solutions describing stationary dissipative structures comparable to Liesegang bands form a new class of stationary composite solutions of differential parabolic equations.

This research direction is connected with the law of V.V. Dokuchaeva on the zonality of the distribution of living organisms in nature: plants, animals, bacteria in the soil, etc. (Mishustin, 1982). It is significant that the diversity of soils and the composition of bacteria in them, even in small areas of the earth's surface, is extremely large (Dobrovolsky, 2001). Zonal (spotty) distribution is also characteristic of plankton. It occurs whenever there are interacting components in the medium. This interaction excites the system, launching self - regulation processes. They lead to the formation of dissipative stable systems. In the previous chapter, it was shown on the example of plankton consisting of phytoplankton and zooplankton that the interaction of these components in the local approximation leads to the achievement of an attractor, that is, an equilibrium position as a point in phase space, or a limit cycle, or a limit set. If we take into account the distribution of the substance (in this case, living matter) in space, then the processes of self - regulation lead to the zone distribution of the components.

Key Terms in this Chapter

Separatrix: A trajectory separating from each other the trajectories of the phase space of a qualitatively different type.

Dissipative Structures: The structures in various mediums that exist in nature are stationary due to the consumption (dissipation) of energy from the environment.

Strange Attractor: A attracting set of unstable trajectories in the phase space of a dissipative dynamical system.

Copitrophic Bacteria: Bacteria living in water that can grow at low nutrient concentrations.

Attractor: A compact subset of the phase space of a dynamic system, all trajectories from a certain neighborhood of which tend to it at a time tending to infinity.

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