Apparently, a network is one of the most widespread forms of organisation of Space in Nature. Note that it can be as real a geometric space as any other; for instance, a space of trophic possibilities.

A propos, what is the dimension of a network? What dimension has a planar cobweb?

From table-talks in Karrebœk.

The network approach is very popular in ecology. Flows of energy, matter and information are described over a network of nodes and links. It began with Lindeman's (1942) classic work. More recently, Patten has devoted much attention to this problem (Patten, 1982, 1991; Patten et al., 1990). Chapter 7 of J0rgensen's (2002b) book "Integration of Ecosystem Theories: A Pattern" contains a very good overview of the concept and an extensive list of references. On the other hand, the so-called "compartmental models" are very popular in biology: see, for example, Straskraba and Gnauk (1985). These models represent typical linear dynamic structures (systems of ordinary differential equations) that correspond to specific graphic structures. In this way they can be interpreted as trophic networks. It is obvious that trophic chains (see Chapter 7) are a special case of trophic networks, which, in turn, can be considered as a combination of several trophic chains.

If we look at numerous publications on biogeochemical cycles in ecosystems we can see that, typically, the systems are presented by a flow-chart diagram with storage of certain elements inside compartments and flows between them.

We also know that the non-linear Volterra equations are classic objects of mathematical ecology and they are often used for the description of ecological dynamics. The following question emerges: "Can we construct some non-contradictory combinations from these two approaches?" Later on (in Section 9.7) we shall attempt to answer the question.

The network description is also popular in biological and chemical sciences. In particular, a typical example of a network description is the description of a metabolic cycle in biophysics (Rubin, 1999). A classic work on the network approach in biophysics is "Network thermodynamics: dynamic modelling of biophysical systems" by Oster et al. (1973). Aoki (1988, 1992) has applied the entropy and exergy concepts to ecological networks at steady state.

Towards a Thermodynamic Theory for Ecological Systems, pp. 221-241 © 2004 Elsevier Ltd. All Rights Reserved.

In Chapter 4, we already dealt with the hierarchy of descriptions of biological communities (or ecosystems) when the first level of description was simply a list of species, and where if n species were indistinguishable then the information per one species was equal to /1 = k ln n, where k is either Boltzmann's constant or its analogue. In the next level of description each species possessed such a property as the number of specimens N,. Then the information per one specimen was equal to /j = -kin= Pi ln pi, Pi = N/Y.n=J N,. It is obvious that maxp, /2 = ln n and /2 # /j. This can be expected, since the "specific" information /j and /2 are calculated for different "agents": species in the first case and specimens in the second. Further complication of the description will lead to a more and more detailed description of specimen characteristics, such as their specific biomass, size, volume, energy, exergy, etc. However, beginning with the first level we could be using another approach, bearing in mind that any ecosystem is not only an entity of different individuals but also a combination of interactions both between different species and between them and the environment. The first obvious step, so common in ecology, is to take a qualitative description of the trophic network as a graph with vertices corresponding to species and arcs indicating the pathways of energy and matter in the system. In this case, we say that any two species are connected by a trophic relationship. Such a type of graph is called "directed" since the directions of energy and matter flows are given. Since all species are exchanging energy and matter with the environment to "complete" the description we have to define the environment as one or several quasi-species. For instance, the environment can be represented as two quasi-species: the first one combines all flows from the environment into the system (solar radiation, nutrients, etc.) while the second combines all flows out of the system (metabolism, dying-off, etc.).

The representation of ecosystem structure as a trophic graph does not exhaust the entire set of interspecies relations. For instance, in the ecosystem structure we could select a group of species as a trophic level. These species are not connected by trophic relations, but they are usually either competing for resources (the relation of competition) or co-operating in their utilisation (the relation of mutualism, or symbiosis).

The classification of all sorts of interspecific relations is based on an idea formulated by Velimir Khlebnikov (1910), a prominent Russian poet. He classified the interaction between species not by their mechanisms in the specimen level (in this case the classification would become boundless), but according to the effect of interaction on the state (in particular, on the size) of both interacting populations. By interpreting Khlebnikov's idea in modern terms we can say that, for instance, if the population growth of one species inhibits the growth of the other, and the growth of the second stimulates the growth of the first, we have the "prey-predator" (or "host-parasite", or "herbivorous-plant") relation. We shall denote the effect of one of the species on another by one of the signs: + (stimulating), — (inhibiting) or 0 (neutral). The full list of pair-wise relations will then consist of the following six types:

+ + symbiosis, or mutualism;

competition;

commensalism;

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