CH11 network enfolding

This property refers to the incorporation of indirect energy and matter flows and storages into empirically observed and measured flows and storages. It is another property of coupled systems elucidated by environ mathematics, and it potentially touches many areas of ecology such as chemical stoichiometry, embodied energy ("emergy"), and ecological indicators. Ecologists observe and measure, for example, the chemical composition of organisms or bulk samples. For an entity to have a "composition" means it is a composite—made up of materials brought to it from wherever its incoming network reaches in the containing system—directly and indirectly. This has consequences for even a seemingly straightforward concept like a "direct" flow where it turns out to be "macroscopically direct" and must be distinguished from "microscopically direct." To illustrate, in Figure 5.2b the flow f21 from compartment 1 to 2 is unambiguously direct. Macroscopically, the link and the process responsible for it (like eating) are direct, and microscopically so are the molecules (food) transferred because it is derived directly from the boundary input z1. This latter directness is due to the fact that flow f21 represents the first transfer of boundary input from the compartment that received it to another; it is uncontaminated by substance from other sources because, in this case, input z2 cannot reach compartment 1. The situation is different in Figure 5.3. The same flow f21 in this figure is still macroscopically adjacent to compartments 1 and 2 (i.e., "direct"), but now it contains composited flow derived from all three inputs. These inputs, the throughflows and storages (not shown) they generate, and also the other three adjacent flows are all complexly enfolded into f21. The enfolding is mutual, and in this case it is universal because all interior network elements are reachable from all the others. The entire system of Figure 5.3 is thus (at steady-state) a composite of itself, which is the ultimate expression of holism. Moreover, this composition property, network enfolding, is true for complex systems generally. If one can imagine empirically sampling this system, f21 (and the other interior flows as well) strike the senses as direct. However, they are not since they contain indirect flows from the other sources a few to many times removed, and so are better considered as "adjacent", or perhaps just "observed." In environ mathematics, this

Figure 5.3 Three-compartment model illustrating network enfolding. Ecologically, the compartment labeled T1 can be taken as producers, T2 as consumers, and T3 as decomposers.

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