## CH10 network boundary amplification

When a compartment within a system brings substance into the system from outside, the importing compartment is favored in development over others that do not do this. The reason is a technical property of both the throughflow- and storage-generating matrices of environ analysis known as diagonal dominance. The throughflow case is easiest to explain. Its generating matrix multiplies the system input vector to produce a through-flow vector. Elements of the generating matrix represent the number of times substance introduced at one compartment will appear in another. First introduction by boundary input constitutes a first "hit" to the importing compartment. Non-importing compartments do not receive such first hits. In matrix multiplication of the generating matrix and input vector, importing compartments line up with their corresponding inputs such that first hits are recorded in diagonal positions; that is, input zi to compartment i appears in the iith position of the generating matrix. This alignment gives the diagonal dominance. Off-diagonal elements represent contributions to i from the other interior compartments, not across the boundary. These do not receive their first-hit from boundary input, but from other interior compartments, and so are correspondingly smaller in numerical value. Storage generation is similar. Elements of storage-generating matrices denote residence times in each compartment of substance derived from other compartments. Diagonal dominance in these generating matrices also associates longer residence times with boundary vs. non-boundary inputs due to the first-hit phenomenon of the throughflow model, and longer residence times result in greater standing stocks. Boundary amplification may offer explanations for many phenomena in ecosystems—edge effects, zona-tion, ecotones, trophic levels, etc. Take the latter as an example. The transfer levels of network unfolding (CH-7) were seen to be non-discrete due to the mixing around of energy matter in the complex network of indefinitely extending pathways. This negates the mainstream Lindeman (1942) conception of discrete trophic levels. Boundary amplification restores discreteness, however, by giving another argument. Solar photons represent a resource initially outside the ecosystem boundary. Green plants bring them in and thus plant life receives the first-hit advantage and ascends to planetary dominance as a discrete trophic level, the primary producers. In a concentric, onion-like construction of the ecosystem, the resultant living plant biomass represents an untapped resource lying outside the possibilities for use (a functional boundary) until cellulose-digesting animals evolve. When they do, the first-hit advantage establishes them as a second discrete level—herbivores. These organisms initially lie outside the boundary of the next level within, until flesh-eating animals can be developed to employ this resource. Their firsthit advantage produces the third trophic level, carnivores. Omnivores evolve to utilize herbivores and carnivores, and at this point the trophic-dynamic model begins to lose its discreteness. All trophic levels produce dead organic residues, and the procaryotes and eucaryotes were already in place over evolutionary time to utilize these; first-hit boundary amplification establishes them as a discrete tropic category also—decomposers. Boundary amplification is a relatively new property in environ theory. It has the potential to explain the emergence of discrete trophic levels within complex reticular networks, and of course the more general property behind this is system openness.

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