CH5 network internal amplification

It is sometimes observed in the environ mathematics of particular networks that substance introduced into one compartment at the boundary will appear more than once at another compartment, despite boundary dissipation in the interim. This is due to recycling, and it is easily seen how progressively diminishing fractions of a unit of introduced substance can cumulatively produce a sum over time in a limit process that exceeds the original amount. The second law cannot be defeated by this means, but energy cycling (Patten, 1985) following from open boundaries can compensate it and make it appear at least challenged in network organization. This is but one of numerous unexpected properties of networks contributed by cyclic interconnection and system openness.

CH-6: network unfolding

Ever since Raymond Lindeman (1942) pursued Charles Elton's original food cycles, but they came out unintendedly as sequential food chains instead, to which ecologists could better relate, mainstream empirical ecology (e.g., food-web theory, biogeochemical cycling) has had a difficult time returning to meaningful analysis of the concept of cycling. The preoccupation with chains prompted Higashi and Burns (Higashi et al., 1989) to develop a methodology for unfolding an arbitrary network into corresponding isomorphic "macrochains." Emanating from boundary points of input and arrayed pyramidally, these resemble the food pyramids of popular textbook depictions. Because the networks are cyclic, however, the macrochains differ from normal acyclic food chains in being indefinite in extent. Network unfolding refers to the indefinite proliferation of substance-transfer levels in ecosystems. The terminology "transfer pathways" and "transfer levels" is preferred to "food chains" and "trophic levels" because non-trophic as well as trophic processes are involved in any realistic ecosystem. Examples of non-trophic processes include import and export, anabolism and catabolism, egestion and excretion, diffusion and convection, sequestering, immobilization, and so on. Whipple subsequently modified the original unfolding methodology to discriminate the various trophic and non-trophic processes involved (Whipple and Patten, 1993; Whipple, 1998). The transfer levels so discriminated are non-discrete in containing contributions from most, if not all, the compartments in a system, and also they continue to increase in accordance with continuation of the limit process that ultimately dissipates all the introduced substance from the system. Exchange across open borders is at the heart of network unfolding.

CH-7: network synergism

The quantitative methodology of environ theory lends itself to development of certain qualitative aspects of the environmental relation with organisms. Energy and matter are objective quantities, but when cast as resources they engender subjective consequences of having or not having them. A concept introduced in game theory (von Neumann and Morgenstern, 1944, 1947) to describe the usefulness of outcomes or payoffs in games is utility. Environ mathematics implements this concept to bridge the gap between objective energy and matter and their subjective value as resources. Utility measures the relative value of absolute quantities; it is subjective information extracted from and added onto objective facts (Patten, 1991, 1992). A zero-sum game is one in which a winner gains exactly what the loser loses. Each conservative transaction in ecosystems is zero-sum, but it's relative benefit to the gainer and loss to loser may be different. Network synergism concerns how non-zero-sum interactions arise ultimately in conservative flow-storage networks whose proximate transactional linkages are zero-sum (Fath and Patten, 1998). Non-zero-sum interactions tend to be positive such that benefit/cost ratios, which equal one in direct transactions, tend in absolute value to exceed one when non-local indirect effects are taken into account. Such network synergism involves huge numbers of pathways (CH-1), dominant indirect effects (CH-2), and an indefinite transfer-level structure that unfolds as a limit process (CH-6)—all features of utility generation that reflect holistic organization in ecosystems, and the ecosphere. Once again—no open boundaries, no interior networks, no transactional or relational (see immediately below) interactions, and thus no non-zero-sum benefits to components. Life in networks is worth living, it can be said, because the key property of openness as a necessary condition has made possible all subsequent properties derived from it.

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