Contribution of throughflows to the ecological interpretation of integral network utility

• We symbolically calculated utility (U) and substituted throughflows where possible.
• U was throughflow-reducible for models having only simple input niches.
• Throughflow reducibility appears sufficient for topologically-determined sign(U).
• Cycles must be multiple or have more than one input for parametric determination.
• A throughflow perspective allows us to identify flows generating nonreducibility.

Ecosystems can be abstracted into models consisting of compartments containing matter or energy, transactional flows of matter or energy between compartments, inputs into the system, and outputs from the system. Although direct transactions are measurable in the field, indirect transactions have been demonstrated to have dominant effects. Integral network utility (U) is a summation of all direct and indirect net transactions in a network presented in matrix format and developed as a feature of Network Environ Analysis (NEA). While U can provide qualitative information about ecological interactions between compartments, the nonzero-sum nature of indirect net transactions has made ecological interpretation of quantitative network utility challenging. Here we aimed to examine U for nine 2- or 3-compartment ecosystem models from a throughflow perspective. For each model, we assigned inputs, outputs, and flows algebraically using flow components traceable across the model, developed corresponding flow (F) and throughflow (T) matrices based on these values, and used symbolic Matlab to calculate the net adjacent flow intensity matrix (D) and U. Substituting algebraic combinations of flow components with corresponding throughflow values allowed us to reduce elements of U to throughflows to the maximum extent possible. Models with only simple input environs were fully throughflow reducible, while models with more complex input environs exhibited one to three nonreducible elements in U. Throughflow reducibility was sufficient, but not necessary, for topological determination of ecological relations of a model, as described by sign(U). Parametrically determined elements of sign(U), along with the specific flow components influencing the sign of that element, could be readily identified based on quantitative consideration of nonreducible flow components. We provide an example showing that considering throughflow as a centrality measure can allow the identification of a quantitative basis for network synergism. By allowing identification of specific subsets of transactional flows relating to ecosystem complexity and qualitative differences between human-designed systems in the conventional industrial model and evolved ecological systems, the throughflow perspective of U opens avenues for designing more sustainable human systems.