Ecosystems are a type of dissipative system in which direct transactions (flows of energy or matter) are otherwise conservative. Direct net transactions in ecosystems are thus zero-sum, while indirect net transactions may be nonzero-sum due to dissipative losses and cycling of energy or matter. A key unresolved feature of Network Environ Analysis (NEA) is the nature of network utility, a summation of all direct and indirect net transactions in a network presented in matrix format. Qualitative interpretation of network utility is relatively tractable and has been described in terms of quantitative benefits (+) and costs (-), with ecological interaction types of the mth order described, for example, as predation (+, –), competition (–, –), or mutualism (+, +). However, the nonzero-sum nature of indirect net transactions has made ecological interpretation of quantitative network utility challenging. Here our objective was to investigate the role of compartment throughflows in determining network utility. We developed nine ecosystem models composed of two or three compartments and assigned elements in each resulting system flow matrix as variables. After algebraically calculating the utility matrix for each model, we reduced elements of the utility matrices to throughflows to the greatest extent possible using symbolic Matlab.
Results/Conclusions
In certain types of ecosystem models, utility matrices contained only elements that could be reduced entirely to algebraic combinations of throughflows. Other models required the inclusion of at least one individual, transactional flow component in the utility matrix. In particular, networks that contained cycles or had multiple inputs to at least one compartment gave rise to the latter type. This model type is also more similar to real-world ecosystems, which tend to contain a much larger number of compartments and encompass multiple cycles. Examining utility from the throughflow perspective helps to reveal the non-reducible nature of complex systems and can reveal which transactional flows are particularly important in a given system. Considering throughflow thereby facilitates an ecological interpretation of the basis for and meaning of the quantitative benefits and costs expressed within utility matrices as well as how these quantitative values translate into qualitative ecological interactions.