## Calculating UEVs from Network Flow Data

For each component in a system (Figure 5 a), five flows describe bilateral transactions (e.g., consumption, gross production, net production/transfer, respiration, and egestion) 1 - Total input

2 - Gross production

3 - Respiration

4 - Transfer (net production)

5 - Unassimilated use (egestion)

1 - Total input

2 - Gross production

3 - Respiration

4 - Transfer (net production)

5 - Unassimilated use (egestion)

To From

Solar emergy Comp. 1 Comp. 2

Comp. n

C2 to Cn

Comp. n

Constraints

Objective

S Constraints = 0

Figure 5 Component diagram and schematic of input/output matrix. (a) Diagram of bilateral and internal energy pathways (flows) compiled for each compartment within a network. (b) Schematic of input/output matrix for calculating UEVs from network flow data. UEVs in the top row represent unknowns in the simultaneous equations defined by the constraints. For fully specified systems (i.e., no. of equations = no. of unknown UEVs), the objective function is redundant (P,- = total component production minus respiration; C,- to Cj = energy transfers from component i to component j).

Cn to C2

n with other components in the system. Energy or material inputs (1) arrive from exogenous sources (e.g., sunlight, hydrologie inputs) or from components within the network (e.g., plant biomass supporting herbivores). Gross production (2) quantifies the portion of that energy that is assimilated, while egestion (5), though required for production and partially processed during digestion, is not incorporated. Respiration (3) represents the metabolic work of each compartment (i.e., internal feedbacks to secure energy), while transfer (4) is the energy that is eventually used by other components in the food web. The 'heat sink' symbol represents energy unavailable to do work (i.e., entropy).

The following rules are applied to network evaluation:

1. The UEV of a compartment is the incoming emergy (sej) flows driving biotic production (Flow 1, Figure 5a) divided by the output (energy or mass) trophic transfer for that compartment (Flow 4 in Figure 5 a).

2. Where flows converge to abiotic components such as the detrital pool, the UEV of the incoming flow is adjusted to that of the detrital pool storage (this is necessary to avoid impact of high emergy values of fecal/senescent biomass contributions to detritus from high-quality components).

A linear optimization technique that manipulates a set of unknowns (UEVs) to meet a set of constraints (emergy inflow = emergy outflow) is used to compute UEVs from network flow data. An example of an optimization table is given in Figure 5b where each row and column represents a system component. The constraints to the right of each row are the constraints that emergy inflow and emergy outflow are equal. Specifically, if the flows are in energy, then the energy inputs multiplied by appropriate transformities (unknowns in this case) equal the net production or transfer of energy multiplied by its transformity:

Emergylnflowy = ^ Xy * r = ^ X]n * r} i i = EmergyOutflowy 

where Xiy is the energy transfer from component i to component y, and r j is the transformity value of respective flows.

The aggregated ecosystem diagram of Silver Springs in Figure 6 shows inflows of solar emergy and kinetic energy of spring flow (expressed in solar emergy) being Silver Springs ecosystem (b) Solution matrix for Silver Springs transformities using the emergy network analysis method Transformity

7172

1137

2129

6324

55 398

1 708 000 17 193 869

7172

1137

2129

6324

55 398

1 708 000 17 193 869

 From: Solar Solar Spring kinetic Albedo GPP NPP Detritus Herbivores Carnivores carnivores To: Solar 0 0 0 0 0 0 0 0 0 0 Hydrology 1.04E+09 0 -1.45E+05 0 0 0 0 0 0 0 Albedo 0 1.95E+07 0 -4.71 E+06 0 0 0 0 0 0 GPP 0 0 3.35E+04 4.71E+06 -2.39E+05 0 1000 100 0 0 NPP 0 0 1.17E+04 0 2.39E+05 -1.67E+05 0 0 0 0 Detritus 0 0 0 0 0 5.78E+04 -3.50E+04 1000 10 1 Herbivores 0 0 0 0 0 1.01E+05 0 -4.19E+03 10 0 Carnivores 0 0 0 0 0 0 0 4.19E+03 -1.51E+02 1.5 Top carnivores 0 0 0 0 0 0 0 0 1.51E+02 -1.50E+01

Figure 6 Highly aggregated systems diagram of Silver Springs, FL, showing primary production and three levels of consumed organisms with flows of energy between compartments (a). The systems diagram is translated into matrix format for calculation of UEVs (transformities in top row) using a linear optimization technique.

transformed by plants into gross production and NPP. NPP is split between an autotrophic pathway and the detrital pool. Organisms form a food chain where top carnivores feed from herbivores and carnivores. Cybernetic feedbacks are included.

Using the data in Figure 6 and the linear optimization technique, the solar transformities for the Silver Springs system given in Table 5 were determined. Since the ecosystem was aggregated to only five compartments, the transformities represent averages for each trophic level.

Solar transformities for a more complex network, the Florida Everglades graminoid marsh (Table 6), were evaluated using the same technique. Network data, consisting of carbon flows (gCyr-1), were compiled from published data. There were 66 ecosystem compartments in the gra-minoid marsh, some of which (ecosystem pools of labile and refractory detritus) were not living. The primary production compartments were partitioned into root and leaf compartments. Many of the lower-trophic-level compartments represent aggregations of species (due to lack of data); for example, mesoinvertebrates, macroinvertebrates, centrarchid fish, snakes, and passerine birds are lumped categories for the marsh system. 