xt thAt

Figures 10a and 10b also demonstrate how intensity (of anything) is injected into a system as a source term and then allocated by internal flows. For energy intensity, the source is the embodied energy in imports of similar entities (competitive imports in economic terminology) plus imports of different entities, here just energy itself.

Gloof embodied in incoming in-system flows during time step At

Gloof embodied in stock at time t

Gloof embodied in output flow during time step At

Gloof embodied in stock at time t + At

Gloof generated internally plus that embodied in import flow during time step At

Figure 9 Generic scheme for allocating the embodied generic system input gloof for the dynamic system. In the underlying dynamics, stock changes over time as Si+Ai = St + (OUTPUT^- P INPUTS^)Af.

=in-sysiem inputs

X et+At XJ+At At i = in-system inputs et+At X}+At At c-t + At Q t

t + At X t + At i = in-system inputs t + At X t + At i = in-system inputs

T t+At I X Xjt+At At + IMPORT, + El j \ i = in-system inputs

T t+At I X Xjt+At At + IMPORT, + El j \ i = in-system inputs

Figure 10 (a) Scheme for calculating dynamic energy intensity. The source term is energy itself plus energy embodied in imports. The energy intensity of imports of type j, £imR/, is specified exogenously. (b) Scheme for calculating dynamic residence time. The source term is the aging of the existing stock in the time period At.

For residence time, the source term is just the aging of the stock; external inputs do not contribute to residence time by definition. Similar comments apply to TP and PL.

Simulations of Dynamic Indicators

Figure 11 shows a two-compartment dynamic model system. Initially the system is at steady state with no feedback (Z = 0), but feedback is switched on (Z = 3) at time = 20 days, and then off again (Z = 0) at 500 days. The details of the underlying model are not important here; it incorporates a nonlinear ratio-dependent feeding response by consumers to abundance of producers, and vice versa when feedback is on. Producer output depends on light level, which is assumed constant, and producer biomass. Simulation is performed using the modeling software STELLA.

Figures 12a-12d show dynamic behavior of four of the indicators calculated here: energy intensity, TP, PL, and residence time. On all graphs, the stock of producers and consumers is shown as well. Immediately after the onset of nonzero feedback, producer stock increases as more material now enters that compartment and consumer stock drops. But then consumer stock increases in response to increased producer stock, and both stocks asymptotically increase. This is reasonable, because along with increased feedback comes decreased loss, as shown in Figure 11 .

Similar to the steady-state calculations, energy intensity, TP, PL, and residence time all increase for both producers and consumers. However, the values are not given exactly by the static equations for Z = 3cald_1.

Figure 11 Model for dynamic simulation. Figures in square brackets are initial steady-state values, before feedback is started. Figures within boxes are stocks; others are flows.

o 100






- /■"


Prod. en. intensity

30 n

0 200 400 600 800 1000 Time (day)

30 n

0 200 400 600 800 1000 Time (day)

400 600 Time (day)

400 600 Time (day)

o 100


Cons. stock


Prod. stock

Prod. path length


0.8 1000


~D t









Cons. stock Prod. stock

Cons. res. time

Prod. res. time

200 400 600 Time (day)


Figure 12 Indicators in dynamic system. Initially system is at steady state with feedback (Z) = 0. Z is increased abruptly to a steady value of 3 cal/day for days 20-500, and then returned abruptly to zero. (a) energy intensity; (b) trophic position; (c) path length; (d) residence time.

This is because in the dynamic model, all flows and stocks change when feedback changes, while in the static model used in the previous two sections all flows except feedback are assumed to remain constant.

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