System Overhead and Its Constitutive Terms

Path multiplicity and low level of flow organization, giving rise to overhead, can be interpreted as system inefficiency in processing material and energy, but, in case of stress and perturbations, they represent an advantage in terms of system adaptability to new threats.

The system overhead can be divided into four separate contributions, each related to a certain form of multiplicity of pathways: input from outside (overhead on imports, $j), exports to other systems (overhead on exports, $E), respirations (dissipative overhead, $D), and internal transfers (redundancy, $R):

JV=1 tiv

where tj stands for a transfer from compartment i to j; t0j depicts imports to j tin+1 and tin+2 denote, respectively, export and respiration flows from node i.

Overhead on inputs summarize the fraction of 'flow inefficiency' related to the number and magnitude distribution of flows coming from the outside of the studied system. When the number of external inputs enlarges and becomes more evenly distributed, overhead on imports increases, measuring a higher inefficiency in getting medium from the outside; nevertheless, the maintenance of an adequate portion of overhead on imports becomes essential to the system survival. in fact, a system depending only on one input would be extremely efficient (in this case overhead on imports is minimized and equal to 0, regardless of the magnitude of the flow) but too fragile, showing risks of catastrophic extinctions in case of external source collapse (with all the inputs to the system occurring via one single arrow).

Overhead on exports quantifies pathway multiplicity for medium exiting the system in a usable form. Like the overhead on inputs, it ranges from a minimum of 0, when all the matter (or energy) leaving the system is concentrated on a single node, to a maximum value achieved in case of exports evenly distributed on each single compartment. When the topology of exports and their relative importance are assigned, this overhead component tends to increase with higher amount of matter exported.

The dissipative overhead is related to the fraction of medium that is modified by internal processes (i.e., respiration in ecosystems) and exiting the system in an unusable form (flows that do not connect boxes). it increases with dissipation intensity and because of ther-modynamic and ecological constraints must always be greater than 0.

The fourth component of overhead is related to the redundancy of pathways within the system. it is a contribution to disorder (inefficiency - disorganization) because sending medium over diverse routes costs more in terms of dissipation than channeling it over few efficient pathways; nevertheless, it becomes absolutely essential to system survival whenever an unexpected perturbation occurs. Under these circumstances, redundancy reflects 'strength in reserve' from which the system can draw to adapt to the new conditions.

The lower limit for redundancy is 0 when no uncertainty is preserved by internal flow structure (this is the unlikely case of a linear chain; see Figure 3d), while the value of its upper bound depends on TST and is associated to a wholly connected topology (maximum of flow uncertainty; see Figure 3 a).

Table 2 summarizes, for 14 real ecosystems, the values of development capacity, ascendency, and overhead, with this latter split into its four constitutive components. Values are also given as percentage of the development capacity.

Additionally, one can also compute an internal development capacity (IC), considering only intercompartmental exchanges. This form finds its counterparts in other internal indices such as internal ascendency (iA) and internal redundancy (IR). Final results will be measured as percentage of the maximum upper bound (internal capacity) (see Table 3):

tijlog i=1 j=1

v i1

Internal capacity aims to define the upper limit to intercompartmental flow organization. While IR and redundancy ($r) coincide, we get a further detail estimating IA, that is, the fraction ofrigidly linked flows between system nodes with respect to the whole ascendency (A).

Table 2 Fourteen real ecosystems are listed with their values of development capacity (C), ascendency (A), and overhead

$E, $D, $R). Data are obtained from Dr. Ulanowicz' database (datall.dat) and processed with NETWRK 4.2b software (http://www.cbl.umces.edu/~ulan/). Flows are measured in mgCm~2d~1 (Charca de Maspalomas, Crystal River Creek control and delta temp., St. Marks River, and Lake Michigan), mgCm~2sum~1 (Chesapeake Bay Mesohaline, Lower, Middle, and Upper Chesapeake Bay), gCm~2yr~1 (Everglades graminoids - wet season, Florida Bay-wet season, and Ythan Estuary), mgCm~2yr~1 (Final Narragansett) and g AFDWm~2yr~1 (Mondego Estuary)

Table 2 Fourteen real ecosystems are listed with their values of development capacity (C), ascendency (A), and overhead

$E, $D, $R). Data are obtained from Dr. Ulanowicz' database (datall.dat) and processed with NETWRK 4.2b software (http://www.cbl.umces.edu/~ulan/). Flows are measured in mgCm~2d~1 (Charca de Maspalomas, Crystal River Creek control and delta temp., St. Marks River, and Lake Michigan), mgCm~2sum~1 (Chesapeake Bay Mesohaline, Lower, Middle, and Upper Chesapeake Bay), gCm~2yr~1 (Everglades graminoids - wet season, Florida Bay-wet season, and Ythan Estuary), mgCm~2yr~1 (Final Narragansett) and g AFDWm~2yr~1 (Mondego Estuary)

C

A

Charca de Maspalomas

39886000

16871 000

2 755 700

906 760

5 538700

13814000

42.30%

6.91%

2.27%

13.89%

34.63%

Chesapeake Bay Mesohaline

19655 000

8593800

1 702 300

79 705

3565200

5 714500

43.72%

8.66%

0.41%

18.14%

29.07%

Crystal River Creek (control)

70 712

28340

3205

6193

18 408

14 566

40.08%

4.53%

8.76%

26.03%

20.60%

Crystal River Creek (delta temp.)

56315

22 434

2 588

3892

15 030

12 372

39.84%

4.60%

6.91%

26.69%

21.97%

Everglades graminoids (wet season)

79572

38643

11 391

675

10181

18 682

48.56%

14.32%

0.85%

12.79%

23.48%

Florida Bay (wet season)

18540

7 004

2 064

53

2 629

6 791

37.78%

11.13%

0.29%

14.18%

36.63%

Lower Chesapeake Bay

7 713 700

2 966 500

633140

81 527

1 271 200

2 761 400

38.46%

8.21%

1.06%

16.48%

35.80%

Middle Chesapeake Bay

9328300

3872 600

634340

37 609

1 548700

3235000

41.51%

6.80%

0.40%

16.60%

34.68%

Upper Chesapeake Bay

4583 700

1 822 300

387190

15 984

791 020

1 567200

39.76%

8.45%

0.35%

17.26%

34.19%

St. Marks River

11 264

3 726

1 488

353

2 267

3432

33.08%

13.21%

3.13%

20.12%

30.47%

Lake Michigan

140690

65 649

10 409

1 814

12013

50805

46.66%

7.40%

1.29%

8.54%

36.11%

Mondego Estuary

39126

16 547

4 799

500

6 932

10 347

42.29%

12.27%

1.28%

17.72%

26.45%

Final Narragansett

20464000

7 506 700

586 940

360110

2 742100

9268300

36.68%

2.87%

1.76%

13.40%

45.29%

Ythan Estuary

23397

8663

1 845

1 363

4158

7 368

37.02%

7.89%

5.82%

17.77%

31.49%

Table 3 Internal capacity (IC), internal ascendency (IA), and internal redundancy (IR) for the 14 ecosystems extracted from Dr. Ulanowicz' database. The last two columns show the percentage of IA and IR with respect to IC

IC

IA

IR

IA (%)

IR (%)

Charca de Maspalomas

25147 000

11 333000

13814000

45.07

54.93

Chesapeake Bay mesohaline

11 584 000

5 869700

5 714500

50.67

49.33

Crystal River Creek (control)

26223

11657

14 566

44.45

55.55

Crystal River Creek (delta temp.)

21 267

8895

12 372

41.83

58.17

Everglades graminoids (wet season)

34090

15 407

18 682

45.20

54.80

Florida Bay (wet season)

11 291

4500

6 791

39.85

60.14

Lower Chesapeake Bay

4 782 000

2 020500

2 761 400

42.25

57.75

Middle Chesapeake Bay

5 867 500

2 632 500

3235 000

44.87

55.13

Upper Chesapeake Bay

2 862 600

1 295400

1 567 200

45.25

54.75

St. Marks River

5 507

2075

3432

37.68

62.32

Lake Michigan

71 540

20735

50 805

28.98

71.02

Mondego Estuary

14285

3938

10 347

27.57

72.43

Final Narragansett

13929000

4661 100

9268300

33.46

66.54

Ythan Estuary

12 805

5437

7 368

42.46

57.54

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