università di padova lasa – laboratorio di analisi dei sistemi ambientali 1 incofish, 14...

INCOFISH, 14 September 2006

L. PalmeriUniversità di PadovaLASA – Laboratorio di Analisi dei Sistemi Ambientali

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Metabolic scaling relations in marine ecosystems trophic networks

Luca Palmeri

Yuri Artioli

Environmental System Analysis LabDepartment of Chemical Processes Engineering

UNIVERSITA’ DI PADOVA

ITALY



2

Quo vadis ecosystem ?

or

where are you going ecosystem ?

Bendoricchio and Palmeri, 2005Ecological Modelling 184: 5–17



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Indicators and Goal Functions

S (IInd TD law) Maximum entropy equilibrium

W (Lotka) Maximum power energy dissipation

p (Prigogine) Minimal entropy production linear regime

Em (Odum) Maximum empower energy quality (solar)

Ex (Jørgensen) Maximum Exergy distance from equilibrium

AMI, NC e Asc

(Ulanowicz) Propensity to maximal Ascendency network organization

Emx (Bastianoni & Marchettini) Minimum Em/Ex cost/benefit

Each indicator gives a different point of view on systems’ state. Goal Functions

are specific (or sectorial), not “global”.



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Ecosystem description (Ecological State) Ecological Ecosystem

Network analysis

(flows and storages)

State a measurable property

System analysis

Holistic indicators from general system properties (e.g. allometries)

Jik flow originated in i and entering k

J32

1 3

2

J13

J31

J21

ki ikJJ,

Total flow (TST)



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Trophic networks



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Ecosystem optimization

Ecosystems try to optimize the flows and biomass

Optimal networks show a balance between flows and biomass (lets say between costs and benefits)



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Network optimization

COST: supply the energy

Increase the quality of energy (higher trophic levels)

Foster energy transport (network articulation)

BENEFIT: respond to energy demand

catabolism

anabolism

development

OPTIMIZATION of Stored energy (Biomass) Supply/demand of resources (metabolites, energy

flowing in the network)



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A General Metabolic Growth Model (von Bertalanffy)

anabolism = metabolism - catabolism

hmkmdt

dmG

kmFmetabolism



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Weight vs. metabolism



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Weight vs. growth



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Allometric Metabolic Scaling

Biomass (B)

Flow out, metabolism (F)

Theorem: Banavar et al. (2002)

for an optimal, balanced

and direct

D-dimensional network

BF

1

D

D



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Supply-demand balance

Cost/Benefit Optimization Supply and Demand scale isometrically

Supply rate

Demand rate

2111

ssD

D

BF

0, 111 sBr s

0, 222 sBr s

1

2

1

D

D

r

rBF



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Allometric Metabolic Scaling can be rewritten asBF

211

2

1 ssBr

rBF

For an optimal network in D dimensions, the Theorem by Banavar et al. (2002) states

1

2121

1

D

D

BFssrr

D

D



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Supply-demand balance

If D = 3

If s1 = 0 from the theorem

If s1 = 0, supply rate independent of Biomass, ´= 2/3

If s1 s2 , less energy is supplied than required, 2/3<

´<3/4

Optimal condition: s1 = s2 , ´= 3/4

If s1 s2 , more energy is supplied than required, ´> 3/4

2143 1 ss

911

2 2 D

s



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as an Indicator of Trophic Network State

For biological systems D=3

Generally: 2/3

For a system,

with B-independent supply: = 2/3

undersupplied: < 3/4

in optimal condition: = 3/4

oversupplied: > 3/4

BF



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One answer might be:

43

BF

• Unfortunately ecosystems are not always represented by direct networks

they usually show feedbacks and matter recycling

• A network with ¾ scaling could not correspond to an optimum and stable state

• In that case the system could not employ overhead supply to compensate vulnerabilities to external pressures



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According to the theoretical framework developed here, high values (greater than 0.75 or close to 1) indicate the subsistence of one or several of the following network characteristics:

1. high supply/demand ratio

2. highly undirected network

3. flows redundancies

4. enhanced recycling

5. greater system resilience to external perturbations

6. high costs of maintainance for the network



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Coversely, low values (say equal to or less than 2/3) may indicate conditions spanning from

ill-defined food web representation

to

undersupplied networks



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From the black book ofChristensen and Pauly (1993)

SDB indicator, calculated for 13 trophic networks:

values in the range 0.29 - 2.50



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Caribbean coral reef trophic network(S. Opitz)



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Lagoon of Venice SDB Indicator

N

January May August YearCa’ Roman 0,92 0,77 0,63 0,79Petta di Bo’ 0,91 0,76 0,67 0,83Sacca Sessola 0,81 0,77 0,66 0,91Fusina 0,80 0,85 0,77 0,94Palude della Rosa 0,89 0,73 0,61 0,88

F = 1,41m 0,88

0,001

0,01

0,1

1

10

100

1000

0,001 0,01 0,1 1 10 100 1000



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N


F = 2,53m 0,76

0,001

0,01

0,1

1

10

100

1000

10000

0,001 0,01 0,1 1 10 100 1000



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N


F = 2,98m 0,66

0,001

0,01

0,1

1

10

100

1000

0,001 0,01 0,1 1 10 100 1000



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N


F = 14,18m 0,86

0,001

0,01

0,1

1

10

100

1000

10000

0,001 0,01 0,1 1 10 100 1000



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Lagoon of Venice SDB Indicator (annual)

N Petta di Bo’

Sacca sessola

Ca’ Roman

Fusina Palude della Rosa

y = 12,50x0,79

0,1

1

10

100

1000

10000

0,00 0,10 10,00 1000,00

y = 13,68x0,83

0,1

1

10

100

1000

10000

0,01 0,1 1 10 100 1000

y = 15,87x0,91

0,1

1

10

100

1000

10000

0,001 0,1 10 1000

y = 16,77x0,94

0,10

1,00

10,00

100,00

1000,00

10000,00

0,01 0,10 1,00 10,00 100,00 1000,00

c y = 13,43x0,88

0,01

0,10

1,00

10,00

100,00

1000,00

10000,00

0,001 0,100 10,000 1000,000



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Lagoon of Venice SDB IndicatorCa’ Roman 0,63

August Petta di Bo’ 0,67 (decaying season) Sacca Sessola 0,66

Fusina 0,77Palude della Rosa 0,61Ca’ Roman 0,77

May Petta di Bo’ 0,76 (growing season) Sacca Sessola 0,77


January Petta di Bo’ 0,91 (dormant season) Sacca Sessola 0,81


Year Petta di Bo’ 0,83(averaged values over the year) Sacca Sessola 0,91

Fusina 0,94Palude della Rosa 0,88

From the Lagoon of Venice Ecosystems (ARTISTA study)

SDB is SENSITIVE

accounting for very little differences in the same type of shallow water ecosystems, in different seasons (Fusina is different !)



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Lagoon of Venice SDB IndicatorCa’ Roman 0,63

August Petta di Bo’ 0,67 (decaying season) Sacca Sessola 0,66


May Petta di Bo’ 0,76 (growing season) Sacca Sessola 0,77


January Petta di Bo’ 0,91 (dormant season) Sacca Sessola 0,81


Year Petta di Bo’ 0,83(averaged values over the year) Sacca Sessola 0,91

Fusina 0,94Palude della Rosa 0,88

From the Lagoon of Venice Ecosystems (ARTISTA study)

SDB reflects DYNAMICS

is able to follow the seasonal succession, i.e. all the networks (except Fusina !) present a similar pattern of variation, i.e.:

1. Oversupplied in January (pp dormant, … ready to burst)

2. Balanced during spring (G&D are at a maximum level)

3. Undersupplied in late summer (decaying season)



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Conclusions Relatively easy to apply to “arbitrarily large”

real networks, without

increasing computational demands

increasing the number of free parameters

Allometric principles provide limit intervals (thresholds) for the indicator values and very general convergence schemes

N


Generality, applicable to very different systems

Sensitivity, distinguishes similar systems



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references• Almaas, E., B. Kovàcs, et al. (2004). “Global organization of metabolic fluxes in the

bacterium Escherichia coli.” Nature 427: 839-843.

• Banavar, J. R., F. Colaiori, et al. (2001). “Scaling, Optimality, and Landscape Evolution.” Journal of Statistical Physics 104(1/2).

• Banavar, J. R., J. Damuth, et al. (2002). “Supply–demand balance and metabolic scaling.” Proceedings of the National Academy of Sciences 99(16).

• Banavar, J. R., A. Maritan, et al. (1999). “Size and form in efficient transportation networks.” Nature 399: 130-132.

• Bendoricchio, G. and Palmeri, L. (2005) “Quo vadis ecosystem?” Ecological Modelling 184: 5–17.

• Garlaschelli, D., G. Caldarelli, et al. (2003). “Universal scaling relations in food webs.” Nature 423: 165-168.

• Niklas, K. J. and B. J. Enquist (2001). “Invariant scaling relationships for interspecific plant biomass productrion rates and body size.” Proceedings of the National Academy of Sciences 98(5): 2922-2927.

• West, G. B., J. H. Brown, et al. (2001). “A general model for ontogenic growth.” Nature 413: 628-631.

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