Metabolic scaling relations in marine
ecosystems trophic networks
Luca Palmeri
Yuri Artioli
Environmental System Analysis Lab
Department of Chemical Processes Engineering
UNIVERSITA’ DI PADOVA
ITALY
Università di Padova
LASA – Laboratorio di Analisi dei Sistemi Ambientali
L. Palmeri
1
INCOFISH, 14 September 2006
Quo vadis ecosystem ?
or
where are you going ecosystem ?
Bendoricchio and Palmeri, 2005
Ecological 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
 Emx
(Ulanowicz) Propensity to maximal Ascendency
 network organization
(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
J13
Network analysis
1
(flows and storages)
J31
J21
Total flow (TST)
J  i ,k J ik
3
2
J32
 Jik flow originated in i and entering k
 State  a measurable property
System analysis
Holistic indicators from general system properties (e.g. allometries)
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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
dm
G
 km  hm 
dt


metabolism  F  km
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Weight vs. metabolism
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Weight vs. growth
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Allometric Metabolic Scaling
 Biomass (B)

FB
 Flow out, metabolism (F)
 Theorem: Banavar et al. (2002)
for an optimal, balanced
and direct
D-dimensional network
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D

D 1
L. Palmeri
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Supply-demand balance
Cost/Benefit Optimization  Supply and Demand
scale isometrically
 Supply rate
r1  B s1 ,
s1  0
 Demand rate
r2  B s2 ,
s2  0
FB
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 r1 
F   B 
 r2 
D
D 1
 D


1

s

s

1
2 
 D 1


L. Palmeri
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INCOFISH, 14 September 2006
Allometric Metabolic Scaling

F  B can be rewritten as

 r1 
 1 s1  s2 


F  B   B
 r2 
 For an optimal network in D dimensions, the Theorem by
Banavar et al. (2002) states
D

D


D 1

F

B
D 1

r1  r2  s1  s2 
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Supply-demand balance
If D = 3 
   34 1  s1  s2 
If s1 = 0 from the theorem s2

1
D2
 19
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
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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
FB
undersupplied:  < 3/4
in optimal condition:  = 3/4
oversupplied:  > 3/4
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INCOFISH, 14 September 2006
Quo vadis ecosystem ?
One answer might be:
FB
3
4
• 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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Quo vadis ecosystem ?
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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Quo vadis ecosystem ?
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 of
Christensen 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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SDB Indicator
Lagoon of Venice
January
0,92
0,91
0,81
0,80
0,89
Ca’ Roman
Petta di Bo’
Sacca Sessola
Fusina
Palude della Rosa
May
0,77
0,76
0,77
0,85
0,73
August
0,63
0,67
0,66
0,77
0,61
Year
0,79
0,83
0,91
0,94
0,88
1000
F = 1,41m 0,88
100
N
10
1
0,1
0,01
0,001
0,001
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0,01
0,1
1
10
100
1000
L. Palmeri
22
INCOFISH, 14 September 2006
SDB Indicator
Lagoon of Venice
Ca’ Roman
Petta di Bo’
Sacca Sessola
Fusina
Palude della Rosa
January
0,92
0,91
0,81
0,80
0,89
May
0,77
0,76
0,77
0,85
0,73
August
0,63
0,67
0,66
0,77
0,61
Year
0,79
0,83
0,91
0,94
0,88
10000
F = 2,53m 0,76
1000
N
100
10
1
0,1
0,01
0,001
0,001
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0,01
0,1
1
10
100
1000
L. Palmeri
23
INCOFISH, 14 September 2006
SDB Indicator
Lagoon of Venice
January
0,92
0,91
0,81
0,80
0,89
Ca’ Roman
Petta di Bo’
Sacca Sessola
Fusina
Palude della Rosa
May
0,77
0,76
0,77
0,85
0,73
August
0,63
0,67
0,66
0,77
0,61
Year
0,79
0,83
0,91
0,94
0,88
1000
F = 2,98m 0,66
100
N
10
1
0,1
0,01
0,001
0,001
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0,01
0,1
1
10
100
1000
L. Palmeri
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INCOFISH, 14 September 2006
SDB Indicator
Lagoon of Venice
January
0,92
0,91
0,81
0,80
0,89
Ca’ Roman
Petta di Bo’
Sacca Sessola
Fusina
Palude della Rosa
10000
May
0,77
0,76
0,77
0,85
0,73
August
0,63
0,67
0,66
0,77
0,61
Year
0,79
0,83
0,91
0,94
0,88
F = 14,18m 0,86
1000
N
100
10
1
0,1
0,01
0,001
0,001
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0,01
0,1
1
10
100
1000
L. Palmeri
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INCOFISH, 14 September 2006
Lagoon of Venice SDB Indicator (annual)
10000,00
10000,00
10000
1000,00
1000,00
1000
100,00
100,00
c
10,00
0,01
1,00
0,10
1,00
0,10
10,00
y = 16,77x0,94
1,00
0,100
0,10
0,001
10,00 100,00 1000,0
0
100
0,88
y = 13,43x
10,000
10
1
1000,000
0,001
0,01
Fusina
Petta di Bo’
10000
1000
1000
100
100
0,1
0,1
10
1000
Ca’ Roman
10
y = 13,68x0,83
y = 12,50x0,79
1
1
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0,1
10000
10
0,01
0,1
Sacca sessola
Palude della Rosa
N
y = 15,87x0,91
1
10
100
1000
0,00
0,10
0,1
10,00
1000,00
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INCOFISH, 14 September 2006
SDB Indicator
Lagoon of Venice
From the Lagoon of Venice Ecosystems (ARTISTA study)
Ca’ Roman
August
Petta di Bo’
(decaying season)
Sacca Sessola
Fusina
Palude della Rosa
Ca’ Roman
Petta di Bo’
May
(growing season)
Sacca Sessola
Fusina
Palude della Rosa
Ca’ Roman
Petta di Bo’
January
(dormant season)
Sacca Sessola
Fusina
Palude della Rosa
Ca’ Roman
Petta di Bo’
Year
(averaged values over the year)
Sacca Sessola
Fusina
Palude della Rosa
 SDB
0,63
0,67
0,66
0,77
0,61
0,77
0,76
0,77
0,85
0,73
0,92
0,91
0,81
0,80
0,89
0,79
0,83
0,91
0,94
0,88
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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INCOFISH, 14 September 2006
SDB Indicator
Lagoon of Venice
From the Lagoon of Venice Ecosystems (ARTISTA study)
Ca’ Roman
August
Petta di Bo’
(decaying season)
Sacca Sessola
Fusina
Palude della Rosa
Ca’ Roman
Petta di Bo’
May
(growing season)
Sacca Sessola
Fusina
Palude della Rosa
Ca’ Roman
Petta di Bo’
January
(dormant season)
Sacca Sessola
Fusina
Palude della Rosa
Ca’ Roman
Petta di Bo’
Year
(averaged values over the year)
Sacca Sessola
Fusina
Palude della Rosa
 SDB
0,63
0,67
0,66
0,77
0,61
0,77
0,76
0,77
0,85
0,73
0,92
0,91
0,81
0,80
0,89
0,79
0,83
0,91
0,94
0,88
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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SDB Indicator
Lagoon of Venice
Conclusions

Relatively easy to apply to “arbitrarily large”
real networks, without
 increasing computational demands
 increasing the number of free parameters
N

Allometric principles provide limit intervals
(thresholds) for the indicator values and very
general convergence schemes
 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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