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The Metabolic Theory of Ecology (MTE) and the theory of Dynamic Energy Budgets (DEB)
(and more)
Jaap van der Meer
Royal Netherlands Institute for Sea Research
New developments in ecology
To our minds, the last decade has seen at least two highly significant broad theoretical developments that address the core principles of ecology. The first of these has been the theory of metabolic scaling developed by G.B. West, J.H. Brown, B.J. Enquist and their colleagues. ...
Gaston and Chown 2005
New developments in ecologyCitations Author Year Journal Title
0 Savage 2006 SCIENCE Comment on "The illusion of invariant quantities in life histories"5 Martiny 2006 NATURE REVIEWS MICROBIOLOGYMicrobial biogeography: putting microorganisms on the map16 Jetz 2004 SCIENCE The scaling of animal space use24 Pimm 2004 SCIENCE Domains of diversity0 Enquist 2003 NATURE Invariant scaling relations across tree-dominated communities (vol 410, pg 655, 2001)1 Gillooly 2003 NATURE Allometry: How reliable is the biological time clock? Reply44 Enquist 2003 NATURE Scaling metabolism from organisms to ecosystems23 West 2003 NATURE Why does metabolic rate scale with body size?2 Brown 2003 SCIENCE Heat and biodiversity - Response4 Huston 2003 SCIENCE Heat and biodiversity0 Allen 2003 SCIENCE Response to comment on "Global biodiversity, biochemical kinetics, and the energetic-equivalence rule"6 West 2002 NATURE Ontogenetic growth - Modelling universality and scaling - Reply26 Enquist 2002 NATURE General patterns of taxonomic and biomass partitioning in extant and fossil plant communities1 Belgrano 2002 NATURE Ecology - Oceans under the macroscope97 Allen 2002 SCIENCE Global biodiversity, biochemical kinetics, and the energetic-equivalence rule0 Enquist 2002 SCIENCE Global allocation rules for patterns of biomass partitioning - Response74 Gillooly 2002 NATURE Effects of size and temperature on developmental time13 Enquist 2002 SCIENCE Modeling macroscopic patterns in ecology75 Enquist 2002 SCIENCE Global allocation rules for patterns of biomass partitioning in seed
117 West 2001 NATURE A general model for ontogenetic growth160 Gillooly 2001 SCIENCE Effects of size and temperature on metabolic rate67 Brown 2001 SCIENCE Complex species interactions and the dynamics of ecological systems: Long-term experiments
109 Enquist 2001 NATURE Invariant scaling relations across tree-dominated communities0 Enquist 2000 NATURE Allometric scaling of production and life-history variation in vascular plants (vol 401, pg 907, 1999)
101 Enquist 1999 NATURE Allometric scaling of production and life-history variation in vascular plants158 West 1999 NATURE A general model for the structure and allometry of plant vascular systems277 West 1999 SCIENCE The fourth dimension of life: Fractal geometry and allometric scaling of organisms6 Enquist 1999 NATURE Plant energetics and population density - Reply
179 Enquist 1998 NATURE Allometric scaling of plant energetics and population density2 Brown 1997 SCIENCE Allometric scaling laws in biology - Response
695 West 1997 SCIENCE A general model for the origin of allometric scaling laws in biology2282
An awful lot of fun
“We are making advances on a broad range of questions almost on a weekly basis,” says James Gillooly … “We’ve been having an awful lot of fun.”
“I’ve never been more excited in my life,” says Hubbell. “Ecology now is like quantum mechanics in the 1930s, we’re on the cusp of some major rearrangements and syntheses. I’m having a lot of fun.”
Whitfield 2004
1 Supply to the cells goes through a fractal-like branching structure, designed such that transport costs are minimal2 Maintenance costs of cells are constant3 Difference is available for somatic growth
The basis of MTE
West et al. 1997, 2001
g
mWaW
dt
dW
43
Fractal-like branching structureASSUMPTIONS• Capillaries do not change• Cross-area preservation• Volume preservation
Dodds et al. 2001; Kozlowski and Konarzewski 2005; Rampal et al. 2006; Chaui-Berlinck 2006
PROBLEMS• Closed branching structures are rare• Cross-area preservation would imply immediate death• Volume preservation lacks any ground • Minimisation procedure is mathematically incorrect and ill-posed
CONCLUSIONThe fractal model lacks self-consistency and correct statement
PROBLEMS• Set of assumptions are inconsistent and violate the second law of thermodynamics• Overhead costs of growth are neglected
CONCLUSIONThe only way out of this ambiguity is skipping the first assumption. This would imply that metabolic rate has an intra-specific scaling coefficient of 1 instead of 3/4
Intra-specific scaling
Makarieva 2004; Van der Meer 2006
ASSUMPTIONS• Metabolic rate equals the supply rate of energy• Metabolic rate equals the ‘metabolic rate of a single cell’ (which is assumed constant) summed over the total number of cells, where the ‘sum is over all types of tissue’ • Difference between supply and maintenance is used for growth, where the energy costs per unit of mass are set equivalent to the energy content of mammalian tissue
g
mWaW
dt
dW
43
PROBLEMS• No data for in vitro cells support the -1/4 scaling of m• MTEs (verbal) prediction that only in vivo cells have to follow the -1/4 scaling of metabolic rate (due to constraints set by the supply rate) suffers from the inconsistent definition of metabolic rate
CONCLUSIONThe fractal-like branching structure does not suffice to explain Kleiber’s law. The questionable assumption of a -1/4 scaling of m is additionally required (but nowhere mentioned in later papers).
Inter-specific scaling
Kleiber’s law
Volume-specific maintenance costs
ASSUMPTIONS• Parameters a and g are independent of ultimate body size• Parameter m has a scaling factor of -1/4 with ultimate body size: maintenance costs of a lizard are much higher than those of a baby crocodile of the same size
g
mWaW
dt
dW
43
Van der Meer 2006
From the individual to the population
A large and growing body of work has sought to explore how, through geometrical constraints on exchange surfaces and distribution networks, relationships arise between body size and metabolic rate, developmental time (Gillooly et al. 2002, Nature 417: 70-73), population growth rate (Savage et al. 2004, American Naturalist 163: 429-441), abundance and biomass (Enquist & Niklas 2001, Nature 410: 655-660),production and population energy use (Ernest et al. 2003, Ecology Letters 6: 990-995), and species diversity.
Gaston and Chown 2005
From the individual to the population: r and K
Gillooly et al. 2002, Ernest et al. 2003; Savage et al. 2004
For small sizes
… , thus development time scales with 1/4… , thus time to maturation scales with 1/4… , thus generation time scales with 1/4… , thus rmax scales with -1/4
4343
Wg
a
g
mWaW
dt
dW
log rmax
log W
From the individual to the population: r and K
Gillooly et al. 2002, Ernest et al. 2003; Savage et al. 2004
Assume that at carrying capacity KB is the same for each population, where K is equilibrium population size and B metabolic rate per individual
Since B scales with 3/4, K must scale with -3/4
log KB
log W
MTE DEB
State variables Body mass … and reservesFeeding module No Yes
WITHIN SPECIESAssimilation A 3/4 2/3Maintenance M 1 1Metabolic rate 3/4 or 1? 2/3 to 1
AMONG SPECIESAssimilation a 0 1/3Maintenance m -1/4 0Costs for growth g energy content … and overheadMetabolic rate 3/4 2/3 to 1
West et al. 1997, 2001; Kooijman 2000; Van der Meer 2006
MTE versus DEB
Log body mass Log structural body volume
(a) (b)
Log
supp
ly o
r m
a in t
e na n
ce r
a te
Slope=3/4 Slope=2/3
Slope=1 Slope=1
From the individual to the population: structured-population
models
ingestion faeces
dissipation
reproductiongrowth
assimilation
VE+ER
DEB-organism
Rate of ingestion in response to densities of a variety of available prey items and a variety of (direct) competitors?
Prey selectionMutual interference
searching rate for prey m2/s
1/ average handling time sS density of searching predators m-2
H density of handling predators m-2
P S+H m-2
D density of prey m-2
W per capita intake rate 1/s
S HD
DEB’s synthesising unit (and Holling’s type II functional
response)
HDSdt
dH
dt
dS
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
food density
ing
estio
n r
ate
a
1/h
K
/inge
stio
n ra
te W
food density D
D
DW
1
1
½
Substitutable preyE.g. one edible and one inedible
prey
F S HD
E
EDD
W
1
Interference
qPDD
W
1
Interference
Beddington’s generalized functional response
…, but competitors do not behave as inedible prey
Beddington 1975
The first ‘mechanistic’ models considered competitors as inedible prey, ...
F S H GD
S S
H
Interference
21
21411
212
1
D
DP
DP
DDW
PDDD
DW
121
1
Ruxton et al. 1992; Van der Meer & Smallegange in prep.
A stochastic version
G2
F2
S2 S1H1 H2 2
2D D
Continuous Time Markov Chain
2
F4 F2G2
6
S2G2
S2F2
S4 S3H1 H4 2
3
4DS2H2
2 3
3DS1H3
4
2D D
H2G2S1H1G2
S1H1F2 H2F2
G4
2
3
4 3
2D D
D2D
2
2
4 predators14 states
Transition rates
State Rate at whichleave
Rate at which enter Relative limitingprobability
1 S2 (2D+) P1 P2 + P4 1
2 S1H1 (D++) P2 2DP1 + 2 P3 + P5 2D/
3 H2 2 P3D P2 (D/)2
4 F2 P4 P1
/
5 G2 P5 P2 2(/)(D/)
1
kQkQ
DkW
D
A stochastic version
1055
844
633
422
2
1215432
2
8
2
4
2
2
28
121432
2
6
2
4
2
2
26
12132
2
4
2
2
26
1212
2
2
24
1212
2
1
k
k
k
k
k
k
kkkk
k
kkkk
k
kkk
k
kk
k
kk
akQ
=2,=4
0 1 2 3 4 5 6 7
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
P
y1
Number of competitors
Inta
ke r
ate
Stochastic
Deterministic
Approximation
A stochastic version
• Solitary animals
• Omnivores and cannibals
• Live in coastal water and estuaries
• Occur from Norway to Mauritania
• Size (males) Puberty at carapace width 20-30 mmReproduce when carapace width ~50 mmMaximum carapace width ~ 90 mm
• Maximum age ~10 years
50 mm
Shore crab Carcinus maenas
Behaviours
Eat
Fight
Search
Total time needed for one prey item
Interference time
Total time
Time in s ± 95% CI
Results
ML parameter estimation
1
1
111
1exp
k
kk
kk
i
iin
kkii
qx
ij
iji q
ML parameter estimators
231221
2312 ˆ
2ˆ
nn
D
yy
nnD
251421
2514 ˆˆ
nnyy
nn
322132
3221ˆ
2ˆ
nnyy
nn
524154
5241 ˆˆ
nnyy
nn
G2
F2
S2 S1H1 H2 2
2D D
1
4
32
5
S2 S1H1 H2 F2 G2 yi (s)
S2 28 48 1819
S1H1 27 30 18 1612
H2 30 732
F2 48 647
G2 18 191
Transitions and stage durations
Feeding rate in min-1 ± 95% CI
2 crabs
4 crabs
Two food patches of 0.25 m2 each
Testing predictions on the distribution of crabs
IFD hypothesisSearchers show infinitely fast movements towards the better patch
Random movementsNo preference for a patch.Only searchers move between patcheswith constant dispersal rate
Model predictions
G2
F2
S2 S1H1 H2 2
2D D
G1G1F1F1 S1S1
S1H1
H1S1
H1H1
S1S2
S1H2
H1S2
H1H2 S2S1
S2H1
H2S1
H2H1
G2G2F2F2 S2S2
S2H2
H2S2
H2H2
G1G1F1F1 S1S1
S1H1
H1S1
H1H1
S1S2
S1H2
H1S2
H1H2 S2S1
S2H1
H2S1
H2H1
G2G2F2F2 S2S2
S2H2
H2S2
H2H2
D1
0.5
0.5
D1
D1 D1
D2
0.5
0.5
D2
D2 D2
D2
D1
D1
D2
D2D1
D1
D2
12
3
4
56
1516
17
18
1920
11
12
14
13
7
8
10
9
Random movement
IFD
IFD
Random movement
crabs ate more from best patch
crabs ate more from best patch
Preliminary conclusions
New ‘generalized functional response’ model is generally applicable for foraging shore crabs
The two ‘dispersion’ models (IFD or Random movement) are not
S
finish handling
find food
find handler
H
Conflict module
discovered
win
lose
Adaptive interference competition
Ownersstrategy
Intrudersstrategy
Random
win
lose
not defend
defend
defend
not defendwin
lose
attack
not attack
timeenergy
probability of outcome j
6
1
6
1
ˆ,
ˆ,ˆ,
jjj
jjj
PP
PP
PP
T
G
TE
GEW
All individuals have a strategy set P
Many residents with strategy , few mutant individuals
Intake rate W of mutants is compared to that of the residents
P̂
Adaptive dynamics
value of food 10 J
cost of fight 1 J
fight time 2 s
handling time 1 s
probability of winning 0.5
low food density
intermediate food density
high food density
food density (#s-1)
fora
ger
dens
ity (
#s-1)
0.1 1 10
1
10
100
def
ense
str
ateg
y
attack strategy
Hawk
attack strategy
def
ense
str
ateg
y
Bourgeois
Anti-Bourgeois
00
AB(1,0) B(0,1)
H(1,1) AB(1,0) B(0,1)
H(1,1)
food density (#s-1)
fora
ger
dens
ity (
#s-1)
0.1 1 10
1
10
100
forager density (#s-1)
0.01 1 100
2
4
6
8
10
Bourgeois (0,1)Anti-Bourgeois (1,0)Hawk (1,1)
inta
ke r
ate
(#s-1
)
Summary
MTE is a failure
Structured-population models may become more general if regularities in the predation process itself (prey selection; interference behaviour; dispersal behaviour) can be found.
Adaptive dynamics may be of help in finding these regularities (listen to Tineke)