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 ecologyTo 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

gmWaW

dtdW

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

gmWaW

dtdW

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

gmWaW

dtdW

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

Wga

gmWaW

dtdW

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 ma i

nten

anc e

rate

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/s1/ 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

dHdtdS

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

food density

inge

stio

n ra

te

a

1/h

K

/inge

stio

n ra

te W

food density D

D

DW

11

½

Substitutable preyE.g. one edible and one inedible

prey

F S HD

E

EDDW

1

Interference

qPDDW

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

21411212

1

DDP

DPDDW

PDDD

DW

1211

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

kQkQDkW

D

A stochastic version

1055

844

633

422

2

1215432

28

24

22

28

121432

26

24

22

26

12132

24

22

26

1212

22

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 ra

te

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

kki

iin

kkii

qx

ij

iji q

ML parameter estimators

231221

2312 ˆ2

ˆnn

Dyy

nnD

251421

2514 ˆˆnnyy

nn

322132

3221ˆ

2ˆ

nnyynn

524154

5241 ˆˆnnyy

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

137

8

10

9

Random movement

IFD

IFD

Random movement

crabs ate more from best patch

crabs ate more from best patch

Preliminary conclusionsNew ‘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

PPPP

T

G

TEGE

W

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 d

ensi

ty (#

s-1)

0.1 1 10

1

10

100

defe

nse

stra

tegy

attack strategy

Hawk

attack strategy

defe

nse

stra

tegy

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 d

ensi

ty (#

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 ra

te (#

s-1)

SummaryMTE 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)