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Dynamic Energy Budget theory

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1 Basic Concepts

2 Standard DEB model

3 Metabolism

4 Univariate DEB models

5 Multivariate DEB models

6 Effects of compounds

7 Extensions of DEB models

8 Co-variation of par values

9 Living together

10 Evolution

11 Evaluation

seeds of heather Calluna vulgaris can germinate after 100 year

Crocodylus johnstoni,

Data from Whitehead 1987

embryo

yolk

O2 consumption, ml/h

weight, g

time, d

time, d

Carettochelys insculpta

Data from Web et al 1986

embryo

yolk

O2 consumption, ml/h

weight, g

time, d

time, d

Sphenodon punctatus (tuatara)

Adult: 45-60 cm, Wm = 0.5 – 1 kg, ♂ larger than ♀

10 eggs/litter, life span 60 - >100 a

Body temp 20-25 °C, ap = 20 a, Wb = 4 g, ab = 450 d.

Salmo trutta

Data from Gray 1926

yolk

embryo

weight, g

time, d

ml O2 d-1

ml CO2 d-1

altricial

Troglodytes aëdon

precocial

Gallus domesticus

age, d

age, d

- Observations: just prior to hatching
- respiration shows a plateau in precocial, not in altricial birds
- pore size and frequency in egg shell is such that O2 flux has constant resistance
- Conclusion: ontogeny is constrained by diffusion limitation in precocial birds (Rahn et al 1990)
- DEB theory: reserve dynamics controls ontogeny (same pattern in species without shells)
- Minimization of water loss causes observed constant flux resistance

scaled length at birth

scaled age at birth

scaled res density at birth

scaled res density at birth

scaled initial reserve

scaled res density at birth

scaled maturity

1

scaled reserve

0.8

0.5

scaled age

scaled age

scaled struct volume

scaled age

Foetusses develop like eggs,

but rate not restricted by reserve

(because supply during development)

Reserve of embryo “added” at birth

Initiation of development

can be delayed by implantation egg cell

Nutritional condition of mother only

affects foetus in extreme situations

weight, g

Mus

musculus

time, d

Data: MacDowell et al 1927

- The routine calculates the initial scaled reserve mass UE0 = ME0/ {JEAm}.
- The constraint [UEb] = f [UEm] applies.
- Inputs:
- n-vector with scaled functional response
- 5-vector with parameters
- VHb, d.mm^2, scaled maturity at birth: M_H^b/ ((1 - kap) {J_EAm}) with kap is fraction allocated to soma
- g, -, energy investment ratio
- kJ, 1/d, maturity maintenance rate coefficient
- kM, 1/d, somatic maintenance rate coefficient
- v, mm/d, energy conductance

- optional scalar or n-vector with initial estimates for Lb
- Outputs:
- n-vector with initial scaled reserve: M_E^0/ {J_EAm}
- n-vector with length at birth Lb
- n-vector with indicators for success (1) or failure (0)
- Example of use (for Daphnia magna at 20 C):
- p_Dm = [.8 .42 1.7 1.7 3.24 .012]; initial_scaled_reserve(1,p_Dm).

Kooijman 2009

J Math Biol58: 377-394

- Obtains scaled length at birth, given the scaled reserve density at birth.
- A Newton Raphson scheme is used with Euler integration, starting from an optional initial value.
- The default initial value is the exact one for maintenance ratio 1.
- Consider the application of get_lb_foetus for an alternative initial value.
- Comparable functions:
- get_lb1 uses a Newton Raphson scheme with advanced integration (but is rather slow),
- get_lb2 uses a shooting method (in one variable; and is faster than get_lb1).
- Inputs
- 3-vector with parameters
- g: energy investment ratio
- k: maintenance ratio kJ/ kM
- vHb: scaled maturity at birth UHb g2 kM3/ ((1 - kap) v2) with kap: fraction of mobilised reserve allocated to soma

- optional scalar with scaled reserve density at birth (default 1)
- optional scalar with initial value for scaled length at birth
- Outputs
- scalar with scaled length at birth: lb = Lb/ Lm
- indicator for success (1) or failure (0)
- An example of use is given in mydata_ue0

Kooijman at al 2008

Biol Rev83: 533-525

- Obtains scaled age at birth, given the scaled reserve density at birth.
- Multiply the result with the somatic maintenance rate coefficient to arrive at age at birth.
- Inputs
- 1- (if third input is specified) or 3 -vector with parameters
- g: energy investment ratio
- k: maintenance ratio kJ/ kM
- vHb: scaled maturity at birth UHb g2 kM3/ ((1 - kap) v2) with kap: fraction of mobilised reserve allocated to soma

- optional scalar with scaled reserve density at birth (default 1)
- optional scalar with scaled length at birth.
- Default calls get_lb but then the first input should have 3 rather than 1 elements.
- Output
- scalar with scaled age at birth: taub = ab kM
- An example of use is given in mydata_ue0

Kooijman at al 2008

Biol Rev83: 533-525

Obtains the scaled length at birth of a foetus, which is not restricted by reserve availability.

Inputs

1 or 3-vector with energy investment ratio g, see get_tb_foetus

optional scalar with scaled age at birth.

Default calls get_tb_foetus but then the input parameter should have 3 elements.

Output

scalar with scaled length at birth: lb = Lb/Lm

An example of use is given in mydata_ue0_foetus

Kooijman at al 2008

Biol Rev83: 533-525

- Obtains scaled age at birth, given the scaled reserve density at birth.
- Multiply the result with the somatic maintenance rate coefficient to arrive at age at birth.
- Inputs
- 3-vector with parameters
- g: energy investment ratio
- k: maintenance ratio kJ/ kM
- vHb: scaled maturity at birth UHb g2 kM3/ ((1 - kap) v2) with kap: fraction of mobilised reserve allocated to soma

- optional scalar with initial value for scaled age at birth.
- Default exact value for maintenance ratio 1.
- Output
- scalar with scaled age at birth: taub = ab kM.
- indicator for succes (1) of failure (0).
- An example of use is given in mydata_ue0_foetus

Kooijman at al 2008

Biol Rev83: 533-525

Examples of vegetative propagation

in mosses (Bryophytes)

From:

Probst, W. 1987 Biologie der Moos- und Farnpflanzen, UTB, Wiesbaden

Examples of vegetative propagation

in ferns (Filicatae)

From:

Probst, W. 1987 Biologie der Moos- und Farnpflanzen, UTB, Wiesbaden

103 eggs

103 eggs

Rana esculenta

Data Günther, 1990

Gobius paganellus

Data Miller, 1961

length, mm

length, mm

Male mammals produce

sperm cells during their

whole adult life,

but

Female mammals produce

new egg cells during their

late foetal period only.

These egg cells still grow

during a much longer period.

From: Mader, S. S.

1993 Biology, WCB

Oikopleura dioica

Oikopleura labradoriensis

- primary parameters determine
- food uptake
- changes of state variables (reserve, maturity, structure)
- compound parameters: functions of primary parameters
- composition parameters
- food, reserve, structure, products (feaces, N-waste)
- thermodynamic parameters
- free energies (chemical potentials)
- entropies
- dissipating heat

time-length-energy

time-length-mass

Maturity: information, not mass or energy

quantified as cumulated mass of reserve that is invested

Scale reserve & maturity

- Given primary parameters:
- get composition parameters
- get mass fluxes (respiration)
- get entropies, free energies

Optimality of life history parameters?

Standard DEB model (isomorph, 1 reserve, 1 structure)

reserve & maturity: hidden variables

measured

for 2 food levels

primaryparameters

Functions get_pars_* obtain compound DEB parameters from easy-to-observe quantities

and the functions iget_pars_* do the reverse, which can be used for checking.

The routines are organized as follows:

get_pars iget_pars

food level one several one several

Constraint kJ = kM kJ != kM kJ = kM kJ = kM kJ != kM kJ = kM

growth get_pars_gget_pars_hget_pars_iiget_pars_giget_pars_higet_pars_i

growth & reprod get_pars_rget_pars_sget_pars_tiget_pars_riget_pars_siget_pars_t

Functions for several food levels do not use age at birth data.

If one food level is available, we have to make use of the assumption

of stage transitions at fixed amounts of structure (k_M = k_J).

If several food levels are available, we no longer need to make this assumption,

but it does simplify matters considerably.

Functions elas_pars_g and elas_pars_r give elasticity coefficients.

Function get_pars_u converts compound parameters

into unscaled primary parameters at abundant food.

Kooijman at al 2008

Biol Rev83: 533-525

g

r

h

get_pars_

s

u

g

r

iget_pars_

h

s

red quantities depend on food level, green do not

Kooijman at al 2008

Biol Rev83: 533-525

weight, g

time since birth, d

Data by Bart Laarhoven

- State variables: structural body mass & reserve & maturity
- structure reserve do not change in composition; maturity is information
- Food is converted into faeces
- Assimilates derived from food are added to reserves,
- which fuel all other metabolic processes
- Three categories of processes:
- Assimilation: synthesis of (embryonic) reserves
- Dissipation: no synthesis of biomass
- Growth: synthesis of structural body mass
- Product formation: included in these processes (overheads)
- Basic life stage patterns
- dividers (correspond with juvenile stage)
- reproducers
- embryo (no feeding
- initial structural body mass is negligibly small
- initial amount of reserves is substantial)
- juvenile (feeding, but no reproduction)
- adult (feeding & male/female reproduction)

- Reserve density hatchling = mother at egg formation
- foetuses: embryos unrestricted by energy reserves
- Stage transitions: cumulated investment in maturation > threshold
- embryo juvenile initiates feeding
- juvenile adult initiates reproduction & ceases maturation
- Somatic maintenance structure volume & maturity maintenance maturity
- (but some somatic maintenance costs surface area)
- maturity maintenance does not increase
- after a given cumulated investment in maturation
- Feeding rate surface area; fixed food handling time
- Body mass does not change at steady state
- Fixed fraction of mobilised reserve is spent on
- somatic maintenance + growth (-rule)
- Starving individuals: priority to somatic maintenance
- do not change reserve dynamics; continue maturation, reprod.
- or change reserve dynamics; cease maturation, reprod.; do or do not shrink in structure

All powers are

cubic polynomials in l

all quantities scaled dimensionless

length l, survival S

reserve density, e

maturity, vH

time,

time,

time,

cum. feeding, reprod.

acceleration, q

hazards, h, hH

time,

time,

time,

1 Basic Concepts

2 Standard DEB model

3 Metabolism

4 Univariate DEB models

5 Multivariate DEB models

6 Effects of compounds

7 Extensions of DEB models

8 Co-variation of par values

9 Living together

10 Evolution

11 Evaluation