Dynamic Energy Budget theory

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 together10 Evolution11 Evaluation

Diapauze 2.6.2c

seeds of heather Calluna vulgaris can germinate after 100 year

Embryonic development 2.6.2d

time, d time, d

wei

ght,

g

O2 c

onsu

mpt

ion,

ml/h

Crocodylus johnstoni,Data from Whitehead 1987

yolk

embryo

Embryonic development 2.6.2e

time, d time, dwei

ght,

g

O2 c

onsu

mpt

ion,

ml/h

Carettochelys insculptaData from Web et al 1986

yolk

embryo

High age at birth 2.6.2f

Sphenodon punctatus (tuatara)Adult: 45-60 cm, Wm = 0.5 – 1 kg, ♂ larger than ♀10 eggs/litter, life span 60 - >100 aBody temp 20-25 °C, ap = 20 a, Wb = 4 g, ab = 450 d.

Embryonic development 2.6.2g

time, d

wei

ght,

g

Salmo truttaData from Gray 1926

yolkembryo

Respiration ontogeny in birds 2.6.2h

age, d age, d

ml C

O2

d-1

ml O

2 d-1

altricialTroglodytes aëdon

precocialGallus domesticus

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 res density at birth

scaled res density at birth

scaled res density at birth

scal

ed le

ngth

at b

irth

scal

ed in

itial

res

erve

scal

ed a

ge a

t birt

h

Effects of nutrition 2.6.2i

Reduction of initial reserve 2.6.2j

1

0.8

0.5scaled age

scaled age

scaled age

scal

ed m

atur

itysc

aled

str

uct v

olum

e

scal

ed r

eser

ve

Foetal development 2.6.2kw

eigh

t, g

time, d

Mus musculus

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 cellNutritional condition of mother only affects foetus in extreme situations

Data: MacDowell et al 1927

DEBtool/animal/initial_scaled_reserve 2.6.2l

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 2009J Math Biol 58: 377-394

DEBtool/animal/get_lb 2.6.2m

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 2008Biol Rev 83: 533-525

DEBtool/animal/get_tb 2.6.2n

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 2008Biol Rev 83: 533-525

DEBtool/animal/get_lb_foetus 2.6.2o

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 2008Biol Rev 83: 533-525

DEBtool/animal/get_tb_foetus 2.6.2p

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 2008Biol Rev 83: 533-525

Reproduction 2.7

Vegetative propagation 2.7a

Examples of vegetative propagation in mosses (Bryophytes)

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

Vegetative propagation 2.7b

Examples of vegetative propagation in ferns (Filicatae)

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

Reproduction at constant food 2.7c

length, mm length, mm

103

eggs

103

eggs

Gobius paganellusData Miller, 1961

Rana esculentaData Günther, 1990

Gametes production 2.7d

From: Mader, S. S. 1993 Biology, WCB

Male mammals producesperm cells during theirwhole adult life,

but

Female mammals producenew egg cells during their late foetal period only.

These egg cells still growduring a much longer period.

Appendicularia 2.7.1

Oikopleura labradoriensis

Oikopleura dioica

DEB parameters 2.8

• 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

Primary DEB parameters 2.8a

time-length-energy time-length-mass

Reserve & maturity: hidden 2.8b

Maturity: information, not mass or energy quantified as cumulated mass of reserve that is invested

Scale reserve & maturity

Primary thermodynamic pars 2.8c

Given primary parameters:

• get composition parameters• get mass fluxes (respiration)• get entropies, free energies

One-sample case 2.8d

Two-sample case: D. magna 20°C 2.8e

Optimality of life history parameters?

measured quantities primary pars 2.8f

Standard DEB model (isomorph, 1 reserve, 1 structure)reserve & maturity: hidden variables

measuredfor 2 food levels primary parameters

DEBtool/animal/get_pars 2.8g

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_parsfood level one several one severalConstraint kJ = kM kJ != kM kJ = kM kJ = kM kJ != kM kJ = kMgrowth get_pars_g get_pars_h get_pars_i iget_pars_g iget_pars_h iget_pars_i growth & reprod get_pars_r get_pars_s get_pars_t iget_pars_r iget_pars_s iget_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 2008Biol Rev 83: 533-525

DEBtool/animal/get_pars 2.8h

g

get_

pars

_ig

et_p

ars_

r

s

h

u

s

h

r

g

red quantities depend on food level, green do not Kooijman at al 2008Biol Rev 83: 533-525

Add_my_pet: Phyton_regius 2.8i

wei

ght,

g

time since birth, d

Data by Bart Laarhoven

General assumptions 2.9

• 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)

Specific assumptions 2.9a

• 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

1E,1V isomorph 2.9b

All powers are cubic polynomials in l

1E,1V isomorph 2.9c

all quantities scaled dimensionless

1E,1V isomorph 2.9d

time, time, time,

time, time, time,

rese

rve

dens

ity,

e

leng

th l,

sur

viva

l S

mat

urit

y, v

H

acce

lera

tion

, q

haza

rds,

h, h

H

cum

. fee

ding

, rep

rod.

Dynamic Energy Budget theory

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 together10 Evolution11 Evaluation