Dynamic Energy
Budget Theory - I

with contributions from :
Tânia Sousa
Bas Kooijman
A DEB organism: growth

Feeding
J XA
ME - Reserve
Mobilisation
J EA
J EC
Assimilation
Offspring
MER
J EJ
J ER
J ES
Maturity
Maintenance
Reproduction
Maturation
MH - Maturity
J EG
Somatic
Maintenance
Growth
MV - Structure
 Metabolism in a DEB
individual.
 Rectangles are state
variables
 Arrows are flows of food
JXA, reserve JEA, JEC, JEM, JET ,
JEG, JER, JEJ or structure JVG.
 Circles are processes
 The full square is a fixed
allocation rule (the kappa
rule)
 The full circles are the
priority maintenance rule.
Von Bertalanffy: growth at constant food

 Von Bertallanffy growth in DEB theory
𝑑𝐿
= 𝑟𝐵 𝐿 − 𝐿
𝑑𝑡
𝑚𝐸 𝑣 𝑀𝑉
𝐽𝐸𝑇
𝑚𝐸 𝑣 𝑀𝑉
𝐿 
−
=
− 𝐿𝑇
𝐽𝐸𝑀
𝐽𝐸𝑀
𝐽𝐸𝑀
𝐽𝐸𝑀
𝑟𝐵 =
3 𝑀𝑉 𝑚𝐸 + 3 𝑀𝑉 𝑦𝐸𝑉
 DEB theory predicts:
 𝐿 decreases with specific maintenance needs and
increases with the reserve density (food level)
 𝑟𝐵 decreases with 𝐿
3𝑣 𝑀𝑉 𝑦𝐸𝑉 + 3𝐿 𝑇 𝐽𝐸𝑀
1 3𝐿
=
+
𝑟𝐵
𝑣
𝑣 𝐽𝐸𝑀
Von Bertalanffy: growth at constant food
length, mm
Von Bert growth rate -1, d

time, d
ultimate length, mm
A lower the food level implies a smaller ultimate size and a shorter time to reach it.
 Empirical fact: organisms of the same species at different food levels exhibit von
Bertallanfy growth rates that are inversely proportional to ultimate length
Extremes in relative growth rate in insects

Buprestis splendens (jewel beetle)
Juveniles live in wood for 20-40 a
Antheraea polyphemus (polyphemus moth)
Juveniles increase weight 80000 × in 48 h
Exercise

 Represent dL/dt as a function of L.
 Represent dL/dt as a funtion of L for two organisms
of the same species living at different food densities.
𝑑𝐿
= 𝑟𝐵 𝐿 − 𝐿
𝑑𝑡
𝐿 
𝑚𝐸 𝑣 𝑀𝑉
𝐽𝐸𝑀
−
𝐽𝐸𝑇
𝐽𝐸𝑀
=
𝑚𝐸 𝑣 𝑀𝑉
𝐽𝐸𝑀
𝑟𝐵 =
3 𝑀𝑉 𝑚𝐸 + 3 𝑀𝑉 𝑦𝐸𝑉
𝐽𝐸𝑀
− 𝐿𝑇
Egg and foetal development: differences
 Growth in DEB:

𝑑𝐿 1 𝑚𝐸 𝑣 𝑀𝑉 − 𝐽𝐸𝑀 𝐿 − 𝐽𝐸𝑇
=
𝑑𝑡 3
𝑚𝐸 𝑀𝑉 + 𝑀𝑉 𝑦𝐸𝑉
 What happens to the reserve density in an egg?
Egg and foetal development: differences
 Growth in DEB:

𝑑𝐿 1 𝑚𝐸 𝑣 𝑀𝑉 − 𝐽𝐸𝑀 𝐿 − 𝐽𝐸𝑇
=
𝑑𝑡 3
𝑚𝐸 𝑀𝑉 + 𝑀𝑉 𝑦𝐸𝑉
 What happens to the reserve density in an egg?
 It decreases in time
 What happens to the reserve density in a foetus?
Egg and foetal development: differences
 Growth in DEB:

𝑑𝐿 1 𝑚𝐸 𝑣 𝑀𝑉 − 𝐽𝐸𝑀 𝐿 − 𝐽𝐸𝑇
=
𝑑𝑡 3
𝑚𝐸 𝑀𝑉 + 𝑀𝑉 𝑦𝐸𝑉
 What happens to the reserve density in an egg?
 It decreases in time
 What happens to the reserve density in a foetus?
 It tends to infinity
 Obtain V(t) for a foetus
Egg and foetal development: differences
 Growth in DEB:

𝑑𝐿 1 𝑚𝐸 𝑣 𝑀𝑉 − 𝐽𝐸𝑀 𝐿 − 𝐽𝐸𝑇
=
𝑑𝑡 3
𝑚𝐸 𝑀𝑉 + 𝑀𝑉 𝑦𝐸𝑉
 What happens to the reserve density in an egg?
 It decreases in time
 What happens to the reserve density in a foetus?
 It tends to infinity
 Obtain V(t) for a foetus
𝑣𝑡
V 𝑡 =
3
3
 Empirical pattern: Volume is proportional to cubed
time in a foetus
Egg & Foetal development

Competition between growth and
somatic maintenance

 If growth is supply driven when does growth stops?
𝑣 1 𝑑𝑉
−
𝐿 𝑉 𝑑𝑡
/
𝐽𝐸𝑆 = 𝐽𝐸𝑀 𝑉 + 𝐽𝐸𝑇 𝑉2 3
𝐽𝐸𝐺 = 𝐽𝐸𝐶 − 𝐽𝐸𝑆
𝑑𝑀𝑉
𝑑𝑉
𝐽𝑉𝐺 = 𝑦𝑉𝐸 𝐽𝐸𝐺 =
= 𝑀𝑉
𝑑𝑡
𝑑𝑡 Offspring
𝐽𝐸𝐶 = 𝑀𝐸
Feeding
J XA
ME - Reserve
Mobilisation
J EA
J EC
Assimilation
J EJ
J ER
J ES
Maturity
Maintenance
MER
Reproduction
Maturation
MH - Maturity
J EG
Somatic
Maintenance
Growth
MV - Structure
Competition between growth and
somatic maintenance

 If growth is supply driven when does growth stops?
 When all the energy that goes for somatic maintenance plus
growth is used for maintenance
Feedin
2/3
𝐽𝐸𝐴 = 𝑦𝐸𝑋 𝐽𝑋𝐴 = 𝑓(𝑋) 𝐽𝐸𝐴𝑚 𝑉
𝑣 1 𝑑𝑉
𝐽𝐸𝐶 = 𝑀𝐸
−
𝐿 𝑉 𝑑𝑡
/
𝐽𝐸𝑆 = 𝐽𝐸𝑀 𝑉 + 𝐽𝐸𝑇 𝑉2 3
𝐽𝐸𝐺 = 𝐽𝐸𝐶 − 𝐽𝐸𝑆
𝑑𝑀𝑉
𝑑𝑉
𝐽𝑉𝐺 = 𝑦𝑉𝐸 𝐽𝐸𝐺 =
= 𝑀𝑉
𝑑𝑡
𝑑𝑡
J XA g
ME - Reserve
Mobilisation
J EA
J EC
Assimilation
Offspring
MER
J EJ
J ER
Reproductio
n
J ES
Maturity
Maintenance
Maturation
MH - Maturity
J EG
Growth
Somatic
Maintenance
MV - Structure
 As the organism gets bigger it gets more food ( to V2/3) but it
grows slower because somatic maintenance ( to V) is competing
with growth
 The higher the specific somatic maintenance needs the lower the
ultimate size
A DEB organism: maturity maintenance

Feeding
J XA
ME - Reserve
Mobilisation
J EA
J EC
Assimilation
Offspring
MER
J EJ
J ER
J ES
Maturity
Maintenance
Reproduction
Maturation
MH - Maturity
J EG
Somatic
Maintenance
Growth
MV - Structure
 Metabolism in a DEB
individual.
 Rectangles are state
variables
 Arrows are flows of food
JXA, reserve JEA, JEC, JEM, JET ,
JEG, JER, JEJ or structure JVG.
 Circles are processes
 The full square is a fixed
allocation rule (the kappa
rule)
 The full circles are the
priority maintenance rule.
Maturity maintenance

 Collection of processes that maintain the level of
maturity
 Defense and regulating systems
 Maturity maintenance is paid from flux (1-)JE,C:
𝐽𝐸𝐽 = 𝑘𝐽 𝑀𝐻
  maturity level
 It does not increase after the onset of reproduction
 Specific maturity maintenance costs are constant because of the
strong homeostasis
 The complexity would decrease in the absence of energy spent
in its maintenance (2nd Law of thermodynamics)
 Empirical pattern: no reproduction occurs at very low food
densities
𝑘𝐽 - maturity maintenance rate coefficient
A DEB organism: maturation/reproduction

Feeding
J XA
ME - Reserve
Mobilisation
J EA
J EC
Assimilation
Offspring
MER
J EJ
J ER
J ES
Maturity
Maintenance
Reproduction
Maturation
MH - Maturity
J EG
Somatic
Maintenance
Growth
MV - Structure
 Metabolism in a DEB
individual.
 Rectangles are state
variables
 Arrows are flows of food
JXA, reserve JEA, JEC, JEM, JET ,
JEG, JER, JEJ or structure JVG.
 Circles are processes
 The full square is a fixed
allocation rule (the kappa
rule)
 The full circles are the
priority maintenance rule.
Maturation/Reproduction

 The use of reserve to increase the state of maturity
(embryo and juvenile) or to reproduce (adult)
 Allocation to maturation in a juvenile (MH <MHp) or
to reproduction in na adult (MH >=MHp) (supply
driven):
𝐽𝐸𝑅 = 1 −  𝐽𝐸𝐶 − 𝐽𝐸𝐽
 Empirical pattern: organisms kept at low food density never
reach puberty implying that they will not reproduce
 Stage transitions should not be linked with size
MHb- threshold of maturity at birth
MHp- threshold of maturity at puberty
Extremes in relative maturity at birth in
mammals

Didelphus marsupiales (Am opossum)
♂, ♀ 0.5 + 0.5 m, 6.5 kg
At birth: <2 g; ab = 8-13 d
10-12 (upto 25) young/litter, 2 litters/a
Ommatophoca rossii (Ross Seal)
♂ 1.7-2.1 m, 129-216 kg
♀ 1.3-2.2 m, 159-204 kg
At birth: 1 m, 16.5 kg; ab = 270 d
Extremes in relative maturity at birth in fish

Mola mola (ocean sunfish)
♂,♀ 4 m, 1500 (till 2300) kg
Egg: 3 1010 eggs in buffer
At birth: 1.84 mm g; ab = ? d
Feeds on jellyfish & combjellies
Latimeria chalumnae (coelacanth)
♂, ♀ 1.9 m, 90 kg
Egg: 325 g
At birth: 30 cm; ab = 395 d
Feeds on fish
Reproduction

 The amount of energy continuously invested in
reproduction is accumulated in a buffer and then it is
converted into eggs providing the initial endowment
of the reserve to the embryo
𝑅 𝐽𝐸𝑅 = 𝑅
𝑅=
𝑅 𝐽𝐸𝑅
𝑀𝐸0
1 −  𝐽𝐸𝐶 − 𝐽𝐸𝐽 =
𝑑𝑀𝐸𝑅
𝑑𝑡
 Initial amount of reserve 𝑀𝐸0 follows from
 Initial structural vol. and maturity are negligibly small and
maturity at birth is given
 Empirical fact: reserve density at birth equals that of mother
at egg formation (egg size covaries with the nutritional state
of the mother)
𝑀𝐸0 - initial amount of reserve of the egg
𝑅 - reproduction efficiency
Reproduction: buffer handling rules

 Rules for handling the reproduction buffer are
species-specific (different evolutionary strategies)
 Some species reproduce when
enough energy for a single egg
has been accumulated
 Some species reproduce a large
clutch (some fishes have thousands
of eggs)
 Some species use environmental triggers for spawning
(e.g., moluscs)
Energy flows vs. Mass flows
 Parameters
 Fluxes
𝑝𝑋 = 𝐽𝑋𝐴 𝜇𝑋 = 𝑓(𝑋) 𝑝𝑋𝑚 𝑉 2/3
𝑝𝑋𝑚 = 𝜇𝑋 𝐽𝑋𝑚
𝑝𝐴 = 𝐽𝐸𝐴 𝜇𝐸 = 𝑓(𝑋) 𝑝𝐴𝑚 𝑉 2/3
𝑣
𝑝𝐶 = 𝐽𝐸𝐶 𝜇𝐸 = 𝐸
−𝑟
𝐿
𝑝𝑆 = 𝐽𝐸𝑆 𝜇𝐸 = 𝑝𝑀 𝑉 + 𝑝𝑇 𝑉 2/3
𝑝𝐴𝑚 = 𝜇𝐸 𝐽𝐴𝑚
𝑝𝐽 = 𝐽𝐸𝐽 𝜇𝐸 = 𝜇𝐸 𝑘𝐽 𝑀𝐻 =𝑘𝐽 𝐸𝐻
𝑝𝑅 = 𝐽𝐸𝑅 𝜇𝐸
𝑝𝐺 = 𝐽𝐸𝐺 𝜇𝐸 = 𝐸𝐺
𝑑𝑉
𝑑𝑡
𝑝𝑀 = 𝜇𝐸 𝐽𝐸𝑀
𝑝𝑇 = 𝜇𝐸 𝐽𝐸𝑇
𝐸𝐺 = 𝑦𝐸𝑉 𝜇𝐸 𝑀𝑉
 State Variables
𝐸 = 𝑀𝐸 𝜇𝐸
𝐸𝐻 = 𝑀𝐻 𝜇𝐸
EHb- threshold of maturity at birth
EHp- threshold of maturity at puberty