1. Dynamic Energy Budget Theory
Ask to draw the scheme
Tânia Sousa
with contributions from : Gonçalo Marques and Bas Kooijman

2. A DEB organism – growth Metabolism in a DEB individual.
Rectangles are state variables
Arrows are flows of food pX, reserve pA, pC, pS, pT , pG, pR, pJ or structure pVG.
Circles are transformations
The kappa rule (a fixed allocation rule)
The priority maintenance rules
Draw the scheme in the blackboard. The frontiers and the state variables.

3. Growth
Growth is the increase of the amount of structure (conversion of reserve into structure)
Allocation to growth (supply driven):
𝑝 𝐺 =𝜅 𝑝 𝐶 − 𝑝 𝑆
𝜅 𝐺 𝑝 𝐺 = 𝑑 𝐸 𝑉 𝑑𝑡 = 𝜇 𝐸 𝑀 𝑉 𝑑𝑉 𝑑𝑡

4. Growth
Growth is the increase of the amount of structure (conversion of reserve into structure)
Allocation to growth (supply driven):
𝑝 𝐺 =𝜅 𝑝 𝐶 − 𝑝 𝑆
𝑑 𝐸 𝑉 𝑑𝑡 = 𝜇 𝐸 𝑀 𝑉 𝑑𝑉 𝑑𝑡 = 𝜅 𝐺 𝑝 𝐺
𝑝 𝐺 = 𝐸 𝐺 𝑑𝑉 𝑑𝑡
Strong homeostasis imposes a fixed conversion efficiency
Strong homeostasis imposes a constant density

5. Growth
Growth is the increase of the amount of structure (conversion of reserve into structure)
Allocation to growth (supply driven):
𝑝 𝐺 =𝜅 𝑝 𝐶 − 𝑝 𝑆
𝑑 𝐸 𝑉 𝑑𝑡 = 𝜇 𝑉 𝑀 𝑉 𝑑𝑉 𝑑𝑡 = 𝜅 𝐺 𝑝 𝐺
𝑝 𝐺 = 𝐸 𝐺 𝑑𝑉 𝑑𝑡
Strong homeostasis imposes a fixed conversion efficiency
Strong homeostasis imposes a constant density
𝐸 𝐺 - reserve energy needed to build a unit of structural body volume 𝜅 𝐺 - efficiency of transformation of reserve into structure

6. A DEB organism – parameters
Primary parameters
Auxiliary parameters
𝜅 𝑋
𝜇 𝑋 , 𝜇 𝐸 , 𝜇 𝑉
𝑝 𝐴𝑚
𝑑 𝑋 , 𝑑 𝐸 , 𝑑 𝑉 𝑣
𝑤 𝑋 , 𝑤 𝐸 , 𝑤 𝑉 𝜅
𝑝 𝑀
𝑝 𝑇
𝐸 𝐺

7. A DEB organism – maturity
Metabolism in a DEB individual.
Rectangles are state variables
Arrows are flows of food pX, reserve pA, pC, pS, pT , pG, pR, pJ or structure pVG.
Circles are transformations
The kappa rule (a fixed allocation rule)
The priority maintenance rules
Draw the scheme in the blackboard. The frontiers and the state variables.
Ask for the parameters

8. Maturity maintenance
Collection of processes that maintain the level of maturity
Defense and regulating systems
Maturity maintenance is paid from flux 1−𝜅 𝑝 𝑐 :
 maturity level
It does not increase after the onset of reproduction
𝑝 𝐽 = 𝑘 𝐽 𝐸 𝐻
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
𝑘 𝐽 - maturity maintenance rate coefficient

9. A DEB organism – maturity
Metabolism in a DEB individual.
Rectangles are state variables
Arrows are flows of food pX, reserve pA, pC, pS, pT , pG, pR, pJ or structure pVG.
Circles are transformations
The kappa rule (a fixed allocation rule)
The priority maintenance rules
Draw the scheme in the blackboard. The frontiers and the state variables.
Ask for the parameters
𝑝 𝐽 = 𝐽 𝐸𝐽 𝜇 𝐸 = 𝑘 𝐽 𝐸 𝐻
𝐽 𝐸𝐽 = 𝑘 𝐽 𝑀 𝐻
𝑘 𝐽 - maturity maintenance rate coefficient

10. A DEB organism – maturity
Metabolism in a DEB individual.
Rectangles are state variables
Arrows are flows of food pX, reserve pA, pC, pS, pT , pG, pR, pJ or structure pVG.
Circles are transformations
The kappa rule (a fixed allocation rule)
The priority maintenance rules
Draw the scheme in the blackboard. The frontiers and the state variables.
Ask for the parameters

11. 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 (EH <EHp) or to reproduction in na adult (EH =EHp) (supply driven):
𝑝 𝑅 = 1−𝜅 𝑝 𝐶 − 𝑝 𝐽
Empirical pattern: organisms kept at low food density never reach puberty implying that they will not reproduce
Stage transitions should not be linked to size
EHb- threshold of maturity at birth
EHp- threshold of maturity at puberty

12. Extremes in relative maturity at birth in fish
Mola mola (ocean sunfish)
♂,♀ 4 m, 1500 (till 2300) kg
Egg: 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

13. A DEB organism – parameters
Primary parameters
Auxiliary parameters
𝜅 𝑋
𝜇 𝑋 , 𝜇 𝐸 , 𝜇 𝑉
𝑝 𝐴𝑚
𝑑 𝑋 , 𝑑 𝐸 , 𝑑 𝑉 𝑣
𝑤 𝑋 , 𝑤 𝐸 , 𝑤 𝑉 𝜅
δ 𝑀
𝑝 𝑀
𝑝 𝑇
𝐸 𝐺
𝑘 𝐽
𝐸 𝐻 𝑏
𝐸 𝐻 𝑝

14. 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
Initial amount of reserve 𝐸 0 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)
 𝑅 𝐽 𝐸𝑅 =  𝑅 1− 𝐽 𝐸𝐶 − 𝐽 𝐸𝐽 = 𝑑 𝑀 𝐸𝑅 𝑑𝑡 𝑅=  𝑅 𝐽 𝐸𝑅 𝑀 𝐸0
Make graph of reserve, structure and reserve density.

15. 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)

16. A DEB organism – parameters
Primary parameters
Auxiliary parameters
𝜅 𝑋
𝜇 𝑋 , 𝜇 𝐸 , 𝜇 𝑉
𝑝 𝐴𝑚
𝑑 𝑋 , 𝑑 𝐸 , 𝑑 𝑉 𝑣
𝑤 𝑋 , 𝑤 𝐸 , 𝑤 𝑉 𝜅
δ 𝑀
𝑝 𝑀
𝑝 𝑇
𝐸 𝐺
𝑘 𝐽
𝐸 𝐻 𝑏
𝐸 𝐻 𝑝
𝜅 𝑅

17. Exercises
Obtain expressions that depend only on state variables and parameters for growth using the following equations
𝑝 𝐺 = 𝜅 𝑝 𝐶 − 𝑝 𝑆
𝑝 𝐺 = 𝐸 𝐺 𝑑𝑉 𝑑𝑡
𝑝 𝐶 =𝐸 𝑣 𝐿 − 1 𝑉 𝑑𝑉 𝑑𝑡
𝑝 𝑆 = 𝑝 𝑀 𝑉+ 𝑝 𝑇 𝑉 2/3

18. Exercises
The expression that depends only on state variables and parameters for growth is
What happens at constant food?
𝑑𝑉 𝑑𝑡 = 𝜅𝐸 𝑣 𝑉 − 𝑝 𝑀 𝑉− 𝑝 𝑇 𝑉 𝜅𝐸 𝑉 + 𝐸 𝐺

19. Exercises
Obtain expressions that depend only on state variables and parameters for growth at constant food (weak homeostasis) using the following definition for energy density:
What happens to 𝐸 at constant food density?
𝐸  𝐸 𝑉 - energy density

20. Exercises Is this Von Bertallanffy growth?
The expressions that depend only on state variables and parameters for growth at constant food density (mE is constant) are:
Is this Von Bertallanffy growth?
𝑑𝑉 𝑑𝑡 = 𝜅 𝐸 𝑉 𝑣 − 𝑝 𝑀 𝑉− 𝑝 𝑇 𝑉 𝐸 𝜅+ 𝐸 𝐺
𝑑𝐿 𝑑𝑡 = 1 3 𝜅 𝐸 𝑣 − 𝑝 𝑀 𝐿− 𝑝 𝑇 𝐸 𝜅+ 𝐸 𝐺
𝑑𝐿 𝑑𝑡 = 𝑟 𝐵 𝐿  −𝐿

21. Exercises Is this Von Bertallanffy growth? Yes, with
How do these Von Bertallanffy parameters depend on energy density and maintenance needs?
𝑑𝐿 𝑑𝑡 = 𝑝 𝑀 3 𝐸 𝜅+ 𝐸 𝐺 𝜅 𝐸 𝑣 − 𝑝 𝑇 𝑝 𝑀 −𝐿
𝐿   𝜅 𝐸 𝑣 − 𝑝 𝑇 𝑝 𝑀 = 𝜅 𝐸 𝑣 𝑝 𝑀 −𝐿 𝑇
𝑟 𝐵 = 𝑝 𝑀 3 𝐸 𝜅+ 𝐸 𝐺
𝑑𝐿 𝑑𝑡 = 𝑟 𝐵 𝐿  −𝐿
𝐿 𝑇  𝑝 𝑇 𝑝 𝑀 - heating length

22. Von Bertalanffy: growth at constant food
Von Bertallanffy growth in DEB theory
DEB theory predicts:
𝐿  decreases with specific maintenance needs and increases with the reserve density (food level)
𝑟 𝐵 decreases with 𝐿 
𝑑𝐿 𝑑𝑡 = 𝑟 𝐵 𝐿  −𝐿
𝐿   𝜅 𝐸 𝑣 − 𝑝 𝑇 𝑝 𝑀 = 𝜅 𝐸 𝑣 𝑝 𝑀 −𝐿 𝑇
𝑟 𝐵 = 𝑝 𝑀 3 𝐸 𝜅+ 𝐸 𝐺
1 𝑟 𝐵 = 3 𝐿  𝑣 𝑣 𝐸 𝐺 + 3𝐿 𝑇 𝑝 𝑀 𝑣 𝑝 𝑀

23. Exercise Represent L(t)
Represent L(t) for 2 organisms of the same species that have grown with different food availabilities
𝐿   𝜅 𝐸 𝑣 − 𝑝 𝑇 𝑝 𝑀 = 𝜅 𝐸 𝑣 𝑝 𝑀 −𝐿 𝑇
𝑟 𝐵 = 𝑝 𝑀 3 𝐸 𝜅+ 𝐸 𝐺

24. Von Bertalanffy: growth at constant food
length, mm
Growth since birth as constant food density is according to the von Bertalanffy growth curve,
but the inverse von Bertalenffy growth rate (the von Bertalanffy time) is linear in the ultimate length for the different food levels.
The slope relates to the energy conductance, the intercept to the somatic maintenance rate coefficient.
If we have no idea about parameter values and use these curves to estimate these two parameters, we have to express energy conductance (units: cm/d) in physical length, rather than in (volumetric) structural length.
These lengths differ by a factor equal to the shape coefficient (which is sensitive for how we take physical length).
time, d
What happens to Von Bertalanffy growth rate?
1 𝑟 𝐵 = 3 𝐿  𝑣 𝑣 𝐸 𝐺 + 3𝐿 𝑇 𝑝 𝑀 𝑣 𝑝 𝑀

25. Von Bertalanffy: growth at constant food
length, mm
Von Bert growth rate -1, d
Growth since birth as constant food density is according to the von Bertalanffy growth curve,
but the inverse von Bertalenffy growth rate (the von Bertalanffy time) is linear in the ultimate length for the different food levels.
The slope relates to the energy conductance, the intercept to the somatic maintenance rate coefficient.
If we have no idea about parameter values and use these curves to estimate these two parameters, we have to express energy conductance (units: cm/d) in physical length, rather than in (volumetric) structural length.
These lengths differ by a factor equal to the shape coefficient (which is sensitive for how we take physical length).
time, d
ultimate length, mm
A lower 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

26. Egg & Foetal development
Stewart JR, Thompson MB. 2004
Placental ontogeny of the Tasmanian scincid lizard, Niveoscincus ocellatus (Reptilia: Squamata).
J Morphol. 259(2):
Stewart JR, Brasch KR. 2003
Ultrastructure of the placentae of the natricine snake, Virginia striatula (Reptilia: Squamata).
J Morphol. 255(2):
Ramírez-Pinilla, M. P. 2006
Placental Transfer of Nutrients During Gestation in an Andean Population of the Highly Matrotrophic Lizard Genus Mabuya (Squamata: Scincidae)
Herpetological Monographs 20 (1):

27. Egg and foetal development: differences
Growth in DEB:
What happens to the reserve density in an egg?
It decreases in time
Exercise:
What happens to the reserve density in a foetus?
Obtain V(t) for a foetus?
𝑑𝑉 𝑑𝑡 = 𝜅 𝐸 𝑉 𝑣 − 𝑝 𝑀 𝑉− 𝑝 𝑇 𝑉 𝐸 𝜅+ 𝐸 𝐺

28. Egg and foetal development: differences
Growth in DEB:
What happens to the reserve density in an egg?
It decreases in time
Exercise:
What happens to the reserve density in a foetus?
It tends to infinity
Obtain V(t) for a foetus
𝑑𝑉 𝑑𝑡 = 𝜅 𝐸 𝑉 𝑣 − 𝑝 𝑀 𝑉− 𝑝 𝑇 𝑉 𝐸 𝜅+ 𝐸 𝐺

29. Egg and foetal development: differences
Growth in DEB:
What happens to the reserve density in an egg?
It decreases in time
Exercise:
What happens to the reserve density in a Foetus?
It tends to infinity
Obtain V(t) for a foetus
𝑑𝑉 𝑑𝑡 = 𝜅 𝐸 𝑉 𝑣 − 𝑝 𝑀 𝑉− 𝑝 𝑇 𝑉 𝐸 𝜅+ 𝐸 𝐺
Empirical fact: Foetal weigth is proportional to cubed time
V 𝑡 = 𝑣 𝑡 3 3

30. Competition between growth and somatic maintenance
If growth is supply driven when does growth stops?
Draw the scheme in the blackboard. The frontiers and the state variables.
Ask for the parameters

31. Competition between growth and somatic maintenance
As the organism gets bigger it gets more food (proportional to V2/3) but it grows slower because somatic maintenance (proportional to V) is competing with growth
The higher the specific somatic maintenance needs the lower the ultimate size
𝑑𝐿 𝑑𝑡 = 𝑟 𝐵 𝐿  −𝐿
𝐿   𝜅 𝐸 𝑣 − 𝑝 𝑇 𝑝 𝑀 = 𝜅 𝐸 𝑣 𝑝 𝑀 −𝐿 𝑇
𝑟 𝐵 = 𝑝 𝑀 3 𝐸 𝜅+ 𝐸 𝐺

32. DEB Dynamics
What are the dynamics of the state-variables?

33. Metabolic rates: the effect of temperature
Why do metabolic rates depend on temperature?

34. Metabolic rates: the effect of temperature
Low metabolism (body temperature between 27ºC – 33ºC)

35. Metabolic rates: the effect of temperature
How do metabolic rates ln k(T) depend on temperature?
Daphnia magna
ln rate
reproduction
young/d
ingestion
106 cells/h
growth, d-1
aging, d-1
104 T-1, K-1

36. Metabolic rates: the effect of temperature
How do metabolic rates depend on temperature?
Daphnia magna
ln rate
reproduction
young/d
ingestion
106 cells/h
growth, d-1
aging, d-1
104 T-1, K-1

37. Metabolic rates: the effect of temperature
How do metabolic rates depend on temperature?
What is the meaning of the slope cte=TA?
Write the expression as a function of T1
Write na expression for k(T)
Daphnia magna
ln rate
reproduction
young/d
ingestion
106 cells/h
growth, d-1
aging, d-1
104 T-1, K-1

38. Metabolic rates: the effect of temperature
Arrhenius relationship:
ln rate
reproduction
young/d
Daphnia magna
ingestion
106 cells/h
growth, d-1
aging, d-1
104 T-1, K-1
The Arrhenius relationship has good empirical support
The Arrhenius temperature is given by minus the slope: the higher the Arrhenius temperature the more sensitive organisms are to changes in temperature

39. Metabolic rates: temperature range
The Arrhenius relationship is valid in the temperature tolerance range
At temperatures too high the organism usually dies
At temperatures too low the rates are usually lower than predicted by the Arrhenius relationship, e.g., at cold temperatures the sloths become letargic
Many extinctions are tought to be related with to changes in temperature
late Pleistocene, 40,000 to 10,000 years ago, North America lost over 50 percent of its large mammal species. These species include mammoths, mastodons, giant ground sloths, among many others.

40. Metabolic rates: the effect of temperature
How do feeding power depends on temperature?

41. Metabolic rates: the effect of temperature
How do feeding depends on temperature?
What is the implication for 𝑝 𝑋𝑚 ?
𝑝 𝑋 =𝑓(𝑋) 𝑝 𝑋𝑚 𝑉 2/3 = 𝑝 𝑋1 𝑒𝑥𝑝 𝑇 𝐴 𝑇 1 − 𝑇 𝐴 𝑇

42. Metabolic rates: the effect of temperature
All parameters that have units time-1 depend on temperature

43. Metabolic rates: the effect of temperature
Endotherms: mantain their body temperature
Most mammals mantain their body mperature at 36ºC - 40ºC
Most birds mantain their body temperature at 40-43ºC
How?
Ectotherms: have a variable temperature mostly dictated by environmental conditions
Any regulation?
Expected differences in metabolism?