The Metabolic Theory of Ecology (MTE) and the theory of Dynamic Energy Budgets (DEB) (and more)

1. 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

2. New developments in ecology To 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

3. New developments in ecology

4. 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

5. The basis of MTE 1 Supply to the cells goes through a fractal-like branching structure, designed such that transport costs are minimal 2 Maintenance costs of cells are constant 3 Difference is available for somatic growth West et al. 1997, 2001

6. Fractal-like branching structure • ASSUMPTIONS • Capillaries do not change • Cross-area preservation • Volume preservation • 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 CONCLUSION The fractal model lacks self-consistency and correct statement Dodds et al. 2001; Kozlowski and Konarzewski 2005; Rampal et al. 2006; Chaui-Berlinck 2006

7. Intra-specific scaling • 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 • PROBLEMS • Set of assumptions are inconsistent and violate the second law of thermodynamics • Overhead costs of growth are neglected CONCLUSION The 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 Makarieva 2004; Van der Meer 2006

8. Inter-specific scaling • 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 Kleiber’s law Volume-specific maintenance costs • 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 CONCLUSION The 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). Van der Meer 2006

9. 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

10. From the individual to the population: r and K For small sizes log rmax … , 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 log W Gillooly et al. 2002, Ernest et al. 2003; Savage et al. 2004

11. From the individual to the population: r and K Assume that at carrying capacity KB 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 KB log W Gillooly et al. 2002, Ernest et al. 2003; Savage et al. 2004

12. MTE DEB State variables Body mass … and reserves Feeding module No Yes WITHIN SPECIES Assimilation A 3/4 2/3 Maintenance M 1 1 Metabolic rate 3/4 or 1? 2/3 to 1 AMONG SPECIES Assimilation a 0 1/3 Maintenance m -1/4 0 Costs for growth g energy content … and overhead Metabolic rate 3/4 2/3 to 1 West et al. 1997, 2001; Kooijman 2000; Van der Meer 2006

13. MTE versus DEB

14. faeces ingestion assimilation V E+ER reproduction growth dissipation DEB-organism From the individual to the population: structured-population models Rate of ingestion in response to densities of a variety of available prey items and a variety of (direct) competitors? Prey selection Mutual interference

15. D S H  DEB’s synthesising unit (and Holling’s type II functional response)

16.   ½ ingestion rate W / food density D

17. E D F S H   Substitutable preyE.g. one edible and one inedible prey

18. Interference

19. Interference The first ‘mechanistic’ models considered competitors as inedible prey, ... Beddington’s generalized functional response …, but competitors do not behave as inedible prey Beddington 1975

20. H S D S F S H G    Interference Ruxton et al. 1992; Van der Meer & Smallegange in prep.

21. F2   2D D S2 S1H1 H2  2   G2 A stochastic version Continuous Time Markov Chain

22. F4 F2G2 2D D S2F2 S1H1F2 H2F2   2 2   6  3    4D 3D 2D D S4 S3H1 S2H2 S1H3 H4  2 2 3 4  3  4  3 2D D S2G2 S1H1G2 H2G2    2 2  G4 4 predators 14 states

23. Transition rates

24. A stochastic version

25. A stochastic version Intake rate Stochastic Deterministic Approximation Number of competitors =2,=4

26.  50 mm Shore crab Carcinus maenas • Solitary animals • Omnivores and cannibals • Live in coastal water and estuaries • Occur from Norway to Mauritania • Size (males) Puberty at carapace width 20-30 mm Reproduce when carapace width ~50 mm Maximum carapace width ~ 90 mm • Maximum age ~10 years

27. Behaviours Search Eat Fight Total time needed for one prey item

28. Results Total time Interference time Time in s ± 95% CI

29. ML parameter estimation

30. F2   2D D S2 S1H1 H2  2   G2 ML parameter estimators 4 2 3 1 5

31. Transitions and stage durations

32. 2 crabs 4 crabs Feeding rate in min-1 ± 95% CI

33. Testing predictions on the distribution of crabs Two food patches of 0.25 m2 each

34. F2   2D D S2 S1H1 H2  2   G2   Model predictions IFD hypothesis Searchers show infinitely fast movements towards the better patch Random movements No preference for a patch. Only searchers move between patcheswith constant dispersal rate

35. 3 S1H1  D1 D1 0.5 2 5    S2H2 S2H1 S1H1 S1H2 F1F1 G1G1 S1S1 H1H1 1 6  D1 D1   4     0.5 H2H2 H2H1 H1H2 H1H1 F2F2 F1F1 S2S2 S2S1 S1S2 S1S1 G2G2 G1G1 H1S1  12 8   S1H2 S2H1  H1S1 H2S2 H2S1 H1S2 D2   D1  14 10 D2  D1  11 7 S1S2 H1H2 S2S1 H2H1 D1 D1 D2 D2 13 9        H1S2 H2S1  17    S2H2  D2 D2   19 16  0.5  F2F2 S2S2 H2H2 G2G2 15 20  D2 D2  18   0.5 H2S2

36. IFD Random movement

37. crabs ate more from best patch IFD Random movement

38. crabs ate more from best patch

39. Preliminary conclusions New ‘generalized functional response’ model is generally applicable for foraging shore crabs The two ‘dispersion’ models (IFD or Random movement) are not

40. lose S Conflict module find handler finish handling find food win H discovered Adaptive interference competition

41. Intruders strategy Owners strategy Random win defend lose attack not defend defend not attack win not defend lose time energy

42. 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 probability of outcome j Adaptive dynamics

43. 100 10 forager density (#s-1) 1 0.1 1 10 food density (#s-1) low food density intermediate food density high food density value of food 10 J cost of fight 1 J fight time 2 s handling time 1 s probability of winning 0.5

44. H(1,1) Hawk H(1,1) AB(1,0) B(0,1) AB(1,0) B(0,1) defense strategy Bourgeois attack strategy defense strategy 0 Anti-Bourgeois 0 attack strategy

45. 100 10 10 forager density (#s-1) 8 1 6 intake rate (#s-1) 4 Bourgeois (0,1) 0.1 1 10 2 Anti-Bourgeois (1,0) food density (#s-1) Hawk (1,1) 0.01 100 1 forager density (#s-1)

46. Summary MTE 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)