Theoretical Ecology

, Volume 7, Issue 1, pp 23–33 | Cite as

Size-based predictions of food web patterns

  • Lai Zhang
  • Martin Hartvig
  • Kim Knudsen
  • Ken H. Andersen
Original Paper


We employ size-based theoretical arguments to derive simple analytic predictions of ecological patterns and properties of natural communities: size-spectrum exponent, maximum trophic level, and susceptibility to invasive species. The predictions are brought about by assuming that an infinite number of species are continuously distributed on a size–trait axis. It is, however, an open question whether such predictions are valid for a food web with a finite number of species embedded in a network structure. We address this question by comparing the size-based predictions to results from dynamic food web simulations with varying species richness. To this end, we develop a new size- and trait-based food web model that can be simplified into an analytically solvable size-based model. We confirm existing solutions for the size distribution and derive novel predictions for maximum trophic level and invasion resistance. Our results show that the predicted size-spectrum exponent is borne out in the simulated food webs even with few species, albeit with a systematic bias. The predicted maximum trophic level turns out to be an upper limit since simulated food webs may have a lower number of trophic levels, especially for low species richness, due to structural constraints. The size-based model possesses an evolutionary stable state and is therefore un-invadable. In contrast, the food web simulations show that all communities, irrespective of number of species, are equally open to invasions. We use these results to discuss the validity of size-based predictions in the light of the structural constraints imposed by food webs.


Biodiversity Food web assembly Individual size distribution Size spectrum Traits Maximum trophic level 



Lai Zhang gratefully acknowledges the financial support from the Swedish Kempe Foundation. Martin Hartvig acknowledges the Danish National Research Foundation for support to the Center for Macroecology, Evolution and Climate. Ken Haste Andersen was financially supported by the European FP7 program MEECE and the VKR center of excellence Ocean Life.


  1. Andersen KH, Beyer JE (2006) Asymptotic body size determines species abundance in the marine size spectrum. Am Nat 168:54–61PubMedCrossRefGoogle Scholar
  2. Andersen KH, Beyer JE, Lundberg P (2009) Trophic and individual efficiencies of size-structured communities. Proc R Soc B 276:109–114PubMedCrossRefGoogle Scholar
  3. Barnes C, Maxwell D, Reuman D, Jennings S (2010) Global patterns in predator-prey size relationships reveal size dependency of trophic transfer efficiency. Ecology 91:222–232PubMedCrossRefGoogle Scholar
  4. Benoît E, Rochet MJ (2004) A continuous model of biomass size spectra governed by predation and the effects of fishing on them. J Theor Biol 226:9–21PubMedCrossRefGoogle Scholar
  5. Boit A, Martinez ND, Williams RJ, Gadeke U (2012) Mechanistic theory and modelling of complex food-web dynamics in Lake Constance. Ecol Lett 15:594–602PubMedCrossRefGoogle Scholar
  6. Borgmann U (1987) Models on the slope of, and biomass flow up, the biomass size spectrum. Can J Fish Aquat Sci 44:136–140CrossRefGoogle Scholar
  7. Brose U, Williams RJ, Martinez ND (2006a) Allometric scaling enhances stability in complex food webs. Ecol Lett 9:1228–1236CrossRefGoogle Scholar
  8. Brose U, Jonsson T, Berlow EL, Warren P, Banasek-Richter C, Bersier LF, Blanchard JL, Brey T, Carpenter SR, Blandenier MF, Cushing L, Dawah HA, Dell T, Edwards F, Harper-Smith S, Jacob U, Ledger ME, Martinez ND, Memmott J, Mintenbeck K, Pinnegar JK, Rall BC, Rayner TS, Reuman DC, Ruess L, Ulrich W, Williams RJ, Woodward G, Cohen JE (2006b) Consumer-resource body-size relationships in natural food webs. Ecology 87:2411–2417CrossRefGoogle Scholar
  9. Chesson P, Kuang JJ (2008) The interaction between predation and competition. Nature 456:235–238PubMedCrossRefGoogle Scholar
  10. Cohen JE, Pimm SL, Yodzis P, Saldaña J (1993) Body sizes of animal predators and animal prey in food webs. J Anim Ecol 62:67–78CrossRefGoogle Scholar
  11. Datta S, Delius GW, Law R, Plank MJ (2011) A stability analysis of the power-law steady state of marine size spectra. J Math Biol 63:779–799PubMedCrossRefGoogle Scholar
  12. Drake JA (1990) The mechanics of community assembly and succession. J Theor Biol 147:213–233CrossRefGoogle Scholar
  13. Giacomini HC, DeMarco P, Petrere M (2009) Exploring community assembly through an individual-based model for trophic interactions. Ecol Model 220:23–39CrossRefGoogle Scholar
  14. Gomez-Canchong P, Quinones RA, Brose U (2012) Robustness of size-structure across ecological networks in pelagic systems. Theor Ecol 6:45–56CrossRefGoogle Scholar
  15. Hartvig M, Andersen KH , Beyer JE (2011) Food web framework for size-structured populations. J Theor Biol 272:113–122PubMedCrossRefGoogle Scholar
  16. Hartvig M (2011) Ecological processes yield complex and realistic food webs. In: Food web ecology—individual life-histories and ecological processes shape complex communities. ISBN 978-91-7473-080-7. Ph.D. thesis. Department of Biology. Lund University, Sweden, pp 75–126Google Scholar
  17. Haskell J, Ritchie M, Olff H (2002) Fractal geometry predicts varying body size scaling relationships for mammal and bird home ranges. Nature 418:527–530PubMedCrossRefGoogle Scholar
  18. Jetz W, Carbone C, Fulford J, Brown JH (2004) The scaling of animal space use. Science 306:266–268PubMedCrossRefGoogle Scholar
  19. Kerr S (1974) Theory of size distribution in ecological communities. J Fish Res Board Can 31:1859–1862CrossRefGoogle Scholar
  20. Kramer D, Chapmanm M (1999) Implications of fish home range size and relocation for marine reserve function. Environ Biol Fish 55:65–79CrossRefGoogle Scholar
  21. Law R, Morton RD (1996) Permanence and the assembly of ecological communities. Ecology 77:762–775CrossRefGoogle Scholar
  22. Law R (1999) Theoretical aspects of community assembly. In: McGlade J (ed) Advanced ecological theory. Blackwell Science, Oxford, pp 143–171CrossRefGoogle Scholar
  23. Levine S (1980) Several measures of trophic structure applicable to complex food webs. J Theor Biol 83:195–207CrossRefGoogle Scholar
  24. Lewis HM, Law R (2007) Effects of dynamics on ecological networks. J Theor Biol 247:64–76PubMedCrossRefGoogle Scholar
  25. Lindeman RL (1942) The trophic-dynamic aspect of ecology. Ecology 4:399–417CrossRefGoogle Scholar
  26. Loeuille N, Loreau M (2005) Evolutionary emergence of size-structured food webs. Proc Natl Acad Sci USA 102:5761–5766PubMedCrossRefGoogle Scholar
  27. MacArthur R, Levins R (1967) The limiting similarity, convergence, and divergence of coexisting species. Am Nat 101:377–385CrossRefGoogle Scholar
  28. Maury Q, Poggiale JC (2013) From individuals to populations to communities: a dynamic energy budget model of marine ecosystem size-spectrum including life history diversity. J Theor Biol . doi: 10.1016/j.jtbi.2013.01.018 PubMedGoogle Scholar
  29. Maynard Smith J, Price GR (1973) The logic of animal conflict. Nature 246:15–18CrossRefGoogle Scholar
  30. Metz JAJ, Nisbet RM, Geritz SAH (1992) How should we define “fitness” for general ecological scenarios?Tree 7:198–202PubMedGoogle Scholar
  31. Morton RD, Law R (1997) Regional species pools and the assembly of local ecological communities. J Theor Biol 187:321–331CrossRefGoogle Scholar
  32. Odum WE, Heald EJ (1975) The detritus-based food web of an estuarine mangrove community. In: Cronin L (ed) Estuarine research. Chemistry biology and the estuarine system, vol 1. Academic, LondonGoogle Scholar
  33. Oksanen L, Fretwell SD, Arruda J, Niemela P (1981) Exploitation ecosystems in gradients of primary productivity. Am Nat 118:240–261CrossRefGoogle Scholar
  34. Post WM, Pimm SL (1983) Community assembly and food web stability. Math Biosci 64:169–182CrossRefGoogle Scholar
  35. Post DM (2002) The long and short of food-chain length. Trends Ecol Evol 17:269–277CrossRefGoogle Scholar
  36. Rooney N, McCann K, Gellner G, Moore JC (2006) Structural asymmetry and the stability of diverse food webs. Nature 442:265–269PubMedCrossRefGoogle Scholar
  37. Reuman DC, Mulder C, Raffaelli D, Cohen JE (2008) Three allometric relations of population density to body mass: theoretical integration and empirical tests in 149 food webs. Ecol Lett 11:1216–1228PubMedCrossRefGoogle Scholar
  38. Rossberg AG, Ishi R, Amemiya T, Itoh K (2008) The top-down mechanism for body-mass-abundance scaling. Ecology 89:567–580PubMedCrossRefGoogle Scholar
  39. Rossberg AG, Brännström Å, Dieckmann U (2010) How trophic interaction strength depends on traits. Theor Ecol 3:13–24CrossRefGoogle Scholar
  40. Rossberg AG (2012) A complete analytic theory for structure and dynamics of populations and communities spanning wide ranges in body size. Adv Ecol Res 46:429–522Google Scholar
  41. Savage VM, Gillooly JF, Brown JH, West GB, Charnov EL (2004) Effects of body size and temperature on population growth. Am Nat 163:429–441PubMedCrossRefGoogle Scholar
  42. Sheldon RW, Prakash A, Sutcliffe WHJ (1972) The size distribution of particles in the ocean. Limnol Oceanogr 17:327–340CrossRefGoogle Scholar
  43. Sheldon RW, Sutcliffe WHJ, Paranjape MA (1977) Structure of pelagic food chain and relationship between plankton and fish production. J Fish Res Board Can 34:2344–2353CrossRefGoogle Scholar
  44. Taylor PJ (1988) The construction and turnover of complex community models having generalized Lotka-Volterra dynamics. J Theor Biol 135:569–588CrossRefGoogle Scholar
  45. Ursin E (1973) On the prey size preferences of cod and dab. Meddelelser fra Dammarks Fiskeri- og Havundersøgelser 7:84–98Google Scholar
  46. Virgo N, Law R, Emmerson M (2006) Sequentially assembled food webs and extremum principles in ecosystem ecology. J Anim Ecol 75:377–386PubMedCrossRefGoogle Scholar
  47. Ware DM (1978) Bioenergetics of pelagic fish: theoretical change in swimming speed and ration with body size. J Fish Res Board Can 35:220–228CrossRefGoogle Scholar
  48. West GB, Brown JH, Enquist BJ (1997) A general model for the origin of allometric scaling laws in biology. Science 276:122–126PubMedCrossRefGoogle Scholar
  49. Yodzis P (1984) Energy flow and the vertical structure of real ecosystems. Oecologia 65:86–88CrossRefGoogle Scholar
  50. Yodzis P, Innes S (1992) Body size and consumer–resource dynamics. Am Nat 6:1151–1175CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Lai Zhang
    • 1
  • Martin Hartvig
    • 2
    • 3
  • Kim Knudsen
    • 1
  • Ken H. Andersen
    • 4
  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkLyngbyDenmark
  2. 2.National Institute of Aquatic ResourcesTechnical University of DenmarkCharlottenlundDenmark
  3. 3.Center for Macroecology, Evolution and ClimateUniversity of CopenhagenCopenhagen ØDenmark
  4. 4.Center for Ocean Life, National Institute of Aquatic ResourcesTechnical University of DenmarkCharlottenlundDenmark

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