Support of the hyperbolic connectance hypothesis by qualitative stability of model food webs

Abstract

Local stability analysis of dynamical models of interacting populations predicted that food web connectance (C) is proportional to 1/S where S is species richness. This “hyperbolic connectance hypothesis” was initially supported by analyses of documented food webs. This study shows that the qualitative global asymptotic stability of the Lotka-Volterra cascade model with a finite number of species predicts a relationship between connectance and species richness that agrees closely with the hyperbolic connectance hypothesis predicted from the analysis of local asymptotic stability. Moreover, the threshold of the qualitative global asymptotic stability in the Lotka-Volterra cascade model separates food webs in constant environments from those in fluctuating environments. The obvious discrepancy between the C-S relationship based on some recent data and that predicted by the dynamical models could be due to the selection of data.

Abbreviations

LVCM:

Lotka-Volterra Cascade Model

LAS:

Local Asymptotic Stability

QGAS:

Qualitative Global Asymptotic Stability

References

  1. Auerbach, M. J. 1984 Stability, probability and the topology of food webs. In: D. R. Strong, Jr. D. Simberloff, L. G. Abele and A. B. Thistle (eds), Ecological Communities: Conceptual Issues and Evidence. Princeton University Press. pp. 412–436.

  2. Bersier, L.-F., P. Dixon and G. Sugihara. 1999. Scale-invariant behavior of the link density property in food webs: a matter of sampling effort? Amer. Nat. 153: 676–682.

    Article  Google Scholar 

  3. Briand, F. 1983. Environmental control of food web structure. Ecology 64: 253–263.

    Article  Google Scholar 

  4. Briand, F. and J. E. Cohen. 1987. Environmental correlates of food chain length. Science 238: 956–960.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  5. Cohen, J.E., R.A. Beaver, S.H. Cousins, D.L. DeAngelis, L. Goldwasser, K.L. Heong, R.D. Holt, A.J. Kohn, J.H. Lawton, N. Martinez, R.O. O’Malley, L.M. Page, B.C. Patten, S.L. Pimm, G.A. Polis, M. Rejmanek, T.W. Schoener, K. Schoenly, W.G. Sprules, J.M. Teal, R.E. Ulanowicz, P.H. Warren, H.M. Wilbur, P. Yodzis. 1993. Improving food webs. Ecology 74: 252–259.

    Article  Google Scholar 

  6. Cohen, J. E. and F. Briand. 1984. Trophic links of community food webs. Proceedings of the Royal Society of London B 224: 421–448.

    Google Scholar 

  7. Cohen, J. E., F. Briand and C. M. Newman. 1986. A stochastic theory of community food webs. III. Predicted and observed length of food chains. Proceedings of the Royal Society of London B 228: 317–353.

    Article  Google Scholar 

  8. Cohen, J. E., F. Briand and C. M. Newman. 1990a. Community Food Webs: Data And Theory. Springer-Verlag, Berlin.

    Book  Google Scholar 

  9. Cohen, J. E., T. Luczak, C. M. Newman and Z.-M. Zhou. 1990b. Stochastic structure and nonlinear dynamics of food webs: qualitative stability in a Lotka-Volterra cascade model. Proceedings of the Royal Society of London B 240: 607–627.

    Article  Google Scholar 

  10. Cohen, J. E. and C. M. Newman. 1984. The stability of large random matrices and their products. Annals of Probability 12: 283–310.

    Article  Google Scholar 

  11. Cohen, J. E. and C. M. Newman. 1985a. When will a large complex system be stable? J. theoret. Biol. 113: 153–156.

    Article  Google Scholar 

  12. Cohen, J. E. and C. M. Newman. 1985b. A stochastic theory of community food webs. I. Models and aggregated data. Proceedings of the Royal Society of London B 224: 421–448.

    Article  Google Scholar 

  13. Cohen, J. E. and C. M. Newman. 1988. Dynamic basis of food web organization. Ecology 69: 1655–1664.

    Article  Google Scholar 

  14. DeAngelis, A. L. 1975. Stability and connectance in food web models. Ecology 56: 238–243.

    Article  Google Scholar 

  15. Gardner, M. R., and W. R. Ashby. 1970. Connectance of large, dynamical (cybernetic) systems: critical value for stability. Nature 228: 784.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  16. Gloss, G. P. and P. S. Lake. 1994. Spatial and temporal variation in the structure of an intermittent-stream food web. Ecol. Monog. 64: 1–21.

    Google Scholar 

  17. Hall, S. J. and D. G. Raffaelli. 1993. Food webs: theory and reality. Adv. Ecol. Res. 24: 187–239.

    Article  Google Scholar 

  18. Havens, K.E. 1992. Scale and structure in natural food webs. Science 257: 1107–1109.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  19. Havens, K. E. 1997. Unique structural properties of pelagic food webs. Oikos 78: 75–80.

    Article  Google Scholar 

  20. Krebs, J. R. and N. B. Davies. 1993. An Introduction to Behavioural Ecology. 3rd ed. Blackwell Science Ltd, Oxford.

    Google Scholar 

  21. Lawlor, L. R. 1978. A comment on randomly constructed ecosystem models. Amer. Nat. 112: 445–447.

    Article  Google Scholar 

  22. Lawton, J. H. 1989. Food webs. In: J. M. Cherrett (ed.), Ecological Concepts: The Contribution of Ecology to an Understanding of the Natural World. Blackwell Scientific, Oxford, UK. pp. 43–78.

    Google Scholar 

  23. Lawton, J. H. 1992. Feeble links in food webs. Nature 355: 19–20.

    Article  Google Scholar 

  24. Logofet, D. O. 1993. Matrices and Graphs-Stability Problems in Mathematical Ecology. CRC Press, Boca Raton.

    Google Scholar 

  25. MacArthur, J. W. 1976. Environmental fluctuations and species diversity. In: M. Cody and J. M. Diamond (eds.), Ecology of species and communities. Harvard University Press, Cambridge, USA. pp. 74–80.

    Google Scholar 

  26. Martinez, N. D. 1991. Artifacts or attributes? Effects of resolution on the Little Rock Lake food web. Ecol. Monog. 61: 367–392.

    Article  Google Scholar 

  27. Martinez, N. D. 1992. Constant connectance in community food webs. Amer. Nat. 139: 1208–1218.

    Article  Google Scholar 

  28. Martinez, N. D., B. A. Hawkins, H. A. Dawah and B. P. Feifarek. 1999. Effects of sampling effort on characterization of food web structure. Ecology 80: 1044–1055.

    Article  Google Scholar 

  29. May, R. M. 1972. Will a large complex system be stable? Nature 238: 413–414.

    Article  CAS  Google Scholar 

  30. May, R. M. 1974. How many species? Some mathematical aspects of the dynamics of populations. In: J. D. Cowan (ed.), Some Mathematical Problems in Biology. American Mathematical Society, Providence, USA. pp. 64–98.

    Google Scholar 

  31. Paine, R. T. 1980. Food webs: linkage, interaction strength and community infrastructure. J. Animal Ecol. 49: 667–685.

    Article  Google Scholar 

  32. Paine, R. T. 1988. On food webs: road maps of interactions or the grist for theoretical development? Ecology 69: 1648–1654.

    Article  Google Scholar 

  33. Paine, R. T. 1992. Food-web analysis through field measurements of per capita interaction strength. Nature 355: 73–75.

    Article  Google Scholar 

  34. Pimm, S. L. 1980. Bounds on food web connectance. Nature 284: 591.

    Article  Google Scholar 

  35. Pimm, S. L. 1982. Food Webs. Chapman and Hall, London, UK.

    Book  Google Scholar 

  36. Pimm, S. L. 1984. The complexity and stability of ecosystems. Nature 307: 321–326.

    Article  Google Scholar 

  37. Polis, G. A. 1991. Complex trophic interactions in deserts: an empirical critique of food web theory. Amer. Nat. 138: 123–155.

    Article  Google Scholar 

  38. Price, P. W. 1984. Insect Ecology. 2nd Edition. John Wiley & Sons. New York.

    Google Scholar 

  39. Rejmanek, M. and P. Stary. 1979. Connectance in real biotic communities and critical values for stability of model ecosystems. Nature 280: 311–313.

    Article  Google Scholar 

  40. Schoener, T. W. 1989. Food webs from the small to the large. Ecology 70: 1559–1589.

    Article  Google Scholar 

  41. Schoenly, K., R. A. Beaver and T. A. Heumier. 1991. On the trophic relations of insects: a food web approach. Amer. Nat. 137: 597–638.

    Article  Google Scholar 

  42. Schoenly, K. and J. E. Cohen. 1991. Temporal variation in food web structure: 16 empirical cases. Ecol. Monog. 61: 267–298.

    Article  Google Scholar 

  43. Warren, P. H. 1990. Variation in food web structure: the determinants of connectance. Amer. Nat. 136: 689–700.

    Article  Google Scholar 

  44. Warren, P. H. 1994. Making connections in food webs. Trends in Ecology and Evolution 9: 136–141.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  45. Williams, R. J. and N. D. Martinez. 2000. Simple rules yield complex food webs. Nature 404: 180–183.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  46. Winemiller, K. O. 1989. Must connectance decrease with species richness? Amer. Nat. 134: 960–968.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to J. E. Cohen.

Rights and permissions

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Cite this article

Chen, X., Cohen, J.E. Support of the hyperbolic connectance hypothesis by qualitative stability of model food webs. COMMUNITY ECOLOGY 1, 215–225 (2000). https://doi.org/10.1556/ComEc.1.2000.2.11

Download citation

Keyword

  • Local asymptotic stability
  • Lotka-Volterra cascade model
  • Qualitative global asymptotic stability