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.
Lotka-Volterra Cascade Model
Local Asymptotic Stability
Qualitative Global Asymptotic Stability
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.
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.
Briand, F. 1983. Environmental control of food web structure. Ecology 64: 253–263.
Briand, F. and J. E. Cohen. 1987. Environmental correlates of food chain length. Science 238: 956–960.
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.
Cohen, J. E. and F. Briand. 1984. Trophic links of community food webs. Proceedings of the Royal Society of London B 224: 421–448.
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.
Cohen, J. E., F. Briand and C. M. Newman. 1990a. Community Food Webs: Data And Theory. Springer-Verlag, Berlin.
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.
Cohen, J. E. and C. M. Newman. 1984. The stability of large random matrices and their products. Annals of Probability 12: 283–310.
Cohen, J. E. and C. M. Newman. 1985a. When will a large complex system be stable? J. theoret. Biol. 113: 153–156.
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.
Cohen, J. E. and C. M. Newman. 1988. Dynamic basis of food web organization. Ecology 69: 1655–1664.
DeAngelis, A. L. 1975. Stability and connectance in food web models. Ecology 56: 238–243.
Gardner, M. R., and W. R. Ashby. 1970. Connectance of large, dynamical (cybernetic) systems: critical value for stability. Nature 228: 784.
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.
Hall, S. J. and D. G. Raffaelli. 1993. Food webs: theory and reality. Adv. Ecol. Res. 24: 187–239.
Havens, K.E. 1992. Scale and structure in natural food webs. Science 257: 1107–1109.
Havens, K. E. 1997. Unique structural properties of pelagic food webs. Oikos 78: 75–80.
Krebs, J. R. and N. B. Davies. 1993. An Introduction to Behavioural Ecology. 3rd ed. Blackwell Science Ltd, Oxford.
Lawlor, L. R. 1978. A comment on randomly constructed ecosystem models. Amer. Nat. 112: 445–447.
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.
Lawton, J. H. 1992. Feeble links in food webs. Nature 355: 19–20.
Logofet, D. O. 1993. Matrices and Graphs-Stability Problems in Mathematical Ecology. CRC Press, Boca Raton.
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.
Martinez, N. D. 1991. Artifacts or attributes? Effects of resolution on the Little Rock Lake food web. Ecol. Monog. 61: 367–392.
Martinez, N. D. 1992. Constant connectance in community food webs. Amer. Nat. 139: 1208–1218.
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.
May, R. M. 1972. Will a large complex system be stable? Nature 238: 413–414.
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.
Paine, R. T. 1980. Food webs: linkage, interaction strength and community infrastructure. J. Animal Ecol. 49: 667–685.
Paine, R. T. 1988. On food webs: road maps of interactions or the grist for theoretical development? Ecology 69: 1648–1654.
Paine, R. T. 1992. Food-web analysis through field measurements of per capita interaction strength. Nature 355: 73–75.
Pimm, S. L. 1980. Bounds on food web connectance. Nature 284: 591.
Pimm, S. L. 1982. Food Webs. Chapman and Hall, London, UK.
Pimm, S. L. 1984. The complexity and stability of ecosystems. Nature 307: 321–326.
Polis, G. A. 1991. Complex trophic interactions in deserts: an empirical critique of food web theory. Amer. Nat. 138: 123–155.
Price, P. W. 1984. Insect Ecology. 2nd Edition. John Wiley & Sons. New York.
Rejmanek, M. and P. Stary. 1979. Connectance in real biotic communities and critical values for stability of model ecosystems. Nature 280: 311–313.
Schoener, T. W. 1989. Food webs from the small to the large. Ecology 70: 1559–1589.
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.
Schoenly, K. and J. E. Cohen. 1991. Temporal variation in food web structure: 16 empirical cases. Ecol. Monog. 61: 267–298.
Warren, P. H. 1990. Variation in food web structure: the determinants of connectance. Amer. Nat. 136: 689–700.
Warren, P. H. 1994. Making connections in food webs. Trends in Ecology and Evolution 9: 136–141.
Williams, R. J. and N. D. Martinez. 2000. Simple rules yield complex food webs. Nature 404: 180–183.
Winemiller, K. O. 1989. Must connectance decrease with species richness? Amer. Nat. 134: 960–968.
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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
- Local asymptotic stability
- Lotka-Volterra cascade model
- Qualitative global asymptotic stability