Skip to main content

Nonlinear Food Web Models and Their Responses to Increased Basal Productivity

  • Chapter
Food Webs

Abstract

We study here general food web models that include non-Lotka-Volterra trophic interactions. We set two logical requirements that any such model must satisfy: (Cl) invariance under aggregation of identical species, and (C2) mathematical separation of disconnected subwebs. We show that these logical requirements are not always satisfied by previously published models. We give the mathematical expressions corresponding to generalizations of three nonlinear functional responses: Rolling (prey-dependent), DeAngelis-Beddington, and Holling-like ratio-dependent. We then address the two questions of species aggregation and of responses to basal production.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  • Abrams, P. A. 1993. Effect of increased productivity on the abundances of trophic levels. American Naturalist 141: 351–371.

    Article  Google Scholar 

  • Arditi, R. and H. R. Akçakaya. 1990. Underestimation of mutual interference of predators. 0ecologia 83: 358–361.

    Google Scholar 

  • Arditi, R. and L. R. Ginzburg. 1989. Coupling in predator-prey dynamics: Ratio-dependence. Journal of Theoretical Biology 139: 311–326.

    Article  Google Scholar 

  • Arditi, R., L. R. Ginzburg, and H. R. Akçakaya. 1991. Variation in plankton densities among lakes: A case for ratio-dependent predation models. American Naturalist 138: 1287–1296.

    Article  Google Scholar 

  • Beddington, J. R. 1975. Mutual interference between parasites or predators and its effect on searching efficiency. Journal of Animal Ecology 44: 331–340.

    Article  Google Scholar 

  • Berryman, A. A. 1992. The origins and evolution of predator-prey theory. Ecology 73: 1530–1535.

    Article  Google Scholar 

  • Berryman, A. A., J. Michalski, A. P. Gutierrez, and R. Arditi. 1995. Logistic theory of food web dynamics. Ecology 76: 336–343.

    Article  Google Scholar 

  • Cyr, H. and M. L. Pace. 1993. Magnitude and patterns of herbivory in aquatic and terrestrial ecosystems. Nature 361: 148–150.

    Article  Google Scholar 

  • DeAngelis, D. L., R. A. Goldstein, and R. V. O’Neill. 1975. A model for trophic interaction. Ecology 56: 881–892.

    Article  Google Scholar 

  • Getz, W. M. 1984. Population dynamics: A per capita resource approach. Journal of Theoretical Biology 108: 623–643.

    Article  Google Scholar 

  • Getz, W. M. 1991. A unified approach to multi-species modeling. Natural Resources Modeling 5: 393–421.

    Google Scholar 

  • Ginzburg, L. R. and H. R. Akçakaya. 1992. Consequences of ratio-dependent predation for steady-state properties of ecosystems. Ecology 73: 1536–1543.

    Article  Google Scholar 

  • Gomatam, J. 1974. A new model for interacting populations. I. Two-species systems. Bulletin of Mathematical Biology 36: 347–353.

    Google Scholar 

  • Hairston, N. G., F. E. Smith, and L. B. Slobodkin, 1960. Community structure, population control and competition. American Naturalist 94: 421–425.

    Article  Google Scholar 

  • Hassell, M. P. and D. J. Rogers. 1972. Insect parasite responses in the development of population models. Journal of Animal Ecology 41: 661–676.

    Article  Google Scholar 

  • Hassell, M. P. and G. C. Varley. 1969. New inductive population model for insect parasites and its bearing on biological control. Nature 223: 1133–1137.

    Article  PubMed  CAS  Google Scholar 

  • Hastings, A. and T. Powell, 1991. Chaos in a three species food chain. Ecology 72: 896–903.

    Article  Google Scholar 

  • Holling, C. S. 1959. The components of predation as revealed by a study of small mammal predation of the European pine sawfly. Canadian Entomologist 91: 293–320.

    Article  Google Scholar 

  • Ivlev, V. S. 1961. Experimental Ecology of the Feeding of Fishes. Yale University Press, New Haven, CT.

    Google Scholar 

  • Jost, J. L., J. F. Drake, H. M. Tsuchiya, and A. G. Fredrickson. 1973. Microbial food chains and food webs. Journal of Theoretical Biology 41: 461–484.

    Article  PubMed  CAS  Google Scholar 

  • Kretzschmar, M., R. M. Nisbet, and E. McCauley. 1993. Zooplankton grazing on competing algal populations. Theoretical Population Biology 44: 32–66.

    Article  Google Scholar 

  • Leibold, M. A. and H. M. Wilbur. 1992. Interactions between food-web structure and nutrients on pond organisms. Nature 360: 341–343.

    Article  Google Scholar 

  • León, J. A. and D. B. Tumpson. 1975. Competition between two species for two complementary or substitutable resources. Journal of Theoretical Biology 50: 185–201.

    Article  PubMed  Google Scholar 

  • May, R. M. 1973. Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton, NJ.

    Google Scholar 

  • McCauley, E., W. W. Murdoch, and S. Watson. 1987. Simple models and variation in plankton densities among lakes. American Naturalist 132: 383–403.

    Google Scholar 

  • McNaughton, S. J., M. Oesterheld, D. A. Frank, and K. J. Williams. 1989. Ecosystem-level patterns of primary productivity and herbivory in terrestrial habitats. Nature 341: 101–105.

    Article  Google Scholar 

  • Michalski, J. and R. Arditi. 1995. Food web structure at equilibrium and far from it: is it the same? Proceedings of the Royal Society of London B 259: 217–222.

    Google Scholar 

  • Moen, J. and L. Oksanen. 1991. Ecosystem trends. Nature 353: 510.

    Article  Google Scholar 

  • Oksanen, L., S. D. Fretwell, J. Arruda, and P. Niemelä. 1981. Exploitation ecosystems in gradients of primary productivity. American Naturalist 118: 240–261.

    Google Scholar 

  • Persson, L., G. Andersson, S. F. Hamrin, and L. Johansson. 1988. Predator regulation and primary production along the productivity gradient of temperate lake ecosystems. In Complex Interactions in Lake Communities, ed. S. R. Carpenter, pp. 45–65. Springer-Verlag, New York.

    Chapter  Google Scholar 

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

    Book  Google Scholar 

  • Pimm, S. L. 1991. The Balance of Nature? Ecological Issues in the Conservation of Species and Communities. University of Chicago Press, Chicago.

    Google Scholar 

  • Pimm, S. L. 1992. Frog ponds and ocean iron.Nature 360:298–299.

    Article  Google Scholar 

  • Pimm, S. L., J. H. Lawton, and J. E. Cohen. 1991. Food web patterns and their consequences. Nature 350: 669–674.

    Article  Google Scholar 

  • Polis, G. A. and R. D. Holt. 1992. Intraguild predation: The dynamics of complex trophic interactions. Trends in Ecology and Evolution 7: 151–154.

    Article  PubMed  CAS  Google Scholar 

  • Real, L. A. 1977. The kinetics of functional re- sponse. American Naturalist 111: 289–300.

    Article  Google Scholar 

  • Rogers, D. J. and M. P. Hassell. 1974. General models for insect parasite and predator searching behaviour: Interference. Journal of Animal Ecology 43:239–253.

    Article  Google Scholar 

  • Rosenzweig, M. L. 1971. Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time. Science 171: 385–387.

    Article  PubMed  CAS  Google Scholar 

  • Slobodkin, L. B., F. E. Smith, and N. G. Hairston. 1967. Regulation in terrestrial ecosystems, and the implied balance of nature. American Naturalist 101: 109–124.

    Article  Google Scholar 

  • Strebel, D. E. and N. S. Goel. 1973. On the isocline methods for analyzing prey-predator interactions. Journal of Theoretical Biology 39: 211–234.

    Article  PubMed  CAS  Google Scholar 

  • Takahashi, F. 1964. Reproduction curve with two equilibrium points: A consideration on the fluctuation of insect population. Researches in Population Ecology (Kyoto) 6:28–36.

    Google Scholar 

  • Tostowaryk, W. 1972. The effect of prey defence on the functional response of Podisus modestus (Hemiptera: Pentatomidae) to densities of the sawflies Neodiprion swainei and N. pratti banksianae (Hymenoptera: Neodiprionidae). Canadian Entomologist 104:61–69.

    Article  Google Scholar 

  • Vandermeer, J. 1993. Loose coupling of predator-prey cycles: Entrainment, chaos, and intermittency in the classic MacArthur consumer-resource equations. American Naturalist 141: 687–716.

    Article  Google Scholar 

  • Watt, K. E. F. 1959. A mathematical model for the effect of densities of attacked and attacking species on the number attacked. Canadian Entomologist 91:129–144.

    Article  Google Scholar 

  • Yodzis, P. 1988. The indeterminacy of ecological interactions as perceived through perturbation experiments. Ecology 69: 508–515.

    Article  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Arditi, R., Michalski, J. (1996). Nonlinear Food Web Models and Their Responses to Increased Basal Productivity. In: Polis, G.A., Winemiller, K.O. (eds) Food Webs. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-7007-3_12

Download citation

  • DOI: https://doi.org/10.1007/978-1-4615-7007-3_12

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4615-7009-7

  • Online ISBN: 978-1-4615-7007-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics