Bulletin of Mathematical Biology

, Volume 62, Issue 6, pp 1137–1162 | Cite as

Stoichiometry in producer-grazer systems: Linking energy flow with element cycling

  • Irakli Loladze
  • Yang Kuang
  • James J. Elser


All organisms are composed of multiple chemical elements such as carbon, nitrogen and phosphorus. While energy flow and element cycling are two fundamental and unifying principles in ecosystem theory, population models usually ignore the latter. Such models implicitly assume chemical homogeneity of all trophic levels by concentrating on a single constituent, generally an equivalent of energy. In this paper, we examine ramifications of an explicit assumption that both producer and grazer are composed of two essential elements: carbon and phosphorous. Using stoichiometric principles, we construct a two-dimensional Lotka-Volterra type model that incorporates chemical heterogeneity of the first two trophic levels of a food chain. The analysis shows that indirect competition between two populations for phosphorus can shift predator—prey interactions from a (+, −) type to an unusual (−, −) class. This leads to complex dynamics with multiple positive equilibria, where bistability and deterministic extinction of the grazer are possible. We derive simple graphical tests for the local stability of all equilibria and show that system dynamics are confined to a bounded region. Numerical simulations supported by qualitative analysis reveal that Rosenzweig’s paradox of enrichment holds only in the part of the phase plane where the grazer is energy limited; a new phenomenon, the paradox of energy enrichment, arises in the other part, where the grazer is phosphorus limited. A bifurcation diagram shows that energy enrichment of producer—grazer systems differs radically from nutrient enrichment. Hence, expressing producer—grazer interactions in stoichiometrically realistic terms reveals qualitatively new dynamical behavior.


Bifurcation Diagram Stable Equilibrium Producer Density Element Cycling Internal Equilibrium 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Agren, G. and E. Bosatta (1996). Theoretical Ecosystem Ecology: Understanding Element Cycles, NY: Cambridge University Press.Google Scholar
  2. Andersen, T. (1997). Pelagic Nutrient Cycles: Herbivores as Sources and Sinks, NY: Springer-Verlag.Google Scholar
  3. Andersen, T. and D. O. Hessen (1991). Carbon, nitrogen, and phosphorus content of freshwater zooplankton. Limnol. Oceanogr. 36, 807–814.CrossRefGoogle Scholar
  4. DeMott, W. R. (1998). Utilization of cyanobacterium and phosphorus-deficient green algae as a complementary resource by daphnids. Ecology 79, 2463–2481.CrossRefGoogle Scholar
  5. Droop, M. R. (1974). The nutrient status of algal cells in continuous culture. J. Mar. Biol. Assoc. UK 55, 825–855.Google Scholar
  6. Edelstein-Keshet, L. and M. D. Rausher (1989). The effects of inducible plant defenses on herbivore populations. Am. Nat. 133, 787–810.CrossRefGoogle Scholar
  7. Elser, J. J., T. H. Chrzanowski, R. W. Sterner and K. H. Mills (1998). Stoichiometric constraints on food web dynamics: a whole-lake experiment on the Canadian shield. Ecosystems 1, 120–136.CrossRefGoogle Scholar
  8. Elser, J. J., E. R. Marzolf and C. R. Goldman (1990). Can. J. Fisheries Aquatic Sci. 47, 1468–1477.CrossRefGoogle Scholar
  9. Elser, J. J. and J. Urabe (1999). The stoichiometry of consumer-driven nutrient recycling: theory, observations, and consequences. Ecology 80, 735–751.CrossRefGoogle Scholar
  10. Hagen, J. B. (1992). An Entangled Bank: The Origins of Ecosystem Ecology, New Brunswick, NJ: Rutgers.Google Scholar
  11. Hessen, D. O. and T. Andersen (1992). The algae—grazer interface: feedback mechanism linked to elemental ratios and nutrient cycling. Archiv Fuer Hydrobiologie Ergebnisse der Limnologie 35, 111–120.Google Scholar
  12. Huxel, G. R. (1999). On the influence of food quality in consumer—resource interactions. Ecology Lett. 2, 256–261.CrossRefGoogle Scholar
  13. Kooijman, S. A. L. M. (2000). Dynamic energy and mass budgets in biological systems, Cambridge, U.K: Cambridge University Press.Google Scholar
  14. Koppel, J., J. Huisman, R. Wal and H. Olff (1996). Patterns of herbivory along productivity gradient: and empirical and theoretical investigation. Ecology 77, 736–745.CrossRefGoogle Scholar
  15. Lindeman, R. L. (1942). The trophic-dynamic aspect of ecology. Ecology 23, 399–418.CrossRefGoogle Scholar
  16. Lotka, A. J. (1925). Elements of Physical Biology, Baltimore: Williams and Wilkins. Reprinted as Elements of Mathematical Biology (1956) New York: Dover.zbMATHGoogle Scholar
  17. McCann, K. S. (2000). The diversity-stability debate. Nature 405, 228–233.CrossRefGoogle Scholar
  18. Odum, E. P. (1959). Fundamentals of Ecology, Philadelphia: W.B. Saunders.Google Scholar
  19. Odum, E. P. (1968). Energy flow in ecosystems: a historical view. Am. Zoologist 8, 11–18.Google Scholar
  20. Odum, H. P. (1957). Trophic structure and productivity of Silver Springs. Ecol. Monographs 27, 55–112.CrossRefGoogle Scholar
  21. Odum, H. P. (1960). Ecological potential and analogue circuits for the ecosystem. Am. Scientist 48, 1–8.Google Scholar
  22. Reiners, W. A. (1986). Complementary models for ecosystems. Am. Nat. 127, 59–73.CrossRefGoogle Scholar
  23. Rosenzweig, M. L. (1971). Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171, 385–387.Google Scholar
  24. Schindler, D. W. (1977). Evolution of phosphorus limitation in lakes. Science 195, 260–262.Google Scholar
  25. Schwinning, S. and A. J. Parsons (1996). Analysis of the coexistence mechanisms for grasses and legumes in grazing systems. J. Ecology 84, 799–813.Google Scholar
  26. Sterner, R. W. (1990). The ratio of nitrogen to phosphorus resupplied by herbivores: zooplankton and the algal competitive arena. Am. Nat. 136, 209–229.CrossRefGoogle Scholar
  27. Sterner, R. W., J. Clasen, W. Lampert and T. Weisse (1998). Carbon: phosphorus stoichiometry and food chain production. Ecology Lett. 1, 146–150.CrossRefGoogle Scholar
  28. Sterner, R. W. and D. O. Hessen (1994). Algal nutrient limitation and the nutrient of aquatic herbivores. Ann. Rev. Ecol. Syst. 25, 1–29.CrossRefGoogle Scholar
  29. Tilman, D. (1982). Resource Competition and Community Structure, Princeton, NJ: Princeton University Press.Google Scholar
  30. Urabe, J. and R. W. Sterner (1996). Regulation of herbivore growth by the balance of light and nutrients. Proc. Natl. Acad. Sci. USA 93, 8465–8469.CrossRefGoogle Scholar
  31. White, T. C. R. (1993). The Inadequate Environment: Nitrogen and the Abundance of Animals, NY: Springer-Verlag.Google Scholar

Copyright information

© Society for Mathematical Biology 2000

Authors and Affiliations

  • Irakli Loladze
    • 1
  • Yang Kuang
    • 1
  • James J. Elser
    • 2
  1. 1.Department of MathematicsArizona State UniversityTempeUSA
  2. 2.Department of BiologyArizona State UniversityTempeUSA

Personalised recommendations