Abstract
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.
Similar content being viewed by others
References
Agren, G. and E. Bosatta (1996). Theoretical Ecosystem Ecology: Understanding Element Cycles, NY: Cambridge University Press.
Andersen, T. (1997). Pelagic Nutrient Cycles: Herbivores as Sources and Sinks, NY: Springer-Verlag.
Andersen, T. and D. O. Hessen (1991). Carbon, nitrogen, and phosphorus content of freshwater zooplankton. Limnol. Oceanogr. 36, 807–814.
DeMott, W. R. (1998). Utilization of cyanobacterium and phosphorus-deficient green algae as a complementary resource by daphnids. Ecology 79, 2463–2481.
Droop, M. R. (1974). The nutrient status of algal cells in continuous culture. J. Mar. Biol. Assoc. UK 55, 825–855.
Edelstein-Keshet, L. and M. D. Rausher (1989). The effects of inducible plant defenses on herbivore populations. Am. Nat. 133, 787–810.
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.
Elser, J. J., E. R. Marzolf and C. R. Goldman (1990). Can. J. Fisheries Aquatic Sci. 47, 1468–1477.
Elser, J. J. and J. Urabe (1999). The stoichiometry of consumer-driven nutrient recycling: theory, observations, and consequences. Ecology 80, 735–751.
Hagen, J. B. (1992). An Entangled Bank: The Origins of Ecosystem Ecology, New Brunswick, NJ: Rutgers.
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.
Huxel, G. R. (1999). On the influence of food quality in consumer—resource interactions. Ecology Lett. 2, 256–261.
Kooijman, S. A. L. M. (2000). Dynamic energy and mass budgets in biological systems, Cambridge, U.K: Cambridge University Press.
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.
Lindeman, R. L. (1942). The trophic-dynamic aspect of ecology. Ecology 23, 399–418.
Lotka, A. J. (1925). Elements of Physical Biology, Baltimore: Williams and Wilkins. Reprinted as Elements of Mathematical Biology (1956) New York: Dover.
McCann, K. S. (2000). The diversity-stability debate. Nature 405, 228–233.
Odum, E. P. (1959). Fundamentals of Ecology, Philadelphia: W.B. Saunders.
Odum, E. P. (1968). Energy flow in ecosystems: a historical view. Am. Zoologist 8, 11–18.
Odum, H. P. (1957). Trophic structure and productivity of Silver Springs. Ecol. Monographs 27, 55–112.
Odum, H. P. (1960). Ecological potential and analogue circuits for the ecosystem. Am. Scientist 48, 1–8.
Reiners, W. A. (1986). Complementary models for ecosystems. Am. Nat. 127, 59–73.
Rosenzweig, M. L. (1971). Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171, 385–387.
Schindler, D. W. (1977). Evolution of phosphorus limitation in lakes. Science 195, 260–262.
Schwinning, S. and A. J. Parsons (1996). Analysis of the coexistence mechanisms for grasses and legumes in grazing systems. J. Ecology 84, 799–813.
Sterner, R. W. (1990). The ratio of nitrogen to phosphorus resupplied by herbivores: zooplankton and the algal competitive arena. Am. Nat. 136, 209–229.
Sterner, R. W., J. Clasen, W. Lampert and T. Weisse (1998). Carbon: phosphorus stoichiometry and food chain production. Ecology Lett. 1, 146–150.
Sterner, R. W. and D. O. Hessen (1994). Algal nutrient limitation and the nutrient of aquatic herbivores. Ann. Rev. Ecol. Syst. 25, 1–29.
Tilman, D. (1982). Resource Competition and Community Structure, Princeton, NJ: Princeton University Press.
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.
White, T. C. R. (1993). The Inadequate Environment: Nitrogen and the Abundance of Animals, NY: Springer-Verlag.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Loladze, I., Kuang, Y. & Elser, J.J. Stoichiometry in producer-grazer systems: Linking energy flow with element cycling. Bull. Math. Biol. 62, 1137–1162 (2000). https://doi.org/10.1006/bulm.2000.0201
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1006/bulm.2000.0201