Bulletin of Mathematical Biology

, Volume 62, Issue 6, pp 1137–1162

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


  • Irakli Loladze
    • Department of MathematicsArizona State University
  • Yang Kuang
    • Department of MathematicsArizona State University
  • James J. Elser
    • Department of BiologyArizona State University

DOI: 10.1006/bulm.2000.0201

Cite this article as:
Loladze, I., Kuang, Y. & Elser, J.J. Bull. Math. Biol. (2000) 62: 1137. doi:10.1006/bulm.2000.0201


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.

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© Society for Mathematical Biology 2000