Bulletin of Mathematical Biology

, Volume 45, Issue 1, pp 103–116 | Cite as

Non-linear growth mechanics—I. Volterrahamilton systems

  • P. L. Antonelli
  • B. H. Voorhees
Article

Abstract

In this, the first of a series of papers on stochastic and deterministic non-linear allometric growth models, a deterministic model is proposed which generalizes the widely applicable classical linear model of Huxley and Needham. There aren types of producers, each type depositing a product which accumulates monotonically in the environment. Producers interact via a mass action law satisfying an optimality condition. Coefficients may be interpreted as competition between the various producer types in the usual Volterra sense. An ideal coral reef is studied in which then species of coral polyps lay down aragonite calcium carbonate in building the reef framework. This deterministic model predicts that younger reefs are strongly unstable relative to initial species abundance, while older reefs grow in the classical sense of Huxley and Needham, asymptotically, as time goes to infinity.

Keywords

Aragonite Sectional Curvature Record Variable Bump Function Coral Polyp 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature

  1. Alexander, R. M. 1979.The Invertebrates. Cambridge: Cambridge University Press.Google Scholar
  2. Gangolli, R. 1964. “On the Construction of Certain Diffusions on a Differentiable Manifold”.Z. Wahrscheinlichkeitstheorie 2, 406–419.MATHMathSciNetCrossRefGoogle Scholar
  3. Huxley, J. 1972.Problems of Relative Growth, 2nd Edn. Tiptree: Dover Press.Google Scholar
  4. Jackson, J., and L. Buss. 1975. “Allelopathy and Spatial Competion Among Coral Reef Invertebrates”.Proc. natn. Acad. Sci. U.S.A. 72, 5160–5163.CrossRefGoogle Scholar
  5. Maragos, J. E. 1978. “Coral Growth: Geometrical Relationship.” InCoral Reefs: Research Methods. Eds. D. R. Stoddard and R. E. Johannes, pp. 543–550, UNESCO Monographs in Occanographic Methodology.Google Scholar
  6. Needham, J. 1934. “Heterogony, a Chemical Groundplan for Development”.Biol. Rev. 9, 79–109.Google Scholar
  7. Rhunau, E. 1977. “Spaces of Locally Constant Connection.” Masters Thesis, Univ of Alberta, pp. 94.Google Scholar
  8. Stearn, C. W. and J. P. Scoffin. 1977. “Carbonate Budget of Fringing Reef, Barbados.” In Proceedings, Third International Coral Reef Symposium.Google Scholar
  9. Volterra, A. 1936. “Principles de Biologie Mathematique”.Acta biotheor. III, 1–36.Google Scholar
  10. Whittaker, R. H. and P. P. Feeny. 1971. “Allelochemics: Chemical Interactions between Species”.Science, N. Y.171, 757–768.Google Scholar

Copyright information

© Society for Mathematical Biology 1983

Authors and Affiliations

  • P. L. Antonelli
    • 1
  • B. H. Voorhees
    • 1
  1. 1.Department of MathematicsThe University of AlbertaEdmontonCanada

Personalised recommendations