Journal of Mathematical Biology

, Volume 10, Issue 4, pp 401–415

The existence of globally stable equilibria of ecosystems of the generalized Volterra type

  • Yasuhiro Takeuchi
  • Norihiko Adachi


In this paper, global asymptotic stability of ecosystems of the generalized Volterra type
$$dx_i /dt = x_i \left( {b_{i - } \mathop \sum \limits_{j = 1}^n a_{ij} x_j } \right),{\text{ }}i = 1,...,n,$$
is investigated. We obtain the conditions for the existence of a nonnegative and stable equilibrium point of the system by applying a result of linear complementarity theory.

The results of this paper show that there exists a class of systems that do not have multiple domains of attractions. This class is defined in terms of the species interactions alone, and does not involve carrying capacities or species net birth rates.

Key words

Stability Volterra ecosystems Linear complementarity theory 


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  1. 1.
    Case, T. J., Casten, R. G.: Global stability and multiple domains of attractions in ecological systems. Amer. Naturalist 113, 705–714 (1979)Google Scholar
  2. 2.
    Cottle, R. W., Dantzig, G. B.: Complementary pivot theory of mathematical programming. Linear Algebra and Its Applic. 1, 103–125 (1968)Google Scholar
  3. 3.
    Gilpin, M. E., Case, T. J.: Multiple domains of attraction in competitive communities. Nature 261, 40–42 (1976)Google Scholar
  4. 4.
    Goel, N. S., Maitra, S. C., Montroll, E. W.: On the Volterra and other nonlinear models of interacting populations. Rev. Modern Phys. 43, 231–276 (1971)Google Scholar
  5. 5.
    Goh, B. S.: Sector stability of a complex ecosystem model. Math. Biosciences 40, 157–166 (1978)Google Scholar
  6. 6.
    Krikorian, N.: The Volterra model for three species predator-prey systems: Boundedness and stability. J. Math. Biol. 7, 117–132 (1979)Google Scholar
  7. 7.
    MacArthur, R.: Species packing and competitive equilibrium for many species. Theoret. Population Biol. 1, 1–11 (1970)Google Scholar
  8. 8.
    May, R. M.: Stability and complexity in model ecosystems. Princeton: Princeton University Press 1973Google Scholar
  9. 9.
    Maybee, J., Quirk, J.: Qualitative problems in matrix theory. SIAM Rev. 11, 30–51 (1969)Google Scholar
  10. 10.
    Murty, K. G.: On the number of solutions to the complementarity problem and spanning properties of complementarity cones. Linear Algebra and Its Applic. 5, 65–108 (1972)Google Scholar
  11. 11.
    Nikaido, H.: Convex structure and economic theory. New York: Academic Press 1968Google Scholar
  12. 12.
    Takeuchi, Y., Adachi, N., Tokumaru, H.: The stability of generalized Volterra equations. J. Math. Anal. Appl. 62, 453–473 (1978)Google Scholar
  13. 13.
    Takeuchi, Y., Adachi, N., Tokumaru, H.: Global stability of ecosystems of the generalized Volterra type. Math. Biosciences 42, 119–136 (1978)Google Scholar
  14. 14.
    Van de Panne, C.: Methods for linear and quadratic programming. New York: American Elsevier P. C. 1975Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Yasuhiro Takeuchi
    • 1
  • Norihiko Adachi
    • 2
  1. 1.Department of Applied Mathematics, Faculty of EngineeringShizuoka UniversityHamamatsuJapan
  2. 2.Department of Applied Mathematics and Physics, Faculty of EngineeringKyoto UniversityKyotoJapan

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