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

DOI: 10.1007/BF00276098

Cite this article as:
Takeuchi, Y. & Adachi, N. J. Math. Biology (1980) 10: 401. doi:10.1007/BF00276098


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

StabilityVolterra ecosystemsLinear complementarity theory

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