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The existence of globally stable equilibria of ecosystems of the generalized Volterra type
 Yasuhiro Takeuchi,
 Norihiko Adachi
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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.
 Case, T. J., Casten, R. G. (1979) Global stability and multiple domains of attractions in ecological systems. Amer. Naturalist 113: pp. 705714
 Cottle, R. W., Dantzig, G. B. (1968) Complementary pivot theory of mathematical programming. Linear Algebra and Its Applic. 1: pp. 103125
 Gilpin, M. E., Case, T. J. (1976) Multiple domains of attraction in competitive communities. Nature 261: pp. 4042
 Goel, N. S., Maitra, S. C., Montroll, E. W. (1971) On the Volterra and other nonlinear models of interacting populations. Rev. Modern Phys. 43: pp. 231276
 Goh, B. S. (1978) Sector stability of a complex ecosystem model. Math. Biosciences 40: pp. 157166
 Krikorian, N. (1979) The Volterra model for three species predatorprey systems: Boundedness and stability. J. Math. Biol. 7: pp. 117132
 MacArthur, R. (1970) Species packing and competitive equilibrium for many species. Theoret. Population Biol. 1: pp. 111
 May, R. M. (1973) Stability and complexity in model ecosystems. Princeton University Press, Princeton
 Maybee, J., Quirk, J. (1969) Qualitative problems in matrix theory. SIAM Rev. 11: pp. 3051
 Murty, K. G. (1972) On the number of solutions to the complementarity problem and spanning properties of complementarity cones. Linear Algebra and Its Applic. 5: pp. 65108
 Nikaido, H. (1968) Convex structure and economic theory. Academic Press, New York
 Takeuchi, Y., Adachi, N., Tokumaru, H. (1978) The stability of generalized Volterra equations. J. Math. Anal. Appl. 62: pp. 453473
 Takeuchi, Y., Adachi, N., Tokumaru, H. (1978) Global stability of ecosystems of the generalized Volterra type. Math. Biosciences 42: pp. 119136
 Panne, C. (1975) Methods for linear and quadratic programming. American Elsevier P. C., New York
 Title
 The existence of globally stable equilibria of ecosystems of the generalized Volterra type
 Journal

Journal of Mathematical Biology
Volume 10, Issue 4 , pp 401415
 Cover Date
 19801201
 DOI
 10.1007/BF00276098
 Print ISSN
 03036812
 Online ISSN
 14321416
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Stability
 Volterra ecosystems
 Linear complementarity theory
 Authors

 Yasuhiro Takeuchi ^{(1)}
 Norihiko Adachi ^{(2)}
 Author Affiliations

 1. Department of Applied Mathematics, Faculty of Engineering, Shizuoka University, 432, Hamamatsu, Japan
 2. Department of Applied Mathematics and Physics, Faculty of Engineering, Kyoto University, 606, Kyoto, Japan