Abstract
In this paper, global asymptotic stability of ecosystems of the generalized Volterra type
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.
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Case, T. J., Casten, R. G.: Global stability and multiple domains of attractions in ecological systems. Amer. Naturalist 113, 705–714 (1979)
Cottle, R. W., Dantzig, G. B.: Complementary pivot theory of mathematical programming. Linear Algebra and Its Applic. 1, 103–125 (1968)
Gilpin, M. E., Case, T. J.: Multiple domains of attraction in competitive communities. Nature 261, 40–42 (1976)
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)
Goh, B. S.: Sector stability of a complex ecosystem model. Math. Biosciences 40, 157–166 (1978)
Krikorian, N.: The Volterra model for three species predator-prey systems: Boundedness and stability. J. Math. Biol. 7, 117–132 (1979)
MacArthur, R.: Species packing and competitive equilibrium for many species. Theoret. Population Biol. 1, 1–11 (1970)
May, R. M.: Stability and complexity in model ecosystems. Princeton: Princeton University Press 1973
Maybee, J., Quirk, J.: Qualitative problems in matrix theory. SIAM Rev. 11, 30–51 (1969)
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)
Nikaido, H.: Convex structure and economic theory. New York: Academic Press 1968
Takeuchi, Y., Adachi, N., Tokumaru, H.: The stability of generalized Volterra equations. J. Math. Anal. Appl. 62, 453–473 (1978)
Takeuchi, Y., Adachi, N., Tokumaru, H.: Global stability of ecosystems of the generalized Volterra type. Math. Biosciences 42, 119–136 (1978)
Van de Panne, C.: Methods for linear and quadratic programming. New York: American Elsevier P. C. 1975
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Takeuchi, Y., Adachi, N. The existence of globally stable equilibria of ecosystems of the generalized Volterra type. J. Math. Biology 10, 401–415 (1980). https://doi.org/10.1007/BF00276098
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DOI: https://doi.org/10.1007/BF00276098