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Transactions of the American Mathematical Society

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From local to global behavior in competitive Lotka-Volterra systems

Authors: E. C. Zeeman and M. L. Zeeman
Journal: Trans. Amer. Math. Soc. 355 (2003), 713-734
MSC (2000): Primary 37N25, 92D25, 34C12, 34D23
Published electronically: October 9, 2002
MathSciNet review: 1932722
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Abstract: In this paper we exploit the linear, quadratic, monotone and geometric structures of competitive Lotka-Volterra systems of arbitrary dimension to give geometric, algebraic and computational hypotheses for ruling out non-trivial recurrence. We thus deduce the global dynamics of a system from its local dynamics.

The geometric hypotheses rely on the introduction of a split Liapunov function. We show that if a system has a fixed point $p\in\operatorname{int}{{\mathbf R}^n_+}$ and the carrying simplex of the system lies to one side of its tangent hyperplane at $p$, then there is no nontrivial recurrence, and the global dynamics are known. We translate the geometric hypotheses into algebraic hypotheses in terms of the definiteness of a certain quadratic function on the tangent hyperplane. Finally, we derive a computational algorithm for checking the algebraic hypotheses, and we compare this algorithm with the classical Volterra-Liapunov stability theorem for Lotka-Volterra systems.

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Additional Information

E. C. Zeeman
Affiliation: Hertford College, Oxford, 0X1 3BW, England

M. L. Zeeman
Affiliation: Department of Applied Mathematics, The University of Texas at San Antonio, San Antonio, Texas 78249-0664

Keywords: Carrying simplex, split Liapunov function, ruling out recurrence, Volterra-Liapunov
Received by editor(s): June 18, 2001
Published electronically: October 9, 2002
Additional Notes: This research was supported in part by NSF grant DMS-9404621, The University of Texas at San Antonio Office of Research Development, The Geometry Center, and The University of Michigan Mathematics Department.
Article copyright: © Copyright 2002 American Mathematical Society