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

DOI:
https://doi.org/10.1090/S0002-9947-02-03103-3

Published electronically:
October 9, 2002

MathSciNet review:
1932722

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Abstract | References | Similar Articles | Additional Information

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 and the carrying simplex of the system lies to one side of its tangent hyperplane at , 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

Email:
zeeman@math.utsa.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-03103-3

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