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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
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 $p\in\operatorname{int}{{\mathbf R}^n_+}$ 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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  • 1. P. Brunovský, Controlling nonuniqueness of local invariant manifolds, J. Reine Angew. Math., 446 (1994), 115-135. MR 94k:58125
  • 2. G. Butler, R. Schmid and P. Waltman, Limiting the complexity of limit sets in self-regulating systems, J. Math. Anal. Appl., 147 (1990), 63-68. MR 91e:58152
  • 3. M. W. Hirsch, Systems of differential equations that are competitive or cooperative. III: competing species, Nonlinearity, 1 (1988), 51-71. MR 90d:58070
  • 4. J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics. (First published as The Theory of Evolution and Dynamical Systems.) Cambridge University Press, Cambridge, 1998. MR 99h:92027
  • 5. J. Hofbauer and J. W.-H. So, Multiple limit cycles for three-dimensional Lotka-Volterra equations, Appl. Math. Lett., 7 (1994), 65-70. MR 96g:34063
  • 6. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985. MR 87e:15001; Corrected reprint of 1985 original. MR 91i:15001
  • 7. M. D. LaMar and M. L. Zeeman, Csimplex: a Geomview module for visualizing the carrying simplex of a competitive Lotka-Volterra system, http://www.math.utsa.edu/zeeman, to appear.
  • 8. S. Levy, T. Munzner, M. Phillips et al., Geomview, http://www.geom.umn.edu, The Geometry Center, Minneapolis, MN, 1996.
  • 9. R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species., SIAM J. Appl. Math., 29 (1975), 243-253. MR 52:12853
  • 10. J. Mierczynski, The $C^1$ property of carrying simplices for a class of competitive systems of ODEs, J. Differential Equations, 111 (1994), 385-409. MR 95g:34066
  • 11. -, On smoothness of carrying simplices, Proc. Amer. Math. Soc., 127 (1999), 543-551. MR 99c:34086
  • 12. -, Smoothness of carrying simplices for three-dimensional competitive systems: a counterexample, Dynam. Contin. Discrete Impuls. Systems, 6 (1999), 147-154. MR 2000a:34095
  • 13. -, On peaks in carrying simplices, Colloq. Math., 81 (1999), 285-292. MR 2000j:37029
  • 14. M. Plank, Bi-Hamiltonian systems and Lotka-Volterra equations: a three-dimensional classification, Nonlinearity, 9 (1996), 887-896. MR 98d:58064
  • 15. -, On the dynamics of Lotka-Volterra equations having an invariant hyperplane, SIAM J. Appl. Math., 59 (1999), 1540-1551. MR 2001a:34011
  • 16. P. van den Driessche and M. L. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234. MR 99g:92026
  • 17. D. Xiao and W. Li, Limit cycles for the competitive three dimensional Lotka-Volterra system, J. Differential Equations, 164 (2000), 1-15. MR 2001d:34051
  • 18. E. C. Zeeman, Two limit cycles in three-dimensional competitive Lotka-Volterra systems, Preprint.
  • 19. E. C. Zeeman and M. L. Zeeman, On the convexity of carrying simplices in competitive Lotka-Volterra systems, in Differential equations, dynamical systems, and control science, Marcel Dekker, Lecture Notes in Pure and Appl. Math., 152 (1994), 353-364. MR 94h:34033
  • 20. -, An n-dimensional competitive Lotka-Volterra system is generically determined by the edges of its carrying simplex, Nonlinearity, to appear.
  • 21. M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems, 8 (1993), 189-217. MR 94j:34044
  • 22. -, Geometric methods in population dynamics, Comparison methods and stability theory (Waterloo, ON, 1993), Marcel Dekker, Lecture Notes in Pure and Appl. Math., 162 (1994), 339-347. CMP 94:17

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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