From local to global behavior in competitive Lotka-Volterra systems
HTML articles powered by AMS MathViewer
- by E. C. Zeeman and M. L. Zeeman PDF
- Trans. Amer. Math. Soc. 355 (2003), 713-734 Request permission
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.References
- Pavol Brunovský, Controlling nonuniqueness of local invariant manifolds, J. Reine Angew. Math. 446 (1994), 115–135. MR 1256150, DOI 10.1515/crll.1994.446.115
- Geoffrey Butler, Rudolf Schmid, and Paul Waltman, Limiting the complexity of limit sets in self-regulating systems, J. Math. Anal. Appl. 147 (1990), no. 1, 63–68. MR 1044686, DOI 10.1016/0022-247X(90)90384-R
- Morris W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species, Nonlinearity 1 (1988), no. 1, 51–71. MR 928948, DOI 10.1088/0951-7715/1/1/003
- Josef Hofbauer and Karl Sigmund, Evolutionary games and population dynamics, Cambridge University Press, Cambridge, 1998. MR 1635735, DOI 10.1017/CBO9781139173179
- J. Hofbauer and J. W.-H. So, Multiple limit cycles for three-dimensional Lotka-Volterra equations, Appl. Math. Lett. 7 (1994), no. 6, 65–70. MR 1340732, DOI 10.1016/0893-9659(94)90095-7
- Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
- 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.
- S. Levy, T. Munzner, M. Phillips et al., Geomview, http://www.geom.umn.edu, The Geometry Center, Minneapolis, MN, 1996.
- Robert M. May and Warren J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29 (1975), no. 2, 243–253. MR 392035, DOI 10.1137/0129022
- Janusz Mierczyński, The $C^1$ property of carrying simplices for a class of competitive systems of ODEs, J. Differential Equations 111 (1994), no. 2, 385–409. MR 1284419, DOI 10.1006/jdeq.1994.1087
- Janusz Mierczyński, On smoothness of carrying simplices, Proc. Amer. Math. Soc. 127 (1999), no. 2, 543–551. MR 1606000, DOI 10.1090/S0002-9939-99-04887-X
- Janusz Mierczyński, Smoothness of carrying simplices for three-dimensional competitive systems: a counterexample, Dynam. Contin. Discrete Impuls. Systems 6 (1999), no. 1, 147–154. MR 1679762
- Janusz Mierczyński, On peaks in carrying simplices, Colloq. Math. 81 (1999), no. 2, 285–292. MR 1715352, DOI 10.4064/cm-81-2-285-292
- Manfred Plank, Bi-Hamiltonian systems and Lotka-Volterra equations: a three-dimensional classification, Nonlinearity 9 (1996), no. 4, 887–896. MR 1399477, DOI 10.1088/0951-7715/9/4/004
- Manfred Plank, On the dynamics of Lotka-Volterra equations having an invariant hyperplane, SIAM J. Appl. Math. 59 (1999), no. 5, 1540–1551. MR 1699027, DOI 10.1137/S0036139997319463
- P. van den Driessche and M. L. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math. 58 (1998), no. 1, 227–234. MR 1610080, DOI 10.1137/S0036139995294767
- Dongmei Xiao and Wenxia Li, Limit cycles for the competitive three dimensional Lotka-Volterra system, J. Differential Equations 164 (2000), no. 1, 1–15. MR 1761415, DOI 10.1006/jdeq.1999.3729
- E. C. Zeeman, Two limit cycles in three-dimensional competitive Lotka-Volterra systems, Preprint.
- E. C. Zeeman and M. L. Zeeman, On the convexity of carrying simplices in competitive Lotka-Volterra systems, Differential equations, dynamical systems, and control science, Lecture Notes in Pure and Appl. Math., vol. 152, Dekker, New York, 1994, pp. 353–364. MR 1243211
- —, An n-dimensional competitive Lotka-Volterra system is generically determined by the edges of its carrying simplex, Nonlinearity, to appear.
- M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems 8 (1993), no. 3, 189–217. MR 1246002, DOI 10.1080/02681119308806158
- —, 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.
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
- 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.
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1932722