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

Full-text PDF Free Access

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.

**1.**Pavol Brunovský,*Controlling nonuniqueness of local invariant manifolds*, J. Reine Angew. Math.**446**(1994), 115–135. MR**1256150**, 10.1515/crll.1994.446.115**2.**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**, 10.1016/0022-247X(90)90384-R**3.**Morris W. Hirsch,*Systems of differential equations which are competitive or cooperative. III. Competing species*, Nonlinearity**1**(1988), no. 1, 51–71. MR**928948****4.**Josef Hofbauer and Karl Sigmund,*Evolutionary games and population dynamics*, Cambridge University Press, Cambridge, 1998. MR**1635735****5.**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**, 10.1016/0893-9659(94)90095-7**6.**Roger A. Horn and Charles R. Johnson,*Matrix analysis*, Cambridge University Press, Cambridge, 1985. MR**832183**

Roger A. Horn and Charles R. Johnson,*Matrix analysis*, Cambridge University Press, Cambridge, 1990. Corrected reprint of the 1985 original. MR**1084815****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.**Robert M. May and Warren J. Leonard,*Nonlinear aspects of competition between three species*, SIAM J. Appl. Math.**29**(1975), no. 2, 243–253. Special issue on mathematics and the social and biological sciences. MR**0392035****10.**Janusz Mierczyński,*The 𝐶¹ property of carrying simplices for a class of competitive systems of ODEs*, J. Differential Equations**111**(1994), no. 2, 385–409. MR**1284419**, 10.1006/jdeq.1994.1087**11.**Janusz Mierczyński,*On smoothness of carrying simplices*, Proc. Amer. Math. Soc.**127**(1999), no. 2, 543–551. MR**1606000**, 10.1090/S0002-9939-99-04887-X**12.**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****13.**Janusz Mierczyński,*On peaks in carrying simplices*, Colloq. Math.**81**(1999), no. 2, 285–292. MR**1715352****14.**Manfred Plank,*Bi-Hamiltonian systems and Lotka-Volterra equations: a three-dimensional classification*, Nonlinearity**9**(1996), no. 4, 887–896. MR**1399477**, 10.1088/0951-7715/9/4/004**15.**Manfred Plank,*On the dynamics of Lotka-Volterra equations having an invariant hyperplane*, SIAM J. Appl. Math.**59**(1999), no. 5, 1540–1551 (electronic). MR**1699027**, 10.1137/S0036139997319463**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), no. 1, 227–234. MR**1610080**, 10.1137/S0036139995294767**17.**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**, 10.1006/jdeq.1999.3729**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*, 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****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), no. 3, 189–217. MR**1246002**, 10.1080/02681119308806158**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