Smoothness of unordered curves in two-dimensional strongly competitive systems

Janusz Mierczyński

ArticleOriginal scientific text

Title

Smoothness of unordered curves in two-dimensional strongly competitive systems

Authors ¹

Affiliations

Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

Abstract

It is known that in two-dimensional systems of ODEs of the form

{\dot{x}}^{i} = x^{if}^i (x)

with

\frac{\partial f^{i}}{\partial x^{j}} < 0

(strongly competitive systems), boundaries of the basins of repulsion of equilibria consist of invariant Lipschitz curves, unordered with respect to the coordinatewise (partial) order. We prove that such curves are in fact of class

C^{1}

.

Keywords

ENG

strongly competitive system of ordinary differential equations, invariant manifold, d-curve, Lotka-Volterra system

M. Benaïm, On invariant hypersurfaces of strongly monotone maps, J. Differential Equations 137 (1997), 302-319.
P. Brunovský, Controlling nonuniqueness of local invariant manifolds, J. Reine Angew. Math. 446 (1994), 115-135.
P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.
M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets, SIAM J. Math. Anal. 13 (1982), 167-179.
M. W. Hirsch, Systems of differential equations that are competitive or cooperative. II. Convergence almost everywhere, ibid. 16 (1985), 423-439.
M. W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species, Nonlinearity 1 (1988), 51-71.
J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, London Math. Soc. Stud. Texts 7, Cambridge Univ. Press, Cambridge, 1988.
M. S. Holtz, The topological classification of two dimensional cooperative and competitive systems, Ph.D. dissertation, Univ. of California, Berkeley, 1987.
A. Kelley, The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations 3 (1967), 546-570; reprinted as Appendix C in the book: R. Abraham and J. W. Robbin, Transversal Mappings and Flows, Benjamin, New York, 1967, 134-154.
J. Mierczyński, The $C^{1}$ property of carrying simplices for a class of competitive systems of ODEs, J. Differential Equations 111 (1994), 385-409.
J. Mierczyński, On smoothness of carrying simplices, Proc. Amer. Math. Soc. 127 (1999), 543-551.
J. Mierczyński, Smoothness of carrying simplices for three-dimensional competitive systems: A counterexample, Dynam. Contin. Discrete Impuls. Systems, in press.
J. Palis and F. Takens, Topological equivalence of normally hyperbolic dynamical systems, Topology 16 (1977), 335-345.
E. Seneta, Non-negative Matrices and Markov Chains, 2nd ed., Springer Ser. Statist., Springer, New York, 1981.
P. Takáč, Convergence to equilibrium on invariant d-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl. 148 (1990), 223-244.
P. Takáč, Domains of attraction of generic ω-limit sets for strongly monotone discrete-time semigroups, J. Reine Angew. Math. 423 (1992), 101-173.
I. Tereščák, Dynamics of $C^{1}$ smooth strongly monotone discrete-time dynamical systems, preprint.
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, Lecture Notes in Pure and Appl. Math. 152, Dekker, New York, 1994, 353-364.

Title

Smoothness of unordered curves in two-dimensional strongly competitive systems

Affiliations

Abstract

Keywords

Bibliography