On smoothness of carrying simplices

Author:
Janusz Mierczynski

Journal:
Proc. Amer. Math. Soc. **127** (1999), 543-551

MSC (1991):
Primary 34C30, 34C35; Secondary 58F12, 92D40

DOI:
https://doi.org/10.1090/S0002-9939-99-04887-X

MathSciNet review:
1606000

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider dissipative strongly competitive systems of ordinary differential equations. It is known that for a wide class of such systems there exists an invariant attracting hypersurface , called the carrying simplex. In this note we give an amenable condition for to be a submanifold-with-corners. We also provide conditions, based on a recent work of M. Benaïm (*On invariant hypersurfaces of strongly monotone maps*, J. Differential Equations **136** (1997), 302-319), guaranteeing that is of class .

**1.**M. Benaïm,*On invariant hypersurfaces of strongly monotone maps*, J. Differential Equations**137**(1997), 302-319. MR**98d:58114****2.**M. Gobbino and M. Sardella,*On the connectedness of attractors for dynamical systems*, J. Differential Equations**133**(1997), 1-14. CMP**97:06****3.**J. K. Hale,*Asymptotic behavior of dissipative systems*, Math. Surveys Monogr.**25**, American Mathematical Society, Providence, R.I., 1988. MR**89g:58059****4.**M. W. Hirsch,*Systems of differential equations which are competitive or cooperative. III. Competing species*, Nonlinearity**1**(1988), 51-71. MR**90d:58070****5.**M. W. Hirsch, C. C. Pugh and M. Shub*Invariant manifolds*, Lecture Notes in Math.**583**, Springer, Berlin-New York, 1977. MR**58:18595****6.**J. Hofbauer,*An index theorem for dissipative semiflows*, Rocky Mountain J. Math.**20**(1990), 1017-1031. MR**92b:58203****7.**J. Hofbauer and K. Sigmund,*The theory of evolution and dynamical systems*, London Mathematical Society Student Texts**7**, Cambridge University Press, Cambridge, 1988. MR**91h:92019****8.**R. Mañé,*Ergodic theory and differentiable dynamics*(translated from the Portuguese by S. Levy), Ergeb. Math. Grenzgeb. (3)**8**, Springer, Berlin-New York, 1987. MR**88c:58040****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. Mierczy\'{n}ski,*The property of carrying simplices for a class of competitive systems of ODEs*, J. Differential Equations**111**(1994), 385-409. MR**95g:34066****11.**S. J. Schreiber,*Expansion rates and Lyapunov exponents*, Discrete Contin. Dynam. Systems**3**(1997), 433-438. MR**98c:58096****12.**H. L. Smith,*Monotone dynamical systems. An introduction to the theory of competitive and cooperative systems*, Math. Surveys Monogr.**41**, Amer. Math. Soc., Providence, R.I., 1995. MR**96c:34002****13.**I. Tere\v{s}\v{c}ák,*Dynamics of smooth strongly monotone discrete-time dynamical systems*, preprint.**14.**A. Tineo,*An iterative scheme for the -competing species problem*, J. Differential Equations**116**(1995), 1-15. MR**95m:92023****15.**M. L. Zeeman,*Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems*, Dynam. Stability Systems**8**(1993), 189-217. MR**94j:34044**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
34C30,
34C35,
58F12,
92D40

Retrieve articles in all journals with MSC (1991): 34C30, 34C35, 58F12, 92D40

Additional Information

**Janusz Mierczynski**

Affiliation:
Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiań- skiego 27, PL-50-370 Wrocław, Poland

Email:
mierczyn@banach.im.pwr.wroc.pl

DOI:
https://doi.org/10.1090/S0002-9939-99-04887-X

Received by editor(s):
June 2, 1997

Additional Notes:
The author’s research was supported by KBN grant 2 P03A 076 08 (1995–97).

Communicated by:
Hal L. Smith

Article copyright:
© Copyright 1999
American Mathematical Society