On smoothness of carrying simplices
Author:
Janusz Mierczynski
Journal:
Proc. Amer. Math. Soc. 127 (1999), 543551
MSC (1991):
Primary 34C30, 34C35; Secondary 58F12, 92D40
MathSciNet review:
1606000
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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 submanifoldwithcorners. We also provide conditions, based on a recent work of M. Benaïm (On invariant hypersurfaces of strongly monotone maps, J. Differential Equations 136 (1997), 302319), guaranteeing that is of class .
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 J. K. Hale, Asymptotic behavior of dissipative systems, Math. Surveys Monogr. 25, American Mathematical Society, Providence, R.I., 1988. MR 89g:58059
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 M. W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species, Nonlinearity 1 (1988), 5171. MR 90d:58070
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 M. W. Hirsch, C. C. Pugh and M. Shub Invariant manifolds, Lecture Notes in Math. 583, Springer, BerlinNew York, 1977. MR 58:18595
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 J. Hofbauer, An index theorem for dissipative semiflows, Rocky Mountain J. Math. 20 (1990), 10171031. 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
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 R. Mañé, Ergodic theory and differentiable dynamics (translated from the Portuguese by S. Levy), Ergeb. Math. Grenzgeb. (3) 8, Springer, BerlinNew York, 1987. MR 88c:58040
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 R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29 (1975), 243253. MR 52:12853
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 J. Mierczy\'{n}ski, The property of carrying simplices for a class of competitive systems of ODEs, J. Differential Equations 111 (1994), 385409. MR 95g:34066
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 S. J. Schreiber, Expansion rates and Lyapunov exponents, Discrete Contin. Dynam. Systems 3 (1997), 433438. MR 98c:58096
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 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
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 I. Tere\v{s}\v{c}ák, Dynamics of smooth strongly monotone discretetime dynamical systems, preprint.
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 A. Tineo, An iterative scheme for the competing species problem, J. Differential Equations 116 (1995), 115. MR 95m:92023
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 M. L. Zeeman, Hopf bifurcations in competitive threedimensional LotkaVolterra systems, Dynam. Stability Systems 8 (1993), 189217. MR 94j:34044
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Additional Information
Janusz Mierczynski
Affiliation:
Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiań skiego 27, PL50370 Wrocław, Poland
Email:
mierczyn@banach.im.pwr.wroc.pl
DOI:
http://dx.doi.org/10.1090/S000299399904887X
PII:
S 00029939(99)04887X
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
