Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

On smoothness of carrying simplices

Author(s): Janusz Mierczynski
Journal: Proc. Amer. Math. Soc. 127 (1999), 543-551.
MSC (1991): Primary 34C30, 34C35; Secondary 58F12, 92D40
MathSciNet review: 1606000
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Abstract: We consider dissipative strongly competitive systems $\dot{x}_{i}=x_{i}f_{i}(x)$ of ordinary differential equations. It is known that for a wide class of such systems there exists an invariant attracting hypersurface $\Sigma$ , called the carrying simplex. In this note we give an amenable condition for $\Sigma$ to be a $C^{1}$ 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 $\Sigma$ is of class $C^{k+1}$ .

References:

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 Equations133 (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 $C^{1}$ 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 $N$ -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

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