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On a Smale theorem and nonhomogeneous equilibria in cooperative systems


Author: German A. Enciso
Journal: Proc. Amer. Math. Soc. 136 (2008), 2901-2909
MSC (2000): Primary 34C12, 35K57
DOI: https://doi.org/10.1090/S0002-9939-08-09346-5
Published electronically: April 11, 2008
MathSciNet review: 2399057
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Abstract: A standard result by Smale states that $ n$ dimensional strongly cooperative dynamical systems can have arbitrary dynamics when restricted to unordered invariant hyperspaces. In this paper this result is extended to the case when all solutions of the strongly cooperative system are bounded and converge towards one of only two equilibria outside of the hyperplane.

An application is given in the context of strongly cooperative systems of reaction diffusion equations. It is shown that such a system can have a continuum of spatially inhomogeneous steady states, even when all solutions of the underlying reaction system converge to one of only three equilibria.


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Additional Information

German A. Enciso
Affiliation: Mathematical Biosciences Institute, 231 W. 18th Avenue, Columbus, Ohio 43215
Address at time of publication: Department of Systems Biology, Harvard Medical School, 200 Longwood Avenue, Boston, Massachusetts 02115
Email: genciso@mbi.osu.edu

DOI: https://doi.org/10.1090/S0002-9939-08-09346-5
Keywords: Monotone systems, reaction diffusion systems
Received by editor(s): June 15, 2007
Published electronically: April 11, 2008
Additional Notes: This material is based upon work supported by the National Science Foundation under Agreement No. 0112050 and by The Ohio State University.
Communicated by: Walter Craig
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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