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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A standard result by Smale states that 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.