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On a Smale Theorem and Nonhomogeneous Equilibria in Cooperative Systems

German A. Enciso
Proceedings of the American Mathematical Society
Vol. 136, No. 8 (Aug., 2008), pp. 2901-2909
Stable URL: http://www.jstor.org/stable/20535495
Page Count: 9
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
On a Smale Theorem and Nonhomogeneous Equilibria in Cooperative Systems
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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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