On a Smale Theorem and Nonhomogeneous Equilibria in Cooperative Systems

TL;DR

This paper extends Smale's theorem to bounded strongly cooperative systems with limited equilibria, demonstrating the existence of multiple steady states in reaction-diffusion systems despite convergence to few equilibria.

Contribution

It generalizes Smale's theorem to bounded systems with two equilibria, and shows such systems can have many spatially inhomogeneous steady states.

Findings

Bounded strongly cooperative systems can have multiple steady states.

Reaction-diffusion systems can exhibit a continuum of inhomogeneous equilibria.

Systems converge to one of only three equilibria despite complex steady states.

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.