On smoothness of carrying simplices

Mierczyński, Janusz

doi:10.1090/S0002-9939-99-04887-X

On smoothness of carrying simplices
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Proc. Amer. Math. Soc. 127 (1999), 543-551 Request permission

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 $C^{k+1}$.

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