On smoothness of carrying simplices

TL;DR

This paper investigates conditions under which the carrying simplex in dissipative strongly competitive systems is a smooth manifold, providing criteria for its differentiability class based on recent theoretical developments.

Contribution

It introduces an amenable condition for the carrying simplex to be a $C^{1}$ submanifold-with-corners and extends criteria for higher smoothness classes using recent monotone map theory.

Findings

Established a condition for $C^{1}$ smoothness of the carrying simplex.

Provided criteria for the carrying simplex to be of class $C^{k+1}$.

Connected the smoothness of invariant hypersurfaces to recent monotone map results.

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. Bena\"{\i}m "On invariant hypersurfaces of strongly monotone maps", J. Differential Equations 137 (1997), 302-319, guaranteeing that $\Sigma$ is of class $C^{k+1}$.