The $C^1$ property of convex carrying simplices for three-dimensional competitive maps

TL;DR

This paper proves that convex carrying simplices in three-dimensional competitive maps are smoothly embedded $C^1$ submanifolds-with-corners within the non-negative octant, enhancing understanding of their geometric structure.

Contribution

It establishes the $C^1$ regularity and neat embedding of convex carrying simplices in three-dimensional competitive maps, a significant geometric property.

Findings

Convex carrying simplices are $C^1$ submanifolds-with-corners.

These simplices are neatly embedded in the non-negative octant.

The result applies specifically to three-dimensional competitive maps.

Abstract

It is proved that a convex carrying simplex for a three-dimensional competitive map is a $C^1$ submanifold-with-corners neatly embedded in the non-negative octant.