# Linearization and invariant manifolds on the carrying simplex for competitive maps

## TL;DR

This paper demonstrates that the local dynamics of interior fixed points on the carrying simplex for competitive maps can be linearized and studied using invariant foliations, despite the non-smooth nature of the manifold.

## Contribution

It proves the topological conjugacy of the map to its pseudo-unstable manifold near interior fixed points and constructs stable and unstable manifolds on the carrying simplex.

## Key findings

- Linearization techniques are applicable on the carrying simplex near interior fixed points.
- The paper constructs stable and unstable manifolds on the carrying simplex.
- Results provide partial answers to the smoothness problem of the carrying simplex.

## Abstract

A folklore result due to M.W. Hirsch states that most competitive maps admit a carrying simplex, i.e., an invariant hypersurface which attracts all nontrivial orbits. The common approach in the study of these maps is to focus on the dynamical behavior on the carrying simplex. However, this manifold is normally non-smooth. Therefore, not every tool coming from Differential Geometry can be applied. In particular, the classical Grobman-Hartman theorem can not be used on the carrying simplex. In this paper, we prove that the restriction of the map to the carrying simplex in a neighborhood of an interior fixed point is topologically conjugate to the restriction of the map to its pseudo-unstable manifold by an invariant foliation. This implies that the linearization techniques are applicable for studying the local dynamics of the interior fixed points on the carrying simplex. We further construct the stable and unstable manifolds on the carrying simplex. On the other hand, our results also give partial responses to Hirsch's problem regarding the smoothness of the carrying simplex. We discuss some applications in classical models of population dynamics. Although we focus on monotone maps, many results of the paper can be applied to maps that admit a non-smooth center-manifold.