Existence of the carrying simplex for a retrotone map

TL;DR

This paper introduces a new dynamical approach to analyze the existence of carrying simplices in retrotone maps, using radial representations and metric convergence techniques.

Contribution

It develops a novel method employing radial representations and metric convergence to establish the existence of carrying simplices in retrotone maps.

Findings

Established Kuratowski convergence of radial representations

Proved the existence of a unique Lipschitz radial representation

Provided a new framework for analyzing unordered attracting manifolds

Abstract

We present a dynamical approach to the study of unordered, attracting manifolds of retrotone maps commonly known as carrying simplices. Our approach is novel in that it uses the radial representation of unordered manifolds over the probability simplex coupled with distances between these manifolds measured by way of the Harnack and Hausdorff metrics. We establish Kuratowski convergence of radial representations of unordered manifolds to a unique function which then provides the locally Lipschitz radial representation of the carrying simplex.