Global attraction and repulsion of a heteroclinic limit cycle in three dimensional Kolmogorov maps

TL;DR

This paper establishes criteria for the existence of globally attracting or repelling heteroclinic limit cycles in three-dimensional Kolmogorov maps, using modified carrying simplex theory, and illustrates these with a concrete competitive model.

Contribution

It introduces new criteria for heteroclinic limit cycles in 3D Kolmogorov maps based on modified carrying simplex theory, with practical application to a specific competitive model.

Findings

Criteria for global attraction of heteroclinic cycles

Criteria for global repulsion of heteroclinic cycles

Application to a concrete competitive model

Abstract

There is a recent development in the carrying simplex theory for competitive maps: under some weaker conditions a map has a modified carrying simplex (one of the author's latest publications). In this paper, as one of the applications of the modified carrying simplex theory, a criterion is established for a three dimensional Kolmogorov map to have a globally repelling (attracting) heteroclinic limit cycle. As a concrete example, a discrete competitive model is investigated to illustrate the above criteria for global repulsion (attraction) of a hetericlinic limit cycle.