Global dynamics of a family of 3D Lotka-Volterra Systems

TL;DR

This paper investigates the global dynamics of a family of 3D Lotka-Volterra systems within a bounded region, revealing simple interior behavior and complex boundary bifurcations, notably the focus-center-focus bifurcation.

Contribution

It provides a comprehensive analysis of the boundary limit sets and identifies a new type of global bifurcation in 3D Lotka-Volterra systems.

Findings

Interior orbits are either periodic or move between boundaries

Complete boundary limit set analysis elucidates bifurcations

Identification of focus-center-focus bifurcation as a key global bifurcation

Abstract

In this paper we analyze the flow of a family of three dimensional Lotka-Volterra systems restricted to an invariant and bounded region. The behaviour of the flow in the interior of this region is simple: either every orbit is a periodic orbit or they move from one boundary to another. Nevertheless the complete study of the limit sets in the boundary allows to understand the bifurcations which take place in the region as a global bifurcation that we denote by focus--center--focus bifurcation.