Planar Kolmogorov systems with infinitely many singular points at infinity

TL;DR

This paper classifies the global phase portraits of a five-parameter family of planar Kolmogorov systems derived from 3D Lotka-Volterra models, revealing 13 distinct topological behaviors including infinitely many singular points at infinity.

Contribution

It provides a complete topological classification of the phase portraits for these systems, including their behavior at infinity, which was previously not understood.

Findings

13 topologically distinct global phase portraits identified

Classification includes systems with infinitely many singular points at infinity

Phase portraits described in the Poincaré disc for comprehensive understanding

Abstract

We classify the global dynamics of the five-parameter family of planar Kolmogorov systems \begin{equation*} \begin{split} \dot{y}&=y \left( b_0+ b_1 y z + b_2 y + b_3 z\right), \dot{z}&=z\left( c_0 + b_1 y z + b_2 y + b_3 z\right), \end{split} \end{equation*} which is obtained from the Lotka-Volterra systems of dimension three. These systems have infinitely many singular points at inifnity. We give the topological classification of their phase portraits in the Poincar\'e disc, so we can describe the dynamics of these systems near infinity. We prove that these systems have 13 topologically distinct global phase portraits.