Phase portraits of a family of Kolmogorov systems depending on six parameters

TL;DR

This paper classifies the phase portraits of a family of three-dimensional Kolmogorov systems, derived from Lotka-Volterra models with specific invariants, depending on six parameters, providing a comprehensive understanding of their dynamical behaviors.

Contribution

It provides a complete classification of phase portraits for a specific family of Kolmogorov systems with six parameters, extending previous analyses of similar dynamical systems.

Findings

Complete phase portrait classification in the Poincaré disc.

Identification of parameter conditions for different dynamical behaviors.

Insights into the bifurcation structure of the systems.

Abstract

Consider a general $3$-dimensional Lotka-Volterra system with a rational first integral of degree two of the form $H=x^i y^j z^k$. The restriction of this Lotka-Volterra system to each surface $H(x,y,z)=h$ varying $h\in \mathbb{R}$ provide Kolmogorov systems. With the additional assumption that they have a Darboux invariant of the form $x^\ell y^m e^{st}$ they reduce to the Kolmogorov systems \begin{equation*} \begin{split} \dot{x}&=x \left( a_0- \mu (c_1 x + c_2 z^2 + c_3 z)\right),\\ \dot{z}&=z\left( c_0+ c_1 x + c_2 z^2 + c_3 z\right). \end{split} \end{equation*} In this paper we classify the phase portraits in the Poincar\'e disc of all these Kolmogorov systems which depend on six parameters.