Stability of equilibriums and bifurcation analysis of two-dimensional autonomous competitive Lotka-Volterra dynamical system

TL;DR

This paper thoroughly examines the stability and bifurcations of a two-dimensional competitive Lotka-Volterra system, providing conditions for equilibrium stability and identifying bifurcation types with detailed visual data.

Contribution

It offers new necessary and sufficient conditions for equilibrium stability and the occurrence of bifurcations in the system, including comprehensive tables and figures.

Findings

Conditions for asymptotic stability of equilibriums

Identification of no interior equilibriums under certain conditions

Detection of four transcritical bifurcations in the system

Abstract

A detailed analysis of the stability of equilibriums and bifurcations of the two-dimensional autonomous competitive Lotka-Volterra dynamical system is performed. Necessary and sufficient conditions are determined for equilibriums (without the origin) to be asymptotically stable or unstable on $\left[0, +\infty\right)^2$. Necessary and sufficient conditions are determined so that the observed dynamical system has no equilibriums in $\left(0, +\infty\right)^2 $. All results are presented in five tables and five figures. We also found that four transcritical bifurcations occur in the observed dynamical system if it is analyzed on $\mathrm{R}^2$.