Stability of Equilibriums and Bifurcation Analysis of Two-dimensional Autonomous Competitive Lotka-Volterra Dynamical 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 [0, +∞)2. Necessary and sufficient conditions are determined so that the observed dynamical system has no equilibriums in (0, +∞)2. All results are presented in five tables and five figures with appropriate ecological interpretation. We also show that four transcritical bifurcations occur in the observed dynamical system if it is analyzed on R2.