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

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 ${[0,+\mathrm{\infty })}^2$. Necessary and sufficient conditions are determined so that the observed dynamical system has no equilibriums in ${(0,+\mathrm{\infty })}^2$. 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 ${\mathrm{R}}^2$.