Dynamics of competitive Lotka-Volterra systems that can be projected to a line

It is known that for a three-species autonomous competitive Lotka-Volterra system where axial fixed points are local attractors, almost all trajectories converge to the axial fixed points. For an n-species autonomous competitive Lotka-Volterra system, we prove that almost all trajectories converge to the axial fixed points if the axial fixed points are local attractors and the system can be radially projected to a line.