Extinction in competitive Lotka-Volterra systems

It is well known that for the two species autonomous competitive Lotka-Volterra model with no fixed point in the open positive quadrant, one of the species is driven to extinction, whilst the other population stabilises at its own carrying capacity. In this paper we prove a generalisation of this result to arbitrary finite dimension. That is, for the n-species autonomous competitive Lotka-Volterra model, we exhibit simple algebraic criteria on the parameters which guarantee that all but one of the species is driven to extinction, whilst the one remaining population stabilises at its own carrying capacity.