Vanishing components in autonomous competitive Lotka–Volterra systems

Sufficient conditions are found for an n-dimensional autonomous competitive Lotka–Volterra system to have a component vanishing in an exponential rate as t→∞. These conditions incorporate a typical known result in the literature as a particular case. Moreover, if the n-dimensional system degenerates asymptotically to an m-dimensional subsystem as t→∞, then, under these conditions on the subsystem, the property that the ith component of every solution of the subsystem vanishes in an exponential rate is also preserved for the n-dimensional system.