Extinction and survival in competitive Lotka-Volterra systems with constant coefficients and infinite delays

The qualitative properties of a nonautonomous competitive Lotka-Volterra system with infinite delays are studied.By using a result of matrix theory and the fluctuation lemma, we establish a series of easily verifiable algebraic conditions on the coefficients and the kernel, which are sufficient to ensure the survival and the extinction of a determined number of species. The surviving part is stabilized around a globally stable critical point of a subsystem of the system under study. These conditions also guarantee the asymptotic behavior of the system.