Average conditions for permanence and extinction in nonautonomous Lotka-Volterra system

An n species nonautonomous competitive Lotka-Volterra system is considered in this paper. The average conditions on the coefficients are given to guarantee that all but one of the species are driven to extinction. The generalization for the result is presented, that is, for each r?n the average conditions on the coefficients are provided to guarantee that r of the species in the system are permanent while the remaining n−r are driven to extinction. It is shown that these average conditions are improvement of those of Ahmad and Montes de Oca Appl. Math. Comput. 90 (1998) 155-166] and Montes de Oca and Zeeman Proc. Amer. Math. Soc. 124 (1996) 3677-3687] and J. Math. Anal. Appl. 192 (1995) 360-370].