Average conditions for permanence and extinction in nonautonomous Gilpin–Ayala competition model

In this paper, we consider a general nonautonomous n  -species Gilpin–Ayala competitive system, which is more general and more realistic than classical Lotka–Volterra competition model. By means of Ahmad and Lazer's definitions of lower and upper averages of a function, we first give the average conditions for the permanence and global attractivity of the system. Next, for each r⩽nr⩽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-rn-r species are driven to extinction. Examples show the feasibility of the main results.