Let the functions and be continuous on and bounded above and below by strictly positive constants and let be positive numbers; . The author studies the -species M. E. Gilpin, F. J. Ayala competitive system [Proc. Natl. Acad. Sci. USA 70, 3590-3593 (1973; Zbl 0272.92016)]
The system (1) is called permanent if for any positive solution with components , there exist positive constants , , such that for one has . It is called globally attractive if the components of any two positive solutions , satisfy
Under some “average conditions” as introduced by S. Ahmad and A. C. Lazer [Nonlinear Anal., Theory Methods Appl. 40, No. 1–8(A), 37–49 (2000; Zbl 0955.34041)] the author proves the permanency and global attractiveness of the system (1). Further, similar results are obtained for the subsets , , with , to the effect that of the species in the system are permanent whereas the remaining ones are exposed to extinction. Inasmuch as system (1) is for reduced to the Lotka-Volterra one, the results obtained here generalize those of several authors for the later system.