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Average conditions for permanence and extinction in nonautonomous Lotka-Volterra system. (English) Zbl 1066.34050

Authors’ abstract: An n-species nonautonomous competitive Lotka-Volterra system is considered. The average conditions on the coefficients are given to guarantee that all but one species are driven to extinction. The generalization for the result is presented, i.e., for each rn the average conditions on the coefficients are presented to guarantee that r of the species of the system are permanent while the remaining n-r species are driven to extinction.

It is shown that these average conditions are improvement of those of S. Ahmad and F. Montes de Oca [Appl. Math. Comp. 90, 155–166 (1998; Zbl 0906.92024)] and F. Montes de Oca and M. L. Zeeman [Proc. Am. Math.Soc. 124, 3677–3687 (1996; Zbl 0866.34029) and J. Math. Anal. Appl. 192, 360–370 (1995; Zbl 0830.34039)].

34D05Asymptotic stability of ODE
92D25Population dynamics (general)
34C29Averaging method