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Average conditions for permanence and extinction in nonautonomous Gilpin-Ayala competition model. (English) Zbl 1119.34038

Let the functions b i (t) and a ij (t) be continuous on [c,) and bounded above and below by strictly positive constants and let d ij be positive numbers; i,j=1,2,...,n. The author studies the n-species M. E. Gilpin, F. J. Ayala competitive system [Proc. Natl. Acad. Sci. USA 70, 3590-3593 (1973; Zbl 0272.92016)]

x i ' (t)=x i (t)b i (t)- j=1 n a ij (t)(x j (t)) d ij ·(1)

The system (1) is called permanent if for any positive solution X(t) with components x i (t), there exist positive constants λ i , k i , T such that for tT one has λ i x i (t)k i . It is called globally attractive if the components of any two positive solutions X(t), Y(t) satisfy

lim t |x i (t)-y i (t)|=0·

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 x i (t), i=1,2,...,r, with r<n, to the effect that r of the species in the system are permanent whereas the remaining n-r ones are exposed to extinction. Inasmuch as system (1) is for d ij =1 reduced to the Lotka-Volterra one, the results obtained here generalize those of several authors for the later system.

MSC:
34D05Asymptotic stability of ODE
92D25Population dynamics (general)
34C60Qualitative investigation and simulation of models (ODE)