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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 $$[c,\infty )$$ and bounded above and below by strictly positive constants and let $$d_{ij}$$ be positive numbers; $$i,j=1,2,\ldots ,n$$. 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)] $x'_{i}(t) = x_{i}(t) \left\{ b_{i}(t)-\sum ^{n}_{j=1} a_{ij}(t) (x_{j}(t)) ^{d_{ij}} \right\} . \eqno{(1)}$ The system (1) is called permanent if for any positive solution $$X(t)$$ with components $$x_{i}(t)$$, there exist positive constants $$\lambda _{i}$$, $$k_{i}$$, $$T$$ such that for $$t\geq T$$ one has $$\lambda _{i}\leq x_{i}(t)\leq k_{i}$$. It is called globally attractive if the components of any two positive solutions $$X(t)$$, $$Y(t)$$ satisfy $\lim _{t\rightarrow \infty } | x_{i}(t)-y_{i}(t)| = 0 .$ 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 $$x_{i}(t)$$, $$i=1,2,\ldots ,r$$, 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:
 34D05 Asymptotic properties of solutions to ordinary differential equations 92D25 Population dynamics (general) 34C60 Qualitative investigation and simulation of ordinary differential equation models
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##### References:
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