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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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