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

Let the functions ${b}_{i}\left(t\right)$ and ${a}_{ij}\left(t\right)$ be continuous on $\left[c,\infty \right)$ 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)]

${x}_{i}^{\text{'}}\left(t\right)={x}_{i}\left(t\right)\left\{{b}_{i}\left(t\right)-\sum _{j=1}^{n}{a}_{ij}\left(t\right){\left({x}_{j}\left(t\right)\right)}^{{d}_{ij}}\right\}·\phantom{\rule{2.em}{0ex}}\left(1\right)$

The system (1) is called permanent if for any positive solution $X\left(t\right)$ with components ${x}_{i}\left(t\right)$, there exist positive constants ${\lambda }_{i}$, ${k}_{i}$, $T$ such that for $t\ge T$ one has ${\lambda }_{i}\le {x}_{i}\left(t\right)\le {k}_{i}$. It is called globally attractive if the components of any two positive solutions $X\left(t\right)$, $Y\left(t\right)$ satisfy

$\underset{t\to \infty }{lim}|{x}_{i}\left(t\right)-{y}_{i}\left(t\right)|=0·$

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 ${x}_{i}\left(t\right)$, $i=1,2,...,r$, with $r, 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 stability of ODE 92D25 Population dynamics (general) 34C60 Qualitative investigation and simulation of models (ODE)