*(English)*Zbl 1119.34038

Let the functions ${b}_{i}\left(t\right)$ and ${a}_{ij}\left(t\right)$ be continuous on $[c,\infty )$ and bounded above and below by strictly positive constants and let ${d}_{ij}$ be positive numbers; $i,j=1,2,...,n$. 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)]

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

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 $r<n$, 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 stability of ODE |

92D25 | Population dynamics (general) |

34C60 | Qualitative investigation and simulation of models (ODE) |