zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

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