zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Permanence and extinction for dispersal population systems. (English) Zbl 1073.34052

The system of differential equations x ˙=f(t,x), x n , is said to be permanent if there exists a compact set Kint + n such that all solutions starting in int + n ultimately enter and remain in K.

The authors study the following predator-pray model in a patchy environment

x ˙ 1 =x 1 [b 1 (t)-a 1 (t)x 1 -yϕ(t,x 1 )]+ j=1 n (D 1j (t)x j -D j1 (t)x 1 ),x ˙ i =x i [b i (t)-a i (t)x i ]+ j=1 n (D ij (t)x j -D ji (t)x i ),i=2,...,n,y ˙=y[-d(t)+e(t)x 1 ϕ(t,x 1 )-f(t)y],

where x i denotes the species x in patch i; d(t)>0, e(t)>0, b i (t)>0, a i (t)>0, D ij (t)0 are continuous ω-periodic functions; D ij (t) is the dispersal coefficient of the species from patch j to patch i, D ii (t)0; the predator functional response x 1 ϕ(t,x 1 ) is bounded as x 1 , ϕ(t,x 1 )0, ϕ(t,x 1 )/x 1 0 and (x 1 ϕ(t,x 1 ))/x 1 0. Necessary and sufficient conditions for the permanence of the above system are presented. Sufficient conditions for the permanence of the single-species system in the absence of a predator (y=0) is also obtained. The approach is based on the well-known properties of the periodic logistic model z ˙=z(b(t)-a(t)z).

The biological interpretion of the main results is given.

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