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)
Long-time behaviour of a stochastic prey–predator model. (English) Zbl 1075.60539

A stochastic version

dX t =(αX t -βX t Y t -μX t 2 )dt+σX t dW t ,dY t =(-γY t +δX t Y t -νY t 2 )dt+ρY t dW t (1)

of the Lotka-Volterra system is studied, where α, β, γ, δ, μ, ν, ρ and σ are positive constants, and W is a standard Wiener process. By setting X t =exp(ξ t ) and Y t =exp(η t ) the equations (1) are transformed to

dξ t =(α-σ 2 /2-μe ξ t -βe η t )dt+σdW t ,dη t =(-γ-ρ 2 /2+δe ξ t -νe η t )dt+ρdW t ·(2)

Let us set c 1 =α-σ 2 /2, c 2 =γ+ρ 2 /2. Let (ξ,η) be an arbitrary solution to (2). It is proven that if c 1 >0 and μc 2 <δc 1 , then there exists a unique invariant probability measure m * for (2) and the distribution of (ξ t ,η t ) converges to m * as t in the total variation norm. If c 1 >0 and μc 2 >δc 1 , then lim t η t =- almost surely, while the law of ξ t converges weakly to a measure having density Cexp(2c 1 σ -2 x-2μσ -2 e x ). Finally, if c 1 <0, then both ξ t and η t converge to - as t almost surely. In the course of proofs, it is shown that the laws of both (ξ t ,η t ) and m * have density with respect to two-dimensional Lebesgue measure, hence recent results on long-time behaviour of integral Markov semigroups [see e.g.K.Pichór and R.Rudnicki, J. Math. Anal. Appl. 249, 668–685 (2000; Zbl 0965.47026)] may be applied.

60H10Stochastic ordinary differential equations
47D07Markov semigroups of linear operators and applications to diffusion processes
60J60Diffusion processes
92D25Population dynamics (general)