# 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)
Stochastic delay Lotka–Volterra model. (English) Zbl 1043.92034

Summary: We reveal in this paper that the environmental noise will not only suppress a potential population explosion in the stochastic delay Lotka–Volterra model but will also make the solutions to be stochastically ultimately bounded. To reveal these interesting facts, we stochastically perturb the delay Lotka-Volterra model

$\stackrel{˙}{x}\left(t\right)=\text{diag}\left({x}_{1}\left(t\right),\cdots ,{x}_{n}\left(t\right)\right)\left[b+Ax\left(t-\tau \right)\right]$

into the Itô form

$dx\left(t\right)=\text{diag}\left({x}_{1}\left(t\right),\cdots ,{x}_{n}\left(t\right)\right)\left[\left(b+Ax\left(t-\tau \right)\right)dt+\sigma x\left(t\right)dw\left(t\right)\right],$

and show that although the solution to the original delay equation may explode to infinity in finite time, with probability one that of the associated stochastic delay equation does not. We also show that the solution of the stochastic equation will be stochastically ultimately bounded without any additional condition on the matrix $A$.

##### MSC:
 92D40 Ecology 60H30 Applications of stochastic analysis 92D25 Population dynamics (general) 60H10 Stochastic ordinary differential equations