## Abstract and Applied Analysis

### Dynamical Behavior of a Stochastic Food-Chain System with Beddington-DeAngelis Functional Response

#### Abstract

We investigate a stochastic Food-Chain System $dx(t)=[{r}_{1}(t)-{a}_{11}(t)x-({a}_{12}(t)y/(1+{\beta }_{1}(t)x+{\gamma }_{1}(t)y))]$ $xdt+{\sigma }_{1}(t)xd{B}_{1}(t)$, $dy(t)=[{r}_{2}(t)-{a}_{21}(t)y+({a}_{22}(t)x/(1+{\beta }_{1}(t)x+{\gamma }_{1}(t)y))-({a}_{23}(t)z/(1+{\beta }_{2}(t)y+{\gamma }_{2}(t)z))]$ $ydt+{\sigma }_{2}(t)yd{B}_{2}(t)$, $dz(t)=[-{r}_{3}(t)+({a}_{31}(t)y/(1+{\beta }_{2}(t)y+{\gamma }_{2}(t)z))-{a}_{32}(t)z]zdt+{\sigma }_{3}(t)zd{B}_{3}(t)$, where ${B}_{i}(t)$, $i$ = $1,2,3,$ is a standard Brownian motion. Firstly, the existence, the uniqueness, and the positivity of the solution are proved. Secondly, the stochastically ultimate boundedness of the system is investigated. Thirdly, the boundedness of moments and upper-growth rate of the solution are obtained. Then the global attractivity of the system is discussed. Finally, the main results are illustrated by several examples.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 426702, 11 pages.

Dates
First available in Project Euclid: 3 October 2014

https://projecteuclid.org/euclid.aaa/1412361153

Digital Object Identifier
doi:10.1155/2014/426702

Mathematical Reviews number (MathSciNet)
MR3224311

Zentralblatt MATH identifier
07022371

#### Citation

Ge, Yanming; Song, Zigen. Dynamical Behavior of a Stochastic Food-Chain System with Beddington-DeAngelis Functional Response. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 426702, 11 pages. doi:10.1155/2014/426702. https://projecteuclid.org/euclid.aaa/1412361153

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