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.