Stochastic Lotka-Volterra Competitive Systems with Variable Delay

In this paper we reveal that the environmental noise will not only suppress a potential population explosion in the stochastic Lotka-Volterra competitive systems with variable delay, but also make the solutions to be stochastically ultimately bounded. To reveal these interesting facts, we stochastically perturb the Lotka-Volterra competitive systems with variable delay ẋ(t)=diag(x1(t),...,xn(t))[b+Ax(t−δ(t))] into the Ito form dx(t)=diag(x1(t),...,xn(t))[b+Ax(t−δ(t))]dt+σx(t)dw(t) and show that although the solution to the original delay systems may explode to infinity in a finite time, with probability one that of the associated stochastic delay systems do not. We also show that the stochastic systems will be stochastically ultimately bounded without any additional conditions on the matrix A.