Stochastic Kolmogorov-type population dynamics with variable delay

In general, population systems are often subject to environmental noise. To examine whether the presence of such noise affects population systems significantly, we stochastically perturb the delay Kolmogorov-type system dot x(t)=diag(x1(t),, xn(t))f(x(t), x(t-(t))) into the stochastic delay differential equation dx(t)=diag(x1(t),, xn(t))[f(x(t), x(t-(t)))dt+g(x(t), x(t-(t)))dw(t)]. Under the traditional diagonal dominance condition, we study the existence and uniqueness of the global positive solution of this stochastic system, and its asymptotic-bound properties. These properties are natural requirements from the biological point of view. As the special cases, we also discuss some stochastic Lotka-Volterra systems.