Stochastic functional Kolmogorov-type population dynamics

In this paper we stochastically perturb the functional Kolmogorov-type system (x) over dot(t) = diag(x(1)(t).....x(n)(t))f(x(t)) into the stochastic functional differential equation dx(t) = diag(x(1)(t)....x(n)(t))[f(x(t))dt + g(x(t))dw(t)]. This paper studies existence and uniqueness of the global positive solution, and its asymptotic bound properties and moment average in time. These properties are natural requirements from the biological point of view. As the special cases, we discuss the various stochastic Lotka-Volterra systems.