Unbounded delay stochastic functional Kolmogorov-type system

In general, population systems are often subject to environmental noise. To examine whether the presence of such noise affects these systems significantly, we perturb the functional Kolmogorov-type system x (t)= diag(x1(t), a??, xn (t))f(xt)into the stochastic functional differential equation dx(t) = diag(x1(t), a??, xn(t))[f(x t) dt + g(xt) dw(t)]. We show that different environmental noise structures have different effects on the population system with unbounded delay. Under two classes of different environmental noise perturbations, we establish existence theorems of the global positive solution to the unbounded delay stochastic functional Kolmogorov-type system. As the desired results for population dynamics, we also examine asymptotic boundedness, including the moment boundedness, stochastically ultimate boundedness and the moment average boundedness in time. To illustrate our idea more clearly, as a special case we also discuss a Lotka-Volterra system with unbounded delay.