Asymptotic Properties of Stochastic Functional Kolmogorov-Type System

Abstract

In general, population systems are often subject to environmental noise. To examine whether the presence of such noise affects population systems significantly, this paper stochastically perturbs the functional Kolmogorov-type system $$\dot{x}(t)=\mbox{diag}(x_{1}(t),\ldots,x_{n}(t))f(x_{t})$$ into the stochastic functional differential equation $$dx(t)=\mbox{diag}(x_{1}(t),\ldots,x_{n}(t))[f(x_{t})dt+g(x(t))dw(t)].$$ This paper studies existence and uniqueness of the global positive solution of this stochastic system, its asymptotic boundedness and gives asymptotic pathwise estimation. These properties are natural requirements from the biological point of view.