Existence, uniqueness, stochastic persistence and global stability of positive solutions of the logistic equation with random perturbation

This paper discusses a randomized logistic equation dx(t)=x(t)[(a-bx(t))dt+x(theta)(t)dB(t)] with initial value x(0)=x0>0, where B(t) is a standard one-dimension Brownian motion, and (theta)(epsilon)(0, 0.5). We show that the positive solution of the stochastic differential equation does not explode at any finite time under certain conditions. In addition, we study the existence, uniqueness, boundedness, stochastic persistence and global stability of the positive solution.