Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation

This paper discusses a randomized non-autonomous logistic equation d N (t) = N (t) [(a(t) - b(t) N(t)) dt + alpha(t) dB(t)], where B(t) is a 1-dimensional standard Brownian motion. In [D.Q. Jiang, N.Z. Shi, A note on non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl. 303 (2005) 164-172], the authors show that E[1/N(t)] has a unique positive T-periodic solution E[1/N-p(t)] provided a(t), b(t) and alpha(t) are continuous T-periodic functions, a(t) > 0, b(t) > 0 and integral(T)(0) [a(s) - alpha(2) (s)] ds > 0. We show that this equation is stochastically permanent and the solution N-p(t) is globally attractive provided a(t), b(t) and a(t) are continuous T-periodic functions, a(t) > 0, b(t) > 0 and min(t epsilon[0,T]) a(t) > maxt(t epsilon[0,T]) alpha(2)(t). By the way, the similar results of a generalized non-autonomous logistic equation with random perturbation are yielded. (c) 2007 Elsevier Inc. All rights reserved.