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

This paper discusses a randomized non-autonomous logistic equation dN(t) = N(t)[(a(t) −b(t)N(t))dt +α(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/Np(t)] provided a(t), b(t) and α(t) are continuous T -periodic functions, a(t) > 0, b(t) > 0 and T 0 [a(s) − α2(s)]ds > 0. We show that this equation is stochastically permanent and the solution Np(t) is globally attractive provided a(t), b(t) and α(t) are continuous T -periodic functions, a(t) > 0, b(t) > 0 and mint ∈[0,T ] a(t) > maxt∈[0,T ] α2(t). By the way, the similar results of a generalized non-autonomous logistic equation with random perturbation are yielded. © 2007 Elsevier Inc. All rights reserved.