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

AbstractThis 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 ∫0T[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