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

Abstract

This paper discusses a randomized non-autonomous logistic equation d N ( t ) = N ( t ) [ ( a ( t ) − b ( t ) N ( t ) ) d t + α ( t ) d B ( 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 α ( t ) are continuous T-periodic functions, a ( t ) > 0 , b ( t ) > 0 and ∫ 0 T [ a ( s ) − α 2 ( s ) ] d s > 0 . We show that this equation is stochastically permanent and the solution N p ( t ) is globally attractive provided a ( t ) , b ( t ) and α ( t ) are continuous T-periodic functions, a ( t ) > 0 , b ( t ) > 0 and min t ∈ [ 0 , T ] a ( t ) > max t ∈ [ 0 , T ] α 2 ( t ) . By the way, the similar results of a generalized non-autonomous logistic equation with random perturbation are yielded.