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

Abstract

This paper discusses a randomized non-autonomous logistic equationstandard 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 T 0 [a(s) − α 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 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.

Partial permanence and extinction on stochastic Lotka-Volterra competitive systems

“…Author details 1 College of Science, Hohai University, Nanjing, Jiangsu 210098, China. 2 College of Energy and Electrical Engineering, Hohai University, Nanjing, Jiangsu 210098, China.…”

On the periodic solution of a class of stochastic nonlinear system with delays

“…However, few results have been obtained in the direction of the periodically stochastic differential equations. Till now, we only find that very few results for periodic solution of stochastic differential equations have been published in [5,6,19]. Recently, some results have been established for the periodically stochastic differential equations.…”

Dynamical Analysis of a Class of Prey-Predator Model with Beddington-DeAngelis Functional Response, Stochastic Perturbation, and Impulsive Toxicant Input

“…Recently, various models based on stochastic differential equations (SDEs) have extensively been paid the attention of the researchers (see, e.g., [28][29][30][31][32][33][34][35][36][37]). Parameter perturbation induced by white noise is an important and common form to describe the effect of stochasticity (see, e.g., [37][38][39][40][41][42][43][44][45][46][47][48]). In this paper, we consider the white noise perturbation for the intrinsic growth rates of the prey and predator; that is, 1 → 1 + 11 ( ) and 2 → 2 + 22 ( ), where 1 ( ), 2 ( ) are mutually independent Brownian motions and 1 , 2 denote the intensities of the white noise.…”