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Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation. (English) Zbl 1140.60032
This paper deals with stochastic differential equation dN(t)=N(t)[(a(t)-b(t)N(t))dt+α(t)dB(t)], where B(t) is the one-dimensional standard Brownian motion, N(0)=N 0 >0. It is assumed that a(t),b(t),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). The authors show that considered equation is stochastically permanent and the positive solution N p (t) is globally attractive. The similar results for a generalized non-autonomous logistic equation dN(t)=N(t)[(a(t)-b(t)N θ (t))dt+α(t)dB(t)], where θ>0 is an odd number, are presented.
60H10Stochastic ordinary differential equations