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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 $$dN(t)=N(t)[(a(t)-b(t)N(t))dt+\alpha(t)dB(t)]$$, where $$B(t)$$ is the one-dimensional standard Brownian motion, $$N(0)=N_{0}>0$$. It is assumed that $$a(t),b(t),\alpha(t)$$ are continuous $$T$$-periodic functions, $$a(t)>0$$, $$b(t)>0$$ and $$\min_{t\in [0,T]}a(t)>\max_{t\in[0,T]}\alpha^2(t)$$. 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 $$dN(t)=N(t)[(a(t)-b(t)N^{\theta}(t))dt+\alpha(t)dB(t)]$$, where $$\theta>0$$ is an odd number, are presented.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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