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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, $N(0)=N\sb{0}>0$. It is assumed that $a(t),b(t),\alpha(t)$ are continuous $T$-periodic functions, $a(t)>0$, $b(t)>0$ and $\min\sb{t\in [0,T]}a(t)>\max\sb{t\in[0,T]}\alpha\sp2(t)$. The authors show that considered equation is stochastically permanent and the positive solution $N\sb{p}(t)$ is globally attractive. The similar results for a generalized non-autonomous logistic equation $dN(t)=N(t)[(a(t)-b(t)N\sp{\theta}(t))dt+\alpha(t)dB(t)]$, where $\theta>0$ is an odd number, are presented.

MSC:
60H10Stochastic ordinary differential equations
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