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.


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