Almost sure permanence of stochastic single species models

A new definition of almost sure permanence of stochastic population systems is proposed in this paper. We consider the stochastic logistic model dx(t)=x(t)[a(t)-b(t)x^θ(t)]dt+σ(t)x(t)dB(t), where B(t) is a standard Brownian motion and θ is a positive constant. Under a simple assumption, a interesting result is obtained as follows:. 0<lim_t→∞inf x(t) ≤ lim_t→∞supx(t) <∞a.s. The result is analogous to the result in the deterministic case. And some numerical simulations are introduced to support our main results at the end.