For $\ x(0)\equiv x_0>0\ $ and a system governed by $\dot x(t)=-k\ x(t),$ I find that $x(t)>0\ \ \ \forall\ t.$ (Because the solution is $x(t)=e^{-kt}x_0$.)
For which $f$ and $\dot x(t)=f(x(t)),$ does this property hold?
I don't know how difficult this is, but the question might be generalized to functions $f(x(t),t)$ and/or higher order ODEs.