x)=u(t+T, x),
\end{cases}$$ $$\label{random-periodic-op-eq}
\begin{cases}
 -\p_t u(t, x)+\Delta u(t, x)+\lambda_3 m_3(t,x)u(t, x) =\mu_3u(t, x)\, & x\in  \R^N,\\
 u(t+T, x)=u(t, x+p_j{\bf e_j})=u(t, x).
\end{cases}$$ For $i=1, 2, 3$, and any given $\lambda_i$, let $\mu^{r, p}_{i}(\lambda_i)$ be the principal eigenvalue of , , and , respectively. Then $\lambda_1^{r,p}$ (resp. $\lambda_2^{r,p}$, $\lambda_3^{r,p}$) is a principal eigenvalues of (resp. , ) if and only if $\mu^r_1(\lambda_1^{r,p})=0$ (resp. $\mu^r_2(\lambda_2^{r,p})=0$, $\mu^r_3(\lambda_3^{r,p})=0$).

Thanks to its applications in the nonlinear mathematical models, in particular, the population dynamics in biology, the existence of positive principal eigenvalues of , , and is of particular interest and has been well studied. The time independent version was first studied by Manes and Micheletti in [@MaMi]. Then, Hess and Kato in [@HeKo] and Brown and Lin in [@BrLi] obtained some results for elliptic operators in Dirichlet boundary case. The proofs of Brown and Lin and Manes and Micheletti are based on the variational characterization of the principal eigenvalue; the proof of Hess and Kato uses Krein-Rutman’s theorem. For more general elliptic operators subject to various boundary conditions, we refer to [@Boc; @Bo; @BrCoFl; @BrLi; @CaCo1; @GoLa; @He; @HeKo; @Lo-Go0; @Lo-Go; @SeHe] and references therein, and for the applications in population dynamics, such as 