$$\label{neumann-ori}
\begin{cases}
 -\p_t u(t, x)+\int_D \kappa(y-x)[u(t, y)-u(t, x)]dy+\lambda_2 m_2(t,x)u(t, x)=0,\quad x\in \bar D,\\
u(t+T, x)=u(t, x),
\end{cases}$$ where $D\in \R^N$ and $m_2(t, x)$ are as ; and the eigenvalue problem subject to periodic boundary conditions considered in this paper reads as $$\label{periodic-ori}
\begin{cases}
 -\p_t u(t, x)+\int_{\R^N} \kappa(y-x)[u(t, y)-u(t, x)]dy+\lambda_3 m_3(t,x)u(t, x)=0,\quad x\in \R^N,\\
u(t+T, x)=u(t, x+p_j{\bf e_j})=u(t, x),
\end{cases}$$ where $p_j>0$, ${\bf e_j}=(\delta_{j_1}, \delta_{j_2}, \cdots, \delta_{jN})$ ($\delta_{jk}=1$ if $j=k$ and $\delta_{jk}=0$ if $j\neq k$), and $m_3(t, x)$ is a continuous function with $m_3(t+T, x)=m_3(t, x+p_j{\bf e_j})=m_3(t, x)$ for $j=1, 2, \cdots, N$. Throughout the paper, we assume that the nonlocal kernel function $\kappa(\cdot)$ in - satisfies the following condition \[K\] ()C\_c(\^N),  (0)&gt;0, (-z)=(z), \_[[$\mathbb{R}$]{}\^N]{} (z)dz=1.

The eigenvalue problems , , and can be viewed as the nonlocal dispersal counterparts of the following eigenvalue problems associated to random dispersal operators with time periodic indefinite functions and Dirichlet, Neumann, and periodic boundary conditions, $$\label{random-dirichlet-op}
\begin{cases}
 -\p_t u(t, x)+\Delta u(t, x)+\lambda_1m_1(t,x)u(t, x)=0 \, & x\in  D,\\
 u(t, x)=0, & x\in \p D,\\
 u(t, x)=u(t+T, x),
\end{cases}$$ $$\label{random-neumann-op}
\begin{cases}
 -\p_t u(t, x)+\Delta u(t, x)+\lambda_2 m_2(t,x)u(t, x)=0 \, & x\in  D,\\
 \frac{\p u}{\p {\bf n}}(t, x)=0, & x\in \p D,\\
 u(t, x)=u(t+T, x),
\end{cases}$$ and 