\[PSP-neum\] Suppose that $\kappa(\cdot)$ satisfies (K), $m_2(\cdot, \cdot)\in\mathcal X_2$ and $m_2(t, x)\not \equiv m_2(t)$. The eigenvalue problem has exactly one positive principal spectrum point, denoted by $\lambda_2^p$, if and only if \[N\] P (m\_2)&gt;0 \_[D\_2]{} \_0\^Tm\_2(t, x)dtdx&lt;0. More precisely, we show the following.

-   There is a $\lambda_2^p>0$ such that $\mu_2^n(\lambda_2^p)=0$ if and only if (N) holds.

-   If $\lambda_2^{p, 1}, \lambda_2^{p, 2}>0$ are such that $\mu_2^n(\lambda_2^{p, 1})=\mu_2^n(\lambda_2^{p, 2})=0$, then $\lambda_2^{p, 1}=\lambda_2^{p, 2}$.

\[PSP-neum-coro\]

-   If $m_2(t,x)\equiv m_2(t)$, then $$\mu_2^n(\lambda_2)=\lambda_2\widehat m_2.$$ It then follows that, if $\int_0^T m_2(t)dt\not =0$, then there is no positive principal spectrum point of , and if $\int_0^T m_2(t)dt=0$, then any $\lambda_2>0$ is a positive principal spectrum point of .

-   If $m_2(t, x)\equiv m_2(x)$, consider $$\label{autonomous-neumann-ori}
    \int_D\kappa(y-x)[u(y)-u(x)]dy+\lambda_2m_2(x)u(x)=0,\quad x\in\bar D.$$ There is exactly one positive principal spectrum point $\lambda_2^p$ of if and only if $$m_2(x_0)>0\quad \text{for some}\,\, x_0\in D_2\quad {\rm and}\,\,\int_{D_2} m_2(x)dx<0.$$

\[PSP-neum-remark\] Theorem \[PSP-neum\] and Corollary \[PSP-neum-coro\] extend the principal eigenvalue theory for random dispersal operators with time independent or time periodic indefinite weight functions subject to Neumann boundary condition to nonlocal dispersal operators with time independent or time periodic indefinite weight functions subject to Neumann type boundary condition.

\[PSP-peri\] Suppose that $\kappa(\cdot)$ satisfies (K), 