and sufficient conditions for the existence and uniqueness of positive principal spectrum points of , , and .

\[PSP-diri\] Suppose $\kappa(\cdot)$ satisfies (K) and $m_1\in\mathcal X_1$. The eigenvalue problem has exactly one positive principal spectrum point, denoted by $\lambda_1^p$, if and only if \[D\] P(m\_1)&gt;0. More precisely, we show the following.

-   There is $\lambda_1^p>0$ such that $\mu_1^n(\lambda_1^p)=0$ if and only if (D) holds.

-   If $\lambda_1^{p, 1},\lambda_1^{p, 2}>0$ are such that $\mu_1^n(\lambda_1^{p, 1})=\mu_1^n(\lambda_1^{p, 2})=0$, then $\lambda_1^{p, 1}=\lambda_1^{p, 2}$.

\[PSP-diri-coro\] $\quad$

-   If $m_1(t, x)\equiv m_1(t)$, then has exactly one positive principal spectrum point $\lambda_1^p$ if and only if $$\int_0^T m_1(t)dt>0.$$ Moreover, $\lambda_1^p$ is a principal eigenvalue of and $\ds \lambda_1^p=\frac{-\bar\lambda_1}{\widehat m_1}$, where $\bar \lambda_1(<0)$ is the principal eigenvalue of $K_1-B_1(=K_1-\mathcal I)$.

-   If $m_1(t, x)\equiv m_1(x)$ , consider $$\label{autonomous-dirichlet-ori}
    \left[\int_D\kappa(y-x)dy-u(x)\right]+\lambda_1m_1(x)u(x)=0,\quad x\in\bar D.$$ There exists a unique positive principal spectrum point $\lambda_1^p$ of if and only if $m_1(x_0)> 0$ for $x_0\in  D_1$.

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

