, , and , respectively. We also investigate the upper bounds of principal spectrum points and the sufficient conditions for the principal spectrum points to be principal eigenvalues. As an application, we study asymptotic dynamics of , , and by the spectral theory of weighted nonlocal dispersal operators.

The rest of this paper is organized as follows. In section 2, we introduce notations, definitions and state the main results of the paper. In section 3, we present some preliminary materials to be used in the proof of the main results. We prove the main results and discuss the application of the main results in section 4.

Notations, Definitions, and Main Results
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In this section, we introduce notations, definitions, and state the main results.

We first introduce some standing notations and the concept of principal spectrum points of nonlocal dispersal operators with time periodic indefinite weight functions.

Let \[space-t12\] X\_1=X\_2={uC(|D, )| u(t+T, x)=u(t, x)} with norm $\|u\|_{\mathcal X_i}=\sup_{t\in \R, x\in \bar D}|u(t, x)| (i=1, 2)$, \[space-t3\] X\_3={uC(\^N, )|u(t+T, x)=u(t, x+p\_j[**e\_j**]{})=u(t, x) j=1,2,,N} with norm $\|u\|_{\mathcal X_3}=\sup_{t\in\R, x\in\R^N}|u(t, x)|$. Set $$\mathcal X_i^+=\{u\in \mathcal X_i|u\geq 0\}$$ and $$\mathcal X_i^{++}=\text{Int}{X_i^+}=\{\phi\in \mathcal X_i, \phi>0\}$$ ($i=1, 2, 3$). We define the integral operator $K_i$, the 