K_iu-B_iu+\lambda_i\widehat m_i u=\mu_i(\lambda_i) u$$ in $X_i$.

\[upper-bounds\] $\quad$

-   (Upper bounds) If has a unique positive principal spectrum point $\lambda_i^p(\widehat m_i)$, then has also a unique positive principal spectrum point $\lambda_i^p(m_i)$ and $$\lambda_i^p(m_i)\le \lambda_i^p(\widehat m_i).$$

-   (Sufficient conditions of the existence of positive principal eigenvalues)

-   Assume that $N=1$ or $2$, $b_i+\widehat m_i$ is $C^N$, and has a unique positive principal spectrum point $\lambda_i^p$, then $\lambda_i^p$ is a positive principal eigenvalue of .

-   For $N\geq 1$, and $i=1$ or $3$, assume that $\ds\int_{D_i}\frac{1}{\widehat m_{i,\max} -\widehat m_i(x)}dx=\infty$ and has a unique positive principal spectrum point $\lambda_i^p$, then $\lambda_i^p$ is a positive principal eigenvalue of .

\[average-pev-remark\] Assume that $\widehat m_i(x_0)>0$ for some $x_0\in D_i$, and in addition, $\int_{D_i}\widehat m_i(x)dx<0$ in the case $i=2,3$. Then has a unique positive principal spectrum point $\lambda_i^p(\widehat m_i)$.

Finally, we consider the applications of the principal spectrum point theory of nonlocal dispersal operators with time periodic indefinite weight function to the KPP type equations , , and . And we prove the following result.

\[KPP\] Assume (K) and (F) hold. Denote the principal spectrum point of the eigenvalue problem with $m_i(t, x)= f_i(t, x, 0)$ by $\mu_i^n(\lambda_i)$. Assume in addition that $f_1(t, x, 0)$, $ 