the optimization of spatial arrangement of favorable and unfavorable regions for a species to survive in biological context, we refer to [@CaCo; @CaCo2; @Co; @CoCuPo; @HiKaLa; @KaLoYa; @LoYa], etc.

Among others, in the time independent case, it is proved that with $m_1(t, x)=m_1(x)$ admits a unique positive principal eigenvalue if and only if \[random-auto-diri-condi\] m(x\_0)&gt;0 x\_0D (see [@GoLa; @HeKo; @He]), and with $m_2(t,x)=m_2(x)$ admits a unique positive principal eigenvalue if and only if $$\label{random-auto-neum-condi}
m_2(x_0)>0 \quad\text{for some}\,\, x_0\in D\,\,\, \text{and}\,\,\int_D m_2(x)dx<0$$ (see [@He]). In the time periodic case, it is proved that has a unique positive principal eigenvalue if and only if \[R-D\] \_0\^T\_[x|D]{}m\_1(t, x)dt&gt;0 (see [@He0]), and has a unique positive principal eigenvalue if \[R-N\] \_0\^T\_[x|D]{}m\_2(t, x)dt&gt;0 \_D\_0\^Tm\_2(t, x)dtdx&lt;0 (see [@He0]). There is no result for random dispersal operator subject to the spatial periodic boundary condition so far. But as we can see in the proofs for nonlocal dispersal operator, results in spatial periodic boundary case are very similar to those in Neumann boundary case.

The study of , , and is of great interest in its own and will also have important applications in the study of many nonlinear mathematical models with nonlocal dispersal from applied science, including 