the following time periodic dispersal evolution equations, $$\label{n-kpp-diri}
 \begin{cases}
 \p_t u=\int_{D} \kappa(y-x)u(t,y)dy-u(t,x)+\lambda_1 uf_1(t,x,u),\quad x\in \bar D,\\
 u(t+T, x)=u(t, x),\\
 u(0, x)\geq 0, u(0, x)\not\equiv 0,
 \end{cases}$$ $$\label{n-kpp-neum}
 \begin{cases}
 \p_t u=\int_{D} k(y-x)[u(t,y)-u(t,x)]dy+\lambda_2 uf_2(t,x,u),\quad x\in \bar D,\\
 u(t+T, x)=u(t, x),\\
 u(0, x)\geq 0, u(0, x)\not\equiv 0,
 \end{cases}$$ and $$\label{n-kpp-peri}
\begin{cases}
 \p_t u=\int_{\R^N} k(y-x)u(t,y)dy-u(t,x)+\lambda_3 uf_3(t,x,u),\quad x\in \R^N,\cr
u(t+T, x+p_i{\bf e_i})=u(t, x+p_i{\bf e_i})=u(t, x),\\
 u(0, x)\geq 0, u(0, x)\not\equiv 0,
\end{cases}$$ where $u(0, x)$ is continuous and bounded, $\lambda_i >0$, and $f_i(t, x, u)$ satisfies the following condition $(i=1,2,3)$,

[**(F)**]{} [*$f_i$ is $C^1$ in $t\in{\ensuremath{\mathbb{R}}}$ and $C^3$ in $(x,u)\in {\ensuremath{\mathbb{R}}}^N\times{\ensuremath{\mathbb{R}}}$; $f_i(t, x, u)<0$ for $u\gg 1$ and $\partial _u f_i(t, x, u)<0$ for $u\geq0$*]{}; $f_i(t+T,x,u)=f_i(t,x,u)$; and when $i=3$, $f_i(t+T, x,u)=f_i(t, x+p_j{\bf e_j}, u)=f(t,x,u)$ for $j=1, 2, \cdots, N$.

In , , and , $u(t, x)$ represents the density of a species at location $x$ and time $t$, and $\lambda_i f(t, x, 0) (i=1, 2, 3)$ represents the intrinsic growth rate of a species. Hence only non-negative solutions of , , and are of interest. Notice that $u\equiv 0$ is a solution to , , and . It is of great interest to know for which $\lambda_1$ (resp. $\lambda_2$, $\lambda_3$), there is a positive solution 