u_0, m_3):=(\Phi_3(t, s; m_3) u_0)(x)$) is the unique solution of (resp. , ) with $u(s, x; s, u_0, m_1)=u_0(x)$ (resp. $u(s, x; s, u_0, m_2)=u_0(x)$, $u(s, x; s, u_0, m_3)=u_0(x)$).

\[super-sub-sol\] A bounded measurable function $u(t, x)$ on $[0, T)\times \bar D$ is called a [*super-solution*]{} (or [*sub-solution*]{}) of if for any $x\in \bar D$, $u(t, x)$ is differentiable for all but finite many $t$’s in $[0, T)$ and satisfies that for each $x\in \bar D$, $$\p_t u\geq (\text{ or } \leq) \int_D \kappa(y-x)u(t, y)dy-u(t, x)+ m_1(t,x)u(t, x)$$ for all but finite many $t$’s in $[0, T)$.

[*Super-solutions and sub-solutions*]{} of and $\eqref{periodic-ori-evol}$ are defined in an analogous way.

\[comparison\] $\quad$

-   If $u^1(t, x)$ and $u^2(t, x)$ are bounded sub- and super- solution of (resp. , ) on $[s, T)$, respectively, and $u^1(s, \cdot)\leq u^2(s, \cdot)$, then $u^1(t, \cdot)\leq u^2(t, \cdot)$ for $t\in [s, T)$.

-   If $u^1, u^2\in X_i$, $u^1\leq u^2$ and $u^1\neq u^2$, then $$\Phi_i(t, s; m_i) u^1\ll \Phi_i(t, s; m_i) u^2 \text{ for all } t>s.$$

-   If $u_0\in X_i^+$, and $m_i^1, m_i^2\in \mathcal X_i$, if $m_i^1\leq m_i^2$, then $$\Phi_i(t, s; m_i^1)u_0\leq \Phi_i(t, s; m_i^2)u_0 \text{ for all } t>s.$$

\(1) follows from the arguments in 