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The current paper is devoted to the study of semilinear dispersal evolutionequations of the form $$ u_t(t,x)=(\mathcal{A}u)(t,x)+u(t,x)f(t,x,u(t,x)),\quadx\in\mathcal{H}, $$ where $\mathcal{H}=\RR^N$ or $\ZZ^N$, $\mathcal{A}$ is arandom dispersal operator or nonlocal dispersal operator in the case$\mathcal{H}=\RR^N$ and is a discrete dispersal operator in the case$\mathcal{H}=\ZZ^N$, and $f$ is periodic in $t$, asymptotically periodic in $x$(i.e. $f(t,x,u)-f_0(t,x,u)$ converges to 0 as $\|x\|\to\infty$ for some timeand space periodic function $f_0(t,x,u)$), and is of KPP type in $u$. It isproved that Liouville type property for such equations holds, that is, timeperiodic strictly positive solutions are unique. It is also proved that if$u\equiv 0$ is a linearly unstable solution to the time and space periodiclimit equation of such an equation, then it has a unique stable time periodicstrictly positive solution and has a spatial spreading speed in everydirection.

Liouville Type Property and Spreading Speeds of KPP Equations in Periodic Media with Localized Spatial Inhomogeneity