  This paper is devoted to the study of the persistence versus extinction of
species in the reaction-diffusion equation: \begin{equation} u_t-\Delta
u=f(t,x_1-ct,y,u) \quad\quad t>0,\ x\in\Omega,\nonumber \end{equation} where
$\Omega$ is of cylindrical type or partially periodic domain, $f$ is of
Fisher-KPP type and the scalar $c>0$ is a given forced speed. This type of
equation originally comes from a model in population dynamics (see
\cite{BDNZ},\cite{PL},\cite{SK}) to study the impact of climate change on the
persistence versus extinction of species. From these works, we know that the
dynamics is governed by the traveling fronts $u(t,x_1,y)=U(x_1-ct,y)$, thus
characterizing the set of traveling fronts plays a major role. In this paper,
we first consider a more general model than the model of \cite{BDNZ} in higher
dimensional space, where the environment is only assumed to be globally
unfavorable with favorable pockets extending to infinity. We consider in two
frameworks: the reaction term is time-independent or time-periodic dependent.
For the latter, we study the concentration of the species when the environment
outside $\Omega$ becomes extremely unfavorable and further prove a symmetry
breaking property of the fronts.
