  We study a hybrid impulsive reaction-advection-diffusion model given by a
reaction-advection-diffusion equation composed with a discrete-time map in
space dimension $n\in\mathbb N$. The reaction-advection-diffusion equation
takes the form \begin{equation*}\label{} u^{(m)}_t = \text{div}(A\nabla
u^{(m)}-q u^{(m)}) + f(u^{(m)}) \quad \text{for} \ \ (x,t)\in\mathbb R^n \times
(0,1] , \end{equation*} for some function $f$, a drift $q$ and a diffusion
matrix $A$. When the discrete-time map is local in space we use $N_m(x)$ to
denote the density of population at a point $x$ at the beginning of
reproductive season in the $m$th year and when the map is nonlocal we use
$u_m(x)$. The local discrete-time map is \begin{eqnarray*}\label{}\left\{
\begin{array}{lcl} u^{(m)}(x,0) = g(N_m(x)) \quad \text{for} \ \ x\in \mathbb
R^n , \\ N_{m+1}(x):=u^{(m)}(x,1) \quad \text{for} \ \ x\in \mathbb R^n ,
\end{array}\right. \end{eqnarray*} for some function $g$. The nonlocal discrete
time map is \begin{eqnarray*}\label{}\left\{ \begin{array}{lcl} u^{(m)}(x,0) =
u_{m}(x) \quad \text{for} \ \ x\in \mathbb R^n , \\ \label{mainb2} u_{m+1}(x)
:= g\left(\int_{\mathbb R^n} K(x-y)u^{(m)}(y,1) dy\right) \quad \text{for} \ \
x\in \mathbb R^n, \end{array}\right. \end{eqnarray*} when $K$ is a nonnegative
normalized kernel. SEE THE ARTICLE FOR COMPLETE ABSTRACT.
