  Assume that $N_m(x)$ denotes the density of the population at a point $x$ at
the beginning of the reproductive season in the $m$th year. We study the
following impulsive reaction-diffusion model for any $m\in \mathbb Z^+$
\begin{eqnarray*}\label{}
  \ \ \ \ \ \left\{ \begin{array}{lcl}
  u^{(m)}_t = div(A\nabla u^{(m)}-a u^{(m)}) + f(u^{(m)}) \quad \text{for} \ \
(x,t)\in\Omega\times (0,1]
  u^{(m)}(x,0)=g(N_m(x)) \quad \text{for} \ \ x\in \Omega
  N_{m+1}(x):=u^{(m)}(x,1) \quad \text{for} \ \ x\in \Omega \end{array}\right.
  \end{eqnarray*} for functions $f,g$, a drift $a$ and a diffusion matrix $A$
and $\Omega\subset \mathbb R^n$. Study of this model requires a simultaneous
analysis of the differential equation and the recurrence relation. When
boundary conditions are hostile we provide critical domain results showing how
extinction versus persistence of the species arises, depending on the size and
geometry of the domain. We show that there exists an {\it extreme volume size}
such that if $|\Omega|$ falls below this size the species is driven extinct,
regardless of the geometry of the domain. To construct such extreme volume
sizes and critical domain sizes, we apply Schwarz symmetrization rearrangement
arguments, the classical Rayleigh-Faber-Krahn inequality and the spectrum of
uniformly elliptic operators. The critical domain results provide qualitative
insight regarding long-term dynamics for the model. Lastly, we provide
applications of our main results to certain biological reaction-diffusion
models regarding marine reserve, terrestrial reserve, insect pest outbreak and
population subject to climate change.
