  In this article, we are concerned with the following eigenvalue problem of a
linear second order elliptic operator:
  \begin{equation}
  \nonumber
  -D\Delta \phi -2\alpha\nabla m(x)\cdot \nabla\phi+V(x)\phi=\lambda\phi\ \
\hbox{ in }\Omega, \end{equation} complemented by a general boundary condition
including Dirichlet boundary condition and Robin boundary condition:
  $$
  \frac{\partial\phi}{\partial n}+\beta(x)\phi=0 \ \ \hbox{ on }\partial\Omega,
  $$ where $\beta\in C(\partial\Omega)$ allows to be positive, sign-changing or
negative, and $n(x)$ is the unit exterior normal to $\partial\Omega$ at $x$.
The domain $\Omega\subset\mathbb{R}^N$ is bounded and smooth, the constants
$D>0$ and $\alpha>0$ are, respectively, the diffusive and advection
coefficients, and $m\in C^2(\bar\Omega),\,V\in C(\bar\Omega)$ are given
functions.
  We aim to investigate the asymptotic behavior of the principal eigenvalue of
the above eigenvalue problem as the diffusive coefficient $D\to0$ or
$D\to\infty$. Our results, together with those of \cite{CL2,DF,Fr} where the
Nuemann boundary case (i.e., $\beta=0$ on $\partial\Omega$) and Dirichlet
boundary case were studied, reveal the important effect of advection and
boundary conditions on the asymptotic behavior of the principal eigenvalue. We
also apply our results to a reaction-diffusion-advection equation which is used
to describe the evolution of a single species living in a heterogeneous stream
environment and show some interesting behaviors of the species persistence and
extinction caused by the buffer zone and small/large diffusion rate.
