  In this article, we study, as the coefficient $s\to\infty$, the asymptotic
behavior of the principal eigenvalue of
$$-\varphi''(x)-2sm'(x)\varphi'(x)+c(x)\varphi(x)=\lambda_s\varphi(x),\ \
0<x<1,$$ supplemented by different boundary conditions. This problem is
relevant to nonlinear propagation phenomena in reaction-diffusion equations.
The main point is that the advection (or drift) term $m$ allows natural
degeneracy. For instance, $m$ could be constant on $[a,b]\subset[0,1]$.
Depending on the behavior of $m$ near the neighbourhood of the end points
$a,\,b$, the limiting value could be the principal eigenvalue of
$$-\varphi''(x)+c(x)\varphi(x)=\lambda\varphi(x),\ \ a<x<b,$$ coupled with
Dirichlet or Newmann boundary condition. A complete understanding of the
limiting behavior of the principal eigenvalue and its eigenfunction is
obtained, and new fundamental effects of large degenerate advection and
boundary conditions on the principal eigenvalue and the principal eigenfunction
are revealed. In one space dimension, the results in the existing literature
are substantially improved.
