The study on the qualitative properties of principal eigenvalue for second order elliptic operators has a long history. In recent years, there are some interest in investigating the monotonicity and asymptotic behaviors of such principal eigenvalue with small diffusion or large advection rates. In this talk, we will discuss some recent works along this line, and try to find some global information on the principal eigenvalue, i.e. the classification of topological structures of the level sets as a function of multiple parameters. This enables us to better understand qualitative properties of the principal eigenvalues.