Effects of Diffusion and Advection on the Smallest Eigenvalue of an Elliptic Operator and Their Applications

We investigate the effects of diffusion and advection on the smallest eigenvalue of an elliptic operator with zero Neumann boundary condition. Various asymptotic behaviors of the smallest eigenvalue, as diffusion and advection coefficients approach zero or infinity, are derived. As an application, these qualitative results yield new insight into the role of turbulent diffusion on the persistence of a sinking phytoplankton species.