  We study the ground states of the one-dimensional non-self-adjoint Jacobi
operators in the almost periodic media by using the method of dynamical
systems. We show the existence of the ground state. Particularly, in the
quasi-periodic media, we show that the lower regularity of coefficients can
guarantee the existence of ground states. Besides that, we give two
applications: the first application is to show the existence and uniqueness of
the positive steady state of the discrete Fisher-KPP type equation; the second
application is to investigate the asymptotic behavior of the discrete
stationary parabolic equation with large lower order terms.
