  We prove Freidlin-Wentzell type large deviation principles for various
rescaled models in populations dynamics that have immigration and possibly
harvesting: birth-death processes, Galton-Watson trees, epidemic SI models, and
prey-predator models.
  The proofs are carried out using a general analytic approach based on the
well-posedness of a class of associated Hamilton-Jacobi equations. The notable
feature for these Hamilton-Jacobi equations is that the Hamiltonian can be
discontinuous at the boundary. We prove a well-posedness result for a large
class of Hamilton-Jacobi equations corresponding to one-dimensional models, and
give partial results for the multi-dimensional setting.
