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We prove Freidlin-Wentzell type large deviation principles for variousrescaled models in populations dynamics that have immigration and possiblyharvesting: birth-death processes, Galton-Watson trees, epidemic SI models, andprey-predator models.  The proofs are carried out using a general analytic approach based on thewell-posedness of a class of associated Hamilton-Jacobi equations. The notablefeature for these Hamilton-Jacobi equations is that the Hamiltonian can bediscontinuous at the boundary. We prove a well-posedness result for a largeclass of Hamilton-Jacobi equations corresponding to one-dimensional models, andgive partial results for the multi-dimensional setting.

Well-posedness of Hamilton-Jacobi equations in population dynamics and applications to large deviations