  We introduce a new optimal transport distance between nonnegative finite
Radon measures with possibly different masses. The construction is based on
non-conservative continuity equations and a corresponding modified
Benamou-Brenier formula. We establish various topological and geometrical
properties of the resulting metric space, derive some formal Riemannian
structure, and develop differential calculus following F. Otto's approach.
Finally, we apply these ideas to identify an ideal free distribution model of
population dynamics as a gradient flow and obtain new long-time convergence
results.
