  In this paper, we show that unbalanced optimal transport provides a
convenient framework to handle reaction and diffusion processes in a unified
metric framework. We use a constructive method, alternating minimizing
movements for the Wasserstein distance and for the Fisher-Rao distance, and
prove existence of weak solutions for general scalar
reaction-diffusion-advection equations. We extend the approach to systems of
multiple interacting species, and also consider an application to a very
degenerate diffusion problem involving a Gamma-limit. Moreover, some numerical
simulations are included.
